Research ArticleCoefficients of Meromorphic Bi-Bazilevic Functions

Jay M Jahangiri1 and Samaneh G Hamidi2

1 Department of Mathematical Sciences Kent State University Burton OH 44021-9500 USA2 Institute of Mathematical Sciences Faculty of Science University of Malaya 50603 Kuala Lumpur Malaysia

Correspondence should be addressed to Jay M Jahangiri jjahangikentedu

Received 19 November 2013 Accepted 6 February 2014 Published 9 March 2014

Academic Editor Yan Xu

Copyright copy 2014 J M Jahangiri and S G Hamidi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A function is said to be bi-Bazilevic in a given domain if both the function and its inverse map are Bazilevic there Applying theFaber polynomial expansions to the meromorphic Bazilevic functions we obtain the general coefficient bounds for bi-Bazilevicfunctions We also demonstrate the unpredictability of the behavior of early coefficients of bi-Bazilevic functions

1 Introduction

Let Σ denote the family of meromorphic functions 119892 of theform

119892 (119911) = 119911 + 1198870+

infin

sum

119899=1

119887119899

1

119911119899 (1)

which are univalent in Δ = 119911 1 lt |119911| lt infin Thecoefficients of ℎ = 119892

minus1 the inverse map of the function 119892 isin Σare given by the Faber polynomial expansion

ℎ (119908) = 119892minus1

(119908) = 119908 + 1198610+

infin

sum

119899=1

119861119899

119908119899

= 119908 minus 1198870minus sum

119899ge1

1

119899119870119899

119899+1(1198870 1198871 119887

119899)

1

119908119899 119908 isin Δ

(2)

where

119870119899

119899+1= 119899119887119899minus1

01198871+ 119899 (119899 minus 1) 119887

119899minus2

01198872

+1

2119899 (119899 minus 1) (119899 minus 2) 119887

119899minus3

0(1198873+ 1198872

1)

+119899 (119899 minus 1) (119899 minus 2) (119899 minus 3)

3119887119899minus4

0(1198874+ 311988711198872)

+ sum

119895ge5

119887119899minus119895

0119881119895

(3)

and119881119895with 5 le 119895 le 119899 is a homogeneous polynomial of degree

119895 in the variables 1198871 1198872 119887

119899(see [1] p 349 or [2ndash4])

For 0 le 120572 lt 1 0 le 120573 lt 1 119892 isin Σ and ℎ = 119892minus1 let 119861(120572 120573)

denote the class of bi-Bazilevic functions of order 120572 and type120573 (see Bazilevic [5]) if and only if

Re(( 119911

119892(119911))

1minus120573

1198921015840(119911)) gt 120572 119911 isin Δ

Re(( 119908

ℎ (119908))

1minus120573

ℎ1015840(119908)) gt 120572 119908 isin Δ

(4)

Estimates on the coefficients of classes of meromorphicunivalent functions were widely investigated in the literatureFor example Schiffer [4] obtained the estimate |119887

2| le 23 for

meromorphic univalent functions 119892 with 1198870= 0 and Duren

([6] or [7]) proved that if 1198871

= 1198872

= sdot sdot sdot = 119887119896

= 0 for1 le 119896 lt 1198992 then |119887

119899| le 2(119899+1) Schober [8] considered the

case where 1198870= 0 and obtained the estimate |119861

2119899minus1| le (2119899 minus

2)(119899(119899minus1)) for the odd coefficients of the inverse functionℎ = 119892minus1 subject to the restrictions 119861

2= 1198614= sdot sdot sdot = 119861

119899minus2= 0 if

119899 is even or that 1198612= 1198614= sdot sdot sdot = 119861

119899minus3= 0 if 119899 is odd Kapoor

and Mishra [9] considered the inverse function ℎ = 119892minus1

where 119892 isin 119861(120572 0) and obtained the bound 2(1 minus 120572)(119899 + 1)

if ((119899 minus 1)119899) le 120572 lt 1 This restriction imposed on 120572 is a verytight restriction since the class 119861(120572 0) shrinks for large valuesof 119899 More recently Hamidi et al [10] (also see [11]) improvedthe coefficient estimate given by Kapoor and Mishra in [9]

Hindawi Publishing CorporationJournal of Complex AnalysisVolume 2014 Article ID 263917 4 pageshttpdxdoiorg1011552014263917

2 Journal of Complex Analysis

The real difficulty arises when the bi-univalency condition isimposed on the meromorphic functions 119892 and its inverse ℎ =

119892minus1 The unexpected and unusual behavior of the coefficients

of meromorphic functions 119892 and their inverses ℎ = 119892minus1 prove

the investigation of the coefficient bounds for bi-univalentfunctions to be very challenging In this paper we extend theresults of Kapoor andMishra [9] andHamidi et al [10 11] to alarger class of meromorphic bi-univalent functions namely119861(120572 120573) We conclude our paper with an examination of theunexpected behavior of the early coefficients ofmeromorphicbi-Bazilevic functions which is the best estimate yet appearedin the literature

2 Main Results

Applying a result of Airault [12] or [1 3] to meromorphicfunctions 119892 of the form (1) for real values of 119901 we can write

(119892 (119911)

119911)

1199011199111198921015840(119911)

119892 (119911)= 1 +

infin

sum

119899=0

(1 minus119899 + 1

119901)119870119901

119899+1

times (1198870 1198871 1198872 119887

119899)

1

119911119899+1

(5)

where

119870119901

119899+1=

119901

(119901 minus (119899 + 1)) (119899 + 1)119887119899+1

0

+119901

(119901 minus 119899) (119899 minus 1)119887119899minus1

01198871

+119901

(119901 minus 119899 + 1) (119899 minus 2)119887119899minus2

01198872

+119901

(119901 minus 119899 + 2) (119899 minus 3)119887119899minus3

0[1198873+119901 minus 119899 + 2

21198872

1]

+119901

(119901 minus 119899 + 3) (119899 minus 4)119887119899minus4

0[1198874+ (119901 minus 119899 + 3) 119887

11198872]

+ sum

119895ge5

119887119899minus119895

0119881119895

(6)

and 119881119895is a homogeneous polynomial of degree 119895 5 le

119895 le 119899 in the variables 1198871 1198872 119887

119899 A simple calculation

reveals that the first three terms of119870119901119899+1

(1198870 1198871 1198872 119887

119899)may

be expressed as

(1 minus1

119901)119870119901

1= (119901 minus 1) 119887

0

(1 minus2

119901)119870119901

2= (119901 minus 2) (

119901 minus 1

21198872

0+ 1198871)

(1 minus3

119901)119870119901

3= (119901 minus 3)

times ((119901 minus 1) (119901 minus 2)

61198873

0+ (119901 minus 1) 119887

01198871+ 1198872)

(7)

In general for any real number 119901 an expansion of 119870119901119899

=

119870119901

119899(1198862 1198863 119886

119899) (eg see [2 equation (4)] or [3]) is given

by

119870119901

119899= 119901119886119899+119901 (119901 minus 1)

21198632

119899+

119901

(119901 minus 3)31198633

119899+ sdot sdot sdot

+119901

(119901 minus 119899)119899119863119899

119899

(8)

where

119863119898

119904+119898= 119863119898

119904+119898(1198861 1198862 119886

119904+119898) =

infin

sum

119904=1

119898(1198861)1205831

sdot sdot sdot (119886119904+119898

)120583119904+119898

1205831 sdot sdot sdot 120583119904+119898

(9)

and 1198861

= 1 Here we note that the sum is taken over allnonnegative integers 120583

1 120583

119904+119898satisfying 120583

1+ 1205832+ sdot sdot sdot +

120583119904+119898

= 119898 and 1205831+ 21205832+ sdot sdot sdot + (119904 +119898)120583

119904+119898= 119904 +119898 Evidently

119863119899

119899(1198861 1198862 119886

119904+119898) = 119886

119899

1 [12] A similar Faber polynomial

expansion formula holds for the coefficients of ℎ the inversemap of 119892 (eg see [1 p 349])

The Faber polynomials introduced by Faber [13] play animportant role in various areas of mathematical sciencesespecially in geometric function theory (Gong [14] ChapterIII and Schiffer [4]) The recent interest in the calculus of theFaber polynomials especially when it involves ℎ = 119892

minus1 theinverse of 119892 (see [1 3 12 15]) beautifully fits our case for themeromorphic bi-univalent functions As a result we are ableto state and prove the following

Theorem 1 For 0 le 120572 lt 1 and 0 le 120573 lt 1 let g isin 119861(120572 120573) andℎ = 119892

minus1isin 119861(120572 120573) If 119887

1= 1198872= sdot sdot sdot = 119887

119899minus1= 0 for 119899 being odd

or if 1198870= 1198871= sdot sdot sdot = 119887

119899minus1= 0 for 119899 being even then

10038161003816100381610038161198871198991003816100381610038161003816 le

2 (1 minus 120572)

119899 + 1 minus 120573 (10)

Proof For 119892 isin 119861(120572 120573) and for ℎ = 119892minus1

isin 119861(120572 120573) there existpositive real part functions 119901(119911) = 1 +sum

infin

119899=1119888119899119911minus119899 and 119902(119908) =

1 + suminfin

119899=1119889119899119908minus119899 in Δ so that

(119911

119892 (119911))

1minus120573

1198921015840(119911) = 120572 + (1 minus 120572) 119901 (119911) = 1 + (1 minus 120572)

infin

sum

119899=1

119888119899

119911119899

(11)

(119908

ℎ (119908))

1minus120573

ℎ1015840(119908) = 120572 + (1 minus 120572) 119902 (119908) = 1 + (1 minus 120572)

infin

sum

119899=1

119889119899

119908119899

(12)

Note that according to the Caratheodory lemma (eg [7])|119888119899| le 2 and |119889

119899| le 2

Journal of Complex Analysis 3

On the other hand by the Faber polynomial expansionwe observe that

(119911

119892 (119911))

1minus120573

1198921015840(119911) = (

119892 (119911)

119911)

120573

(1199111198921015840(119911)

119892 (119911))

= 1 +

infin

sum

119899=0

(1 minus119899 + 1

120573)

times 119870120573

119899+1(1198870 1198871 119887

119899)

1

119911119899+1

(13)

(119908

ℎ (119908))

1minus120573

ℎ1015840(119908) = (

ℎ (119908)

119908)

120573

(119908ℎ1015840(119908)

ℎ (119908))

= 1 +

infin

sum

119899=0

(1 minus119899 + 1

120573)

times 119870120573

119899+1(1198610 1198611 119861

119899)

1

119908119899+1

(14)

Comparing the corresponding coefficients of (11) and (13) weobtain

(1 minus 120572) 119888119899+1

= (1 minus119899 + 1

120573)119870120573

119899+1(1198870 1198871 119887

119899) (15)

Similarly from (12) and (14) we obtain

(1 minus 120572) 119889119899+1

= (1 minus119899 + 1

120573)119870120573

119899+1(1198610 1198611 119861

119899) (16)

For the case 1198871= 1198872= sdot sdot sdot = 119887

119899minus1= 0 (119899 = odd) (15) and (16)

respectively upon using a simple algebraic manipulation andthe fact that 119861

119899= minus119887119899 reduce to

(120573 minus 1) sdot sdot sdot (120573 minus (119899 + 1))

(119899 + 1)119887119899+1

0

+ (120573 minus (119899 + 1)) 119887119899 = (1 minus 120572) 119888119899+1

(17)

(120573 minus 1) sdot sdot sdot (120573 minus (119899 + 1))

(119899 + 1)119887119899+1

0

minus (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119889

119899+1

(18)

Multiplying (18) by minus1 and adding it to (17) we obtain

+ 2 (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) (119888

119899+1minus 119889119899+1

) (19)

For the other case 1198870= 1198871= 1198872= sdot sdot sdot = 119887

119899minus1= 0 (119899 = even)

(15) and (16) respectively reduce to

+ (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119888

119899+1 (20)

minus (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119889

119899+1 (21)

Solving either of (19) (20) or (21) for 119887119899 taking the

absolute values and applying the Caratheodory lemma weobtain |119887

119899| le 2(1 minus 120572)(119899 + 1 minus 120573)

Relaxing the coefficient restrictions imposed onTheorem 1 we experience the unpredictable behavior ofthe coefficients of bi-univalent functions

Theorem 2 Let 119892 isin 119861(120572 120573) 0 le 120572 lt 1 0 le 120573 lt 1 bebi-univalent in Δ Then

(i)

100381610038161003816100381611988701003816100381610038161003816 le

radic4 (1 minus 120572)

(2 minus 120573) (1 minus 120573) 0 le 120572 lt

1

2 minus 120573

2 (1 minus 120572)

1 minus 120573

1

2 minus 120573le 120572 lt 1

(22)

(ii)

100381610038161003816100381611988711003816100381610038161003816 le

2 (1 minus 120572)

2 minus 120573 (23)

(iii)

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816le

2 (1 minus 120572)

2 minus 120573minus(1 minus 120573) (3 minus 120573)

2

100381610038161003816100381611988701003816100381610038161003816

2

if5 minus 4120573 + 120573

2

6 minus 5120573 + 1205732le 120572 lt 1

(24)

Proof (i) For 119899 isin 0 1 2 (15) yields

(1 minus 120572) 1198881= (120573 minus 1) 119887

0 (25)

(1 minus 120572) 1198882 =(120573 minus 1) (120573 minus 2)

21198872

0+ (120573 minus 2) 119887

1 (26)

(1 minus 120572) 1198883=

(120573 minus 1) (120573 minus 2) (120573 minus 3)

31198873

0

+ (120573 minus 3) (120573 minus 1) 11988701198871+ (120573 minus 3) 119887

2

(27)

Similarly for 119899 isin 0 1 2 (16) yields

(1 minus 120572) 1198891= minus (120573 minus 1) 119887

0 (28)

(1 minus 120572) 1198892=

(120573 minus 1) (120573 minus 2)

21198872

0minus (120573 minus 2) 119887

1 (29)

(1 minus 120572) 1198893 = minus(120573 minus 1) (120573 minus 2) (120573 minus 3)

31198873

0

+ (120573 minus 3) (120573 minus 2) 11988701198871minus (120573 minus 3) 119887

2

(30)

From either of the relations (25) or (28) we obtain

100381610038161003816100381611988701003816100381610038161003816 le

2 (1 minus 120572)

1 minus 120573 (31)

On the other hand adding (26) and (29) yields

(1 minus 120572) (1198882+ 1198892) = (120573 minus 1) (120573 minus 2) 119887

2

0 (32)

4 Journal of Complex Analysis

Solve the above equation for 1198870 take the absolute values of

both sides and apply the Caratheodory lemma to obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(1 minus 120572) (

100381610038161003816100381611988821003816100381610038161003816 +

100381610038161003816100381611988921003816100381610038161003816)

(1 minus 120573) (2 minus 120573)le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573) (33)

Now the bounds given in Theorem 2 (i) for |1198870| follow upon

noting that

2 (1 minus 120572)

1 minus 120573le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573)if 1

2 minus 120573le 120572 lt 1 (34)

(ii) Multiply (29) by minus1 and adding it to (26) we obtain

(1 minus 120572) (1198882minus 1198892) = 2 (120573 minus 2) 119887

1 (35)

Solve the above equation for 1198871 take the absolute values of

both sides and apply the Caratheodory lemma to obtain thebound |119887

1| le 2(1 minus 120572)(2 minus 120573)

(iii) From (29) we have

1198871minus 1198872

0=

1 minus 120572

2 minus 120573(1198892minus(2 minus 120573) (3 minus 120573)

2 (1 minus 120572)1198872

0) (36)

Substituting for 1198870= ((1 minus 120572)(1 minus 120573))119889

1and taking the

absolute values of both sides we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573

100381610038161003816100381610038161003816100381610038161003816

1198892minus(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)1198892

1

100381610038161003816100381610038161003816100381610038161003816

=1 minus 120572

2 minus 120573

100381610038161003816100381610038161198892+ 1205821198892

1

10038161003816100381610038161003816

(37)

Using the fact |1198892+1205821198892

1| le 2 +120582|119889

1|2 if 120582 ge minus12which is

due to the first author [16 Lemma 1] and noting that 120582 ge minus12

if 120572 ge (5 minus 4120573 + 1205732)(6 minus 5120573 + 120573

2) we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)

100381610038161003816100381611988911003816100381610038161003816

2) (38)

Now substituting back for 1198891= ((1 minus 120573)(1 minus 120572))119887

0we

obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120573) (2 minus 120573) (3 minus 120573)

2 (1 minus 120572)

100381610038161003816100381611988701003816100381610038161003816

2) (39)

Remark 3 For the special case 119861(120572 0) we obtain the class ofmeromorphic bi-starlike functions Consequently the boundgiven for |119887

0| by our Theorem 2 (i) is an improvement to that

given in Hamidi et al [11 Theorem 2i]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Airault and J Ren ldquoAn algebra of differential operators andgenerating functions on the set of univalent functionsrdquo Bulletindes Sciences Mathematiques vol 126 no 5 pp 343ndash367 2002

[2] H Airault ldquoSymmetric sums associated to the factorization ofGrunsky coefficientsrdquo in Conference Groups and SymmetriesMontreal Canada April 2007

[3] H Airault and A Bouali ldquoDifferential calculus on the Faberpolynomialsrdquo Bulletin des Sciences Mathematiques vol 130 no3 pp 179ndash222 2006

[4] M Schiffer ldquoFaber polynomials in the theory of univalentfunctionsrdquo Bulletin of the American Mathematical Society vol54 pp 503ndash517 1948

[5] I E Bazilevic ldquoOn a case of integrability in quadratures of theLoewner-Kufarev equationrdquo Matematicheskii Sbornik vol 37no 79 pp 471ndash476 1955

[6] P L Duren ldquoCoefficients of meromorphic schlicht functionsrdquoProceedings of the American Mathematical Society vol 28 pp169ndash172 1971

[7] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[8] G Schober ldquoCoefficients of inverses of meromorphic univalentfunctionsrdquo Proceedings of the American Mathematical Societyvol 67 no 1 pp 111ndash116 1977

[9] G P Kapoor and A K Mishra ldquoCoefficient estimates forinverses of starlike functions of positive orderrdquo Journal ofMathematical Analysis and Applications vol 329 no 2 pp 922ndash934 2007

[10] S G Hamidi S A Halim and J M Jahangiri ldquoCoefficientestimates for a class of meromorphic bi-univalent functionsrdquoComptes Rendus Mathematique vol 351 no 9-10 pp 349ndash3522013

[11] S G Hamidi S A Halim and J M Jahangiri ldquoFaber poly-nomial coefficient estimates for meromorphic bi-starlike func-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 498159 4 pages 2013

[12] H Airault ldquoRemarks on Faber polynomialsrdquo InternationalMathematical Forum vol 3 no 9ndash12 pp 449ndash456 2008

[13] G Faber ldquoUber polynomische Entwickelungenrdquo Mathematis-che Annalen vol 57 no 3 pp 389ndash408 1903

[14] S Gong The Bieberbach Conjecture vol 12 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyProvidence RI USA 1999

[15] P G Todorov ldquoOn the Faber polynomials of the univalentfunctions of class Σrdquo Journal of Mathematical Analysis andApplications vol 162 no 1 pp 268ndash276 1991

[16] M Jahangiri ldquoOn the coefficients of powers of a class of Bazile-vic functionsrdquo Indian Journal of Pure and Applied Mathematicsvol 17 no 9 pp 1140ndash1144 1986

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Stochastic AnalysisInternational Journal of

2 Journal of Complex Analysis

The real difficulty arises when the bi-univalency condition isimposed on the meromorphic functions 119892 and its inverse ℎ =

119892minus1 The unexpected and unusual behavior of the coefficients

of meromorphic functions 119892 and their inverses ℎ = 119892minus1 prove

the investigation of the coefficient bounds for bi-univalentfunctions to be very challenging In this paper we extend theresults of Kapoor andMishra [9] andHamidi et al [10 11] to alarger class of meromorphic bi-univalent functions namely119861(120572 120573) We conclude our paper with an examination of theunexpected behavior of the early coefficients ofmeromorphicbi-Bazilevic functions which is the best estimate yet appearedin the literature

2 Main Results

Applying a result of Airault [12] or [1 3] to meromorphicfunctions 119892 of the form (1) for real values of 119901 we can write

(119892 (119911)

119911)

1199011199111198921015840(119911)

119892 (119911)= 1 +

infin

sum

119899=0

(1 minus119899 + 1

119901)119870119901

119899+1

times (1198870 1198871 1198872 119887

119899)

1

119911119899+1

(5)

where

119870119901

119899+1=

119901

(119901 minus (119899 + 1)) (119899 + 1)119887119899+1

0

+119901

(119901 minus 119899) (119899 minus 1)119887119899minus1

01198871

+119901

(119901 minus 119899 + 1) (119899 minus 2)119887119899minus2

01198872

+119901

(119901 minus 119899 + 2) (119899 minus 3)119887119899minus3

0[1198873+119901 minus 119899 + 2

21198872

1]

+119901

(119901 minus 119899 + 3) (119899 minus 4)119887119899minus4

0[1198874+ (119901 minus 119899 + 3) 119887

11198872]

+ sum

119895ge5

119887119899minus119895

0119881119895

(6)

and 119881119895is a homogeneous polynomial of degree 119895 5 le

119895 le 119899 in the variables 1198871 1198872 119887

119899 A simple calculation

reveals that the first three terms of119870119901119899+1

(1198870 1198871 1198872 119887

119899)may

be expressed as

(1 minus1

119901)119870119901

1= (119901 minus 1) 119887

0

(1 minus2

119901)119870119901

2= (119901 minus 2) (

119901 minus 1

21198872

0+ 1198871)

(1 minus3

119901)119870119901

3= (119901 minus 3)

times ((119901 minus 1) (119901 minus 2)

61198873

0+ (119901 minus 1) 119887

01198871+ 1198872)

(7)

In general for any real number 119901 an expansion of 119870119901119899

=

119870119901

119899(1198862 1198863 119886

119899) (eg see [2 equation (4)] or [3]) is given

by

119870119901

119899= 119901119886119899+119901 (119901 minus 1)

21198632

119899+

119901

(119901 minus 3)31198633

119899+ sdot sdot sdot

+119901

(119901 minus 119899)119899119863119899

119899

(8)

where

119863119898

119904+119898= 119863119898

119904+119898(1198861 1198862 119886

119904+119898) =

infin

sum

119904=1

119898(1198861)1205831

sdot sdot sdot (119886119904+119898

)120583119904+119898

1205831 sdot sdot sdot 120583119904+119898

(9)

and 1198861

= 1 Here we note that the sum is taken over allnonnegative integers 120583

1 120583

119904+119898satisfying 120583

1+ 1205832+ sdot sdot sdot +

120583119904+119898

= 119898 and 1205831+ 21205832+ sdot sdot sdot + (119904 +119898)120583

119904+119898= 119904 +119898 Evidently

119863119899

119899(1198861 1198862 119886

119904+119898) = 119886

119899

1 [12] A similar Faber polynomial

expansion formula holds for the coefficients of ℎ the inversemap of 119892 (eg see [1 p 349])

The Faber polynomials introduced by Faber [13] play animportant role in various areas of mathematical sciencesespecially in geometric function theory (Gong [14] ChapterIII and Schiffer [4]) The recent interest in the calculus of theFaber polynomials especially when it involves ℎ = 119892

minus1 theinverse of 119892 (see [1 3 12 15]) beautifully fits our case for themeromorphic bi-univalent functions As a result we are ableto state and prove the following

Theorem 1 For 0 le 120572 lt 1 and 0 le 120573 lt 1 let g isin 119861(120572 120573) andℎ = 119892

minus1isin 119861(120572 120573) If 119887

1= 1198872= sdot sdot sdot = 119887

119899minus1= 0 for 119899 being odd

or if 1198870= 1198871= sdot sdot sdot = 119887

119899minus1= 0 for 119899 being even then

10038161003816100381610038161198871198991003816100381610038161003816 le

2 (1 minus 120572)

119899 + 1 minus 120573 (10)

Proof For 119892 isin 119861(120572 120573) and for ℎ = 119892minus1

isin 119861(120572 120573) there existpositive real part functions 119901(119911) = 1 +sum

infin

119899=1119888119899119911minus119899 and 119902(119908) =

1 + suminfin

119899=1119889119899119908minus119899 in Δ so that

(119911

119892 (119911))

1minus120573

1198921015840(119911) = 120572 + (1 minus 120572) 119901 (119911) = 1 + (1 minus 120572)

infin

sum

119899=1

119888119899

119911119899

(11)

(119908

ℎ (119908))

1minus120573

ℎ1015840(119908) = 120572 + (1 minus 120572) 119902 (119908) = 1 + (1 minus 120572)

infin

sum

119899=1

119889119899

119908119899

(12)

Note that according to the Caratheodory lemma (eg [7])|119888119899| le 2 and |119889

119899| le 2

Journal of Complex Analysis 3

On the other hand by the Faber polynomial expansionwe observe that

(119911

119892 (119911))

1minus120573

1198921015840(119911) = (

119892 (119911)

119911)

120573

(1199111198921015840(119911)

119892 (119911))

= 1 +

infin

sum

119899=0

(1 minus119899 + 1

120573)

times 119870120573

119899+1(1198870 1198871 119887

119899)

1

119911119899+1

(13)

(119908

ℎ (119908))

1minus120573

ℎ1015840(119908) = (

ℎ (119908)

119908)

120573

(119908ℎ1015840(119908)

ℎ (119908))

= 1 +

infin

sum

119899=0

(1 minus119899 + 1

120573)

times 119870120573

119899+1(1198610 1198611 119861

119899)

1

119908119899+1

(14)

Comparing the corresponding coefficients of (11) and (13) weobtain

(1 minus 120572) 119888119899+1

= (1 minus119899 + 1

120573)119870120573

119899+1(1198870 1198871 119887

119899) (15)

Similarly from (12) and (14) we obtain

(1 minus 120572) 119889119899+1

= (1 minus119899 + 1

120573)119870120573

119899+1(1198610 1198611 119861

119899) (16)

For the case 1198871= 1198872= sdot sdot sdot = 119887

119899minus1= 0 (119899 = odd) (15) and (16)

respectively upon using a simple algebraic manipulation andthe fact that 119861

119899= minus119887119899 reduce to

(120573 minus 1) sdot sdot sdot (120573 minus (119899 + 1))

(119899 + 1)119887119899+1

0

+ (120573 minus (119899 + 1)) 119887119899 = (1 minus 120572) 119888119899+1

(17)

(120573 minus 1) sdot sdot sdot (120573 minus (119899 + 1))

(119899 + 1)119887119899+1

0

minus (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119889

119899+1

(18)

Multiplying (18) by minus1 and adding it to (17) we obtain

+ 2 (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) (119888

119899+1minus 119889119899+1

) (19)

For the other case 1198870= 1198871= 1198872= sdot sdot sdot = 119887

119899minus1= 0 (119899 = even)

(15) and (16) respectively reduce to

+ (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119888

119899+1 (20)

minus (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119889

119899+1 (21)

Solving either of (19) (20) or (21) for 119887119899 taking the

absolute values and applying the Caratheodory lemma weobtain |119887

119899| le 2(1 minus 120572)(119899 + 1 minus 120573)

Relaxing the coefficient restrictions imposed onTheorem 1 we experience the unpredictable behavior ofthe coefficients of bi-univalent functions

Theorem 2 Let 119892 isin 119861(120572 120573) 0 le 120572 lt 1 0 le 120573 lt 1 bebi-univalent in Δ Then

(i)

100381610038161003816100381611988701003816100381610038161003816 le

radic4 (1 minus 120572)

(2 minus 120573) (1 minus 120573) 0 le 120572 lt

1

2 minus 120573

2 (1 minus 120572)

1 minus 120573

1

2 minus 120573le 120572 lt 1

(22)

(ii)

100381610038161003816100381611988711003816100381610038161003816 le

2 (1 minus 120572)

2 minus 120573 (23)

(iii)

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816le

2 (1 minus 120572)

2 minus 120573minus(1 minus 120573) (3 minus 120573)

2

100381610038161003816100381611988701003816100381610038161003816

2

if5 minus 4120573 + 120573

2

6 minus 5120573 + 1205732le 120572 lt 1

(24)

Proof (i) For 119899 isin 0 1 2 (15) yields

(1 minus 120572) 1198881= (120573 minus 1) 119887

0 (25)

(1 minus 120572) 1198882 =(120573 minus 1) (120573 minus 2)

21198872

0+ (120573 minus 2) 119887

1 (26)

(1 minus 120572) 1198883=

(120573 minus 1) (120573 minus 2) (120573 minus 3)

31198873

0

+ (120573 minus 3) (120573 minus 1) 11988701198871+ (120573 minus 3) 119887

2

(27)

Similarly for 119899 isin 0 1 2 (16) yields

(1 minus 120572) 1198891= minus (120573 minus 1) 119887

0 (28)

(1 minus 120572) 1198892=

(120573 minus 1) (120573 minus 2)

21198872

0minus (120573 minus 2) 119887

1 (29)

(1 minus 120572) 1198893 = minus(120573 minus 1) (120573 minus 2) (120573 minus 3)

31198873

0

+ (120573 minus 3) (120573 minus 2) 11988701198871minus (120573 minus 3) 119887

2

(30)

From either of the relations (25) or (28) we obtain

100381610038161003816100381611988701003816100381610038161003816 le

2 (1 minus 120572)

1 minus 120573 (31)

On the other hand adding (26) and (29) yields

(1 minus 120572) (1198882+ 1198892) = (120573 minus 1) (120573 minus 2) 119887

2

0 (32)

4 Journal of Complex Analysis

Solve the above equation for 1198870 take the absolute values of

both sides and apply the Caratheodory lemma to obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(1 minus 120572) (

100381610038161003816100381611988821003816100381610038161003816 +

100381610038161003816100381611988921003816100381610038161003816)

(1 minus 120573) (2 minus 120573)le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573) (33)

Now the bounds given in Theorem 2 (i) for |1198870| follow upon

noting that

2 (1 minus 120572)

1 minus 120573le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573)if 1

2 minus 120573le 120572 lt 1 (34)

(ii) Multiply (29) by minus1 and adding it to (26) we obtain

(1 minus 120572) (1198882minus 1198892) = 2 (120573 minus 2) 119887

1 (35)

Solve the above equation for 1198871 take the absolute values of

both sides and apply the Caratheodory lemma to obtain thebound |119887

1| le 2(1 minus 120572)(2 minus 120573)

(iii) From (29) we have

1198871minus 1198872

0=

1 minus 120572

2 minus 120573(1198892minus(2 minus 120573) (3 minus 120573)

2 (1 minus 120572)1198872

0) (36)

Substituting for 1198870= ((1 minus 120572)(1 minus 120573))119889

1and taking the

absolute values of both sides we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573

100381610038161003816100381610038161003816100381610038161003816

1198892minus(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)1198892

1

100381610038161003816100381610038161003816100381610038161003816

=1 minus 120572

2 minus 120573

100381610038161003816100381610038161198892+ 1205821198892

1

10038161003816100381610038161003816

(37)

Using the fact |1198892+1205821198892

1| le 2 +120582|119889

1|2 if 120582 ge minus12which is

due to the first author [16 Lemma 1] and noting that 120582 ge minus12

if 120572 ge (5 minus 4120573 + 1205732)(6 minus 5120573 + 120573

2) we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)

100381610038161003816100381611988911003816100381610038161003816

2) (38)

Now substituting back for 1198891= ((1 minus 120573)(1 minus 120572))119887

0we

obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120573) (2 minus 120573) (3 minus 120573)

2 (1 minus 120572)

100381610038161003816100381611988701003816100381610038161003816

2) (39)

Remark 3 For the special case 119861(120572 0) we obtain the class ofmeromorphic bi-starlike functions Consequently the boundgiven for |119887

0| by our Theorem 2 (i) is an improvement to that

given in Hamidi et al [11 Theorem 2i]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Airault and J Ren ldquoAn algebra of differential operators andgenerating functions on the set of univalent functionsrdquo Bulletindes Sciences Mathematiques vol 126 no 5 pp 343ndash367 2002

[2] H Airault ldquoSymmetric sums associated to the factorization ofGrunsky coefficientsrdquo in Conference Groups and SymmetriesMontreal Canada April 2007

[3] H Airault and A Bouali ldquoDifferential calculus on the Faberpolynomialsrdquo Bulletin des Sciences Mathematiques vol 130 no3 pp 179ndash222 2006

[4] M Schiffer ldquoFaber polynomials in the theory of univalentfunctionsrdquo Bulletin of the American Mathematical Society vol54 pp 503ndash517 1948

[5] I E Bazilevic ldquoOn a case of integrability in quadratures of theLoewner-Kufarev equationrdquo Matematicheskii Sbornik vol 37no 79 pp 471ndash476 1955

[6] P L Duren ldquoCoefficients of meromorphic schlicht functionsrdquoProceedings of the American Mathematical Society vol 28 pp169ndash172 1971

[7] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[8] G Schober ldquoCoefficients of inverses of meromorphic univalentfunctionsrdquo Proceedings of the American Mathematical Societyvol 67 no 1 pp 111ndash116 1977

[9] G P Kapoor and A K Mishra ldquoCoefficient estimates forinverses of starlike functions of positive orderrdquo Journal ofMathematical Analysis and Applications vol 329 no 2 pp 922ndash934 2007

[10] S G Hamidi S A Halim and J M Jahangiri ldquoCoefficientestimates for a class of meromorphic bi-univalent functionsrdquoComptes Rendus Mathematique vol 351 no 9-10 pp 349ndash3522013

[11] S G Hamidi S A Halim and J M Jahangiri ldquoFaber poly-nomial coefficient estimates for meromorphic bi-starlike func-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 498159 4 pages 2013

[12] H Airault ldquoRemarks on Faber polynomialsrdquo InternationalMathematical Forum vol 3 no 9ndash12 pp 449ndash456 2008

[13] G Faber ldquoUber polynomische Entwickelungenrdquo Mathematis-che Annalen vol 57 no 3 pp 389ndash408 1903

[14] S Gong The Bieberbach Conjecture vol 12 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyProvidence RI USA 1999

[15] P G Todorov ldquoOn the Faber polynomials of the univalentfunctions of class Σrdquo Journal of Mathematical Analysis andApplications vol 162 no 1 pp 268ndash276 1991

[16] M Jahangiri ldquoOn the coefficients of powers of a class of Bazile-vic functionsrdquo Indian Journal of Pure and Applied Mathematicsvol 17 no 9 pp 1140ndash1144 1986

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Complex Analysis 3

On the other hand by the Faber polynomial expansionwe observe that

(119911

119892 (119911))

1minus120573

1198921015840(119911) = (

119892 (119911)

119911)

120573

(1199111198921015840(119911)

119892 (119911))

= 1 +

infin

sum

119899=0

(1 minus119899 + 1

120573)

times 119870120573

119899+1(1198870 1198871 119887

119899)

1

119911119899+1

(13)

(119908

ℎ (119908))

1minus120573

ℎ1015840(119908) = (

ℎ (119908)

119908)

120573

(119908ℎ1015840(119908)

ℎ (119908))

= 1 +

infin

sum

119899=0

(1 minus119899 + 1

120573)

times 119870120573

119899+1(1198610 1198611 119861

119899)

1

119908119899+1

(14)

Comparing the corresponding coefficients of (11) and (13) weobtain

(1 minus 120572) 119888119899+1

= (1 minus119899 + 1

120573)119870120573

119899+1(1198870 1198871 119887

119899) (15)

Similarly from (12) and (14) we obtain

(1 minus 120572) 119889119899+1

= (1 minus119899 + 1

120573)119870120573

119899+1(1198610 1198611 119861

119899) (16)

For the case 1198871= 1198872= sdot sdot sdot = 119887

119899minus1= 0 (119899 = odd) (15) and (16)

respectively upon using a simple algebraic manipulation andthe fact that 119861

119899= minus119887119899 reduce to

(120573 minus 1) sdot sdot sdot (120573 minus (119899 + 1))

(119899 + 1)119887119899+1

0

+ (120573 minus (119899 + 1)) 119887119899 = (1 minus 120572) 119888119899+1

(17)

(120573 minus 1) sdot sdot sdot (120573 minus (119899 + 1))

(119899 + 1)119887119899+1

0

minus (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119889

119899+1

(18)

Multiplying (18) by minus1 and adding it to (17) we obtain

+ 2 (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) (119888

119899+1minus 119889119899+1

) (19)

For the other case 1198870= 1198871= 1198872= sdot sdot sdot = 119887

119899minus1= 0 (119899 = even)

(15) and (16) respectively reduce to

+ (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119888

119899+1 (20)

minus (120573 minus (119899 + 1)) 119887119899= (1 minus 120572) 119889

119899+1 (21)

Solving either of (19) (20) or (21) for 119887119899 taking the

absolute values and applying the Caratheodory lemma weobtain |119887

119899| le 2(1 minus 120572)(119899 + 1 minus 120573)

Relaxing the coefficient restrictions imposed onTheorem 1 we experience the unpredictable behavior ofthe coefficients of bi-univalent functions

Theorem 2 Let 119892 isin 119861(120572 120573) 0 le 120572 lt 1 0 le 120573 lt 1 bebi-univalent in Δ Then

(i)

100381610038161003816100381611988701003816100381610038161003816 le

radic4 (1 minus 120572)

(2 minus 120573) (1 minus 120573) 0 le 120572 lt

1

2 minus 120573

2 (1 minus 120572)

1 minus 120573

1

2 minus 120573le 120572 lt 1

(22)

(ii)

100381610038161003816100381611988711003816100381610038161003816 le

2 (1 minus 120572)

2 minus 120573 (23)

(iii)

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816le

2 (1 minus 120572)

2 minus 120573minus(1 minus 120573) (3 minus 120573)

2

100381610038161003816100381611988701003816100381610038161003816

2

if5 minus 4120573 + 120573

2

6 minus 5120573 + 1205732le 120572 lt 1

(24)

Proof (i) For 119899 isin 0 1 2 (15) yields

(1 minus 120572) 1198881= (120573 minus 1) 119887

0 (25)

(1 minus 120572) 1198882 =(120573 minus 1) (120573 minus 2)

21198872

0+ (120573 minus 2) 119887

1 (26)

(1 minus 120572) 1198883=

(120573 minus 1) (120573 minus 2) (120573 minus 3)

31198873

0

+ (120573 minus 3) (120573 minus 1) 11988701198871+ (120573 minus 3) 119887

2

(27)

Similarly for 119899 isin 0 1 2 (16) yields

(1 minus 120572) 1198891= minus (120573 minus 1) 119887

0 (28)

(1 minus 120572) 1198892=

(120573 minus 1) (120573 minus 2)

21198872

0minus (120573 minus 2) 119887

1 (29)

(1 minus 120572) 1198893 = minus(120573 minus 1) (120573 minus 2) (120573 minus 3)

31198873

0

+ (120573 minus 3) (120573 minus 2) 11988701198871minus (120573 minus 3) 119887

2

(30)

From either of the relations (25) or (28) we obtain

100381610038161003816100381611988701003816100381610038161003816 le

2 (1 minus 120572)

1 minus 120573 (31)

On the other hand adding (26) and (29) yields

(1 minus 120572) (1198882+ 1198892) = (120573 minus 1) (120573 minus 2) 119887

2

0 (32)

4 Journal of Complex Analysis

Solve the above equation for 1198870 take the absolute values of

both sides and apply the Caratheodory lemma to obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(1 minus 120572) (

100381610038161003816100381611988821003816100381610038161003816 +

100381610038161003816100381611988921003816100381610038161003816)

(1 minus 120573) (2 minus 120573)le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573) (33)

Now the bounds given in Theorem 2 (i) for |1198870| follow upon

noting that

2 (1 minus 120572)

1 minus 120573le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573)if 1

2 minus 120573le 120572 lt 1 (34)

(ii) Multiply (29) by minus1 and adding it to (26) we obtain

(1 minus 120572) (1198882minus 1198892) = 2 (120573 minus 2) 119887

1 (35)

Solve the above equation for 1198871 take the absolute values of

both sides and apply the Caratheodory lemma to obtain thebound |119887

1| le 2(1 minus 120572)(2 minus 120573)

(iii) From (29) we have

1198871minus 1198872

0=

1 minus 120572

2 minus 120573(1198892minus(2 minus 120573) (3 minus 120573)

2 (1 minus 120572)1198872

0) (36)

Substituting for 1198870= ((1 minus 120572)(1 minus 120573))119889

1and taking the

absolute values of both sides we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573

100381610038161003816100381610038161003816100381610038161003816

1198892minus(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)1198892

1

100381610038161003816100381610038161003816100381610038161003816

=1 minus 120572

2 minus 120573

100381610038161003816100381610038161198892+ 1205821198892

1

10038161003816100381610038161003816

(37)

Using the fact |1198892+1205821198892

1| le 2 +120582|119889

1|2 if 120582 ge minus12which is

due to the first author [16 Lemma 1] and noting that 120582 ge minus12

if 120572 ge (5 minus 4120573 + 1205732)(6 minus 5120573 + 120573

2) we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)

100381610038161003816100381611988911003816100381610038161003816

2) (38)

Now substituting back for 1198891= ((1 minus 120573)(1 minus 120572))119887

0we

obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120573) (2 minus 120573) (3 minus 120573)

2 (1 minus 120572)

100381610038161003816100381611988701003816100381610038161003816

2) (39)

Remark 3 For the special case 119861(120572 0) we obtain the class ofmeromorphic bi-starlike functions Consequently the boundgiven for |119887

0| by our Theorem 2 (i) is an improvement to that

given in Hamidi et al [11 Theorem 2i]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Airault and J Ren ldquoAn algebra of differential operators andgenerating functions on the set of univalent functionsrdquo Bulletindes Sciences Mathematiques vol 126 no 5 pp 343ndash367 2002

[2] H Airault ldquoSymmetric sums associated to the factorization ofGrunsky coefficientsrdquo in Conference Groups and SymmetriesMontreal Canada April 2007

[3] H Airault and A Bouali ldquoDifferential calculus on the Faberpolynomialsrdquo Bulletin des Sciences Mathematiques vol 130 no3 pp 179ndash222 2006

[4] M Schiffer ldquoFaber polynomials in the theory of univalentfunctionsrdquo Bulletin of the American Mathematical Society vol54 pp 503ndash517 1948

[5] I E Bazilevic ldquoOn a case of integrability in quadratures of theLoewner-Kufarev equationrdquo Matematicheskii Sbornik vol 37no 79 pp 471ndash476 1955

[6] P L Duren ldquoCoefficients of meromorphic schlicht functionsrdquoProceedings of the American Mathematical Society vol 28 pp169ndash172 1971

[7] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[8] G Schober ldquoCoefficients of inverses of meromorphic univalentfunctionsrdquo Proceedings of the American Mathematical Societyvol 67 no 1 pp 111ndash116 1977

[9] G P Kapoor and A K Mishra ldquoCoefficient estimates forinverses of starlike functions of positive orderrdquo Journal ofMathematical Analysis and Applications vol 329 no 2 pp 922ndash934 2007

[10] S G Hamidi S A Halim and J M Jahangiri ldquoCoefficientestimates for a class of meromorphic bi-univalent functionsrdquoComptes Rendus Mathematique vol 351 no 9-10 pp 349ndash3522013

[11] S G Hamidi S A Halim and J M Jahangiri ldquoFaber poly-nomial coefficient estimates for meromorphic bi-starlike func-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 498159 4 pages 2013

[12] H Airault ldquoRemarks on Faber polynomialsrdquo InternationalMathematical Forum vol 3 no 9ndash12 pp 449ndash456 2008

[13] G Faber ldquoUber polynomische Entwickelungenrdquo Mathematis-che Annalen vol 57 no 3 pp 389ndash408 1903

[14] S Gong The Bieberbach Conjecture vol 12 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyProvidence RI USA 1999

[15] P G Todorov ldquoOn the Faber polynomials of the univalentfunctions of class Σrdquo Journal of Mathematical Analysis andApplications vol 162 no 1 pp 268ndash276 1991

[16] M Jahangiri ldquoOn the coefficients of powers of a class of Bazile-vic functionsrdquo Indian Journal of Pure and Applied Mathematicsvol 17 no 9 pp 1140ndash1144 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Complex Analysis

Solve the above equation for 1198870 take the absolute values of

both sides and apply the Caratheodory lemma to obtain

100381610038161003816100381611988701003816100381610038161003816 le

radic(1 minus 120572) (

100381610038161003816100381611988821003816100381610038161003816 +

100381610038161003816100381611988921003816100381610038161003816)

(1 minus 120573) (2 minus 120573)le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573) (33)

Now the bounds given in Theorem 2 (i) for |1198870| follow upon

noting that

2 (1 minus 120572)

1 minus 120573le radic

4 (1 minus 120572)

(1 minus 120573) (2 minus 120573)if 1

2 minus 120573le 120572 lt 1 (34)

(ii) Multiply (29) by minus1 and adding it to (26) we obtain

(1 minus 120572) (1198882minus 1198892) = 2 (120573 minus 2) 119887

1 (35)

Solve the above equation for 1198871 take the absolute values of

both sides and apply the Caratheodory lemma to obtain thebound |119887

1| le 2(1 minus 120572)(2 minus 120573)

(iii) From (29) we have

1198871minus 1198872

0=

1 minus 120572

2 minus 120573(1198892minus(2 minus 120573) (3 minus 120573)

2 (1 minus 120572)1198872

0) (36)

Substituting for 1198870= ((1 minus 120572)(1 minus 120573))119889

1and taking the

absolute values of both sides we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573

100381610038161003816100381610038161003816100381610038161003816

1198892minus(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)1198892

1

100381610038161003816100381610038161003816100381610038161003816

=1 minus 120572

2 minus 120573

100381610038161003816100381610038161198892+ 1205821198892

1

10038161003816100381610038161003816

(37)

Using the fact |1198892+1205821198892

1| le 2 +120582|119889

1|2 if 120582 ge minus12which is

due to the first author [16 Lemma 1] and noting that 120582 ge minus12

if 120572 ge (5 minus 4120573 + 1205732)(6 minus 5120573 + 120573

2) we obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120572) (2 minus 120573) (3 minus 120573)

2 (1 minus 120573)

100381610038161003816100381611988911003816100381610038161003816

2) (38)

Now substituting back for 1198891= ((1 minus 120573)(1 minus 120572))119887

0we

obtain

100381610038161003816100381610038161198871minus 1198872

0

10038161003816100381610038161003816=

1 minus 120572

2 minus 120573(2 minus

(1 minus 120573) (2 minus 120573) (3 minus 120573)

2 (1 minus 120572)

100381610038161003816100381611988701003816100381610038161003816

2) (39)

Remark 3 For the special case 119861(120572 0) we obtain the class ofmeromorphic bi-starlike functions Consequently the boundgiven for |119887

0| by our Theorem 2 (i) is an improvement to that

given in Hamidi et al [11 Theorem 2i]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Airault and J Ren ldquoAn algebra of differential operators andgenerating functions on the set of univalent functionsrdquo Bulletindes Sciences Mathematiques vol 126 no 5 pp 343ndash367 2002

[2] H Airault ldquoSymmetric sums associated to the factorization ofGrunsky coefficientsrdquo in Conference Groups and SymmetriesMontreal Canada April 2007

[3] H Airault and A Bouali ldquoDifferential calculus on the Faberpolynomialsrdquo Bulletin des Sciences Mathematiques vol 130 no3 pp 179ndash222 2006

[4] M Schiffer ldquoFaber polynomials in the theory of univalentfunctionsrdquo Bulletin of the American Mathematical Society vol54 pp 503ndash517 1948

[5] I E Bazilevic ldquoOn a case of integrability in quadratures of theLoewner-Kufarev equationrdquo Matematicheskii Sbornik vol 37no 79 pp 471ndash476 1955

[6] P L Duren ldquoCoefficients of meromorphic schlicht functionsrdquoProceedings of the American Mathematical Society vol 28 pp169ndash172 1971

[7] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[8] G Schober ldquoCoefficients of inverses of meromorphic univalentfunctionsrdquo Proceedings of the American Mathematical Societyvol 67 no 1 pp 111ndash116 1977

[9] G P Kapoor and A K Mishra ldquoCoefficient estimates forinverses of starlike functions of positive orderrdquo Journal ofMathematical Analysis and Applications vol 329 no 2 pp 922ndash934 2007

[10] S G Hamidi S A Halim and J M Jahangiri ldquoCoefficientestimates for a class of meromorphic bi-univalent functionsrdquoComptes Rendus Mathematique vol 351 no 9-10 pp 349ndash3522013

[11] S G Hamidi S A Halim and J M Jahangiri ldquoFaber poly-nomial coefficient estimates for meromorphic bi-starlike func-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 498159 4 pages 2013

[12] H Airault ldquoRemarks on Faber polynomialsrdquo InternationalMathematical Forum vol 3 no 9ndash12 pp 449ndash456 2008

[13] G Faber ldquoUber polynomische Entwickelungenrdquo Mathematis-che Annalen vol 57 no 3 pp 389ndash408 1903

[14] S Gong The Bieberbach Conjecture vol 12 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyProvidence RI USA 1999

[15] P G Todorov ldquoOn the Faber polynomials of the univalentfunctions of class Σrdquo Journal of Mathematical Analysis andApplications vol 162 no 1 pp 268ndash276 1991

[16] M Jahangiri ldquoOn the coefficients of powers of a class of Bazile-vic functionsrdquo Indian Journal of Pure and Applied Mathematicsvol 17 no 9 pp 1140ndash1144 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of