critical investigation of varioussubclasses of … · 2019. 5. 13. · critical investigation of...
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Innat. J. Math. Mth. Sci.Vol. 4 No. (1981) 89- 99
89
A CRITICAL INVESTIGATIONOF VARIOUS SUBCLASSES OF FUNCTIONS-
WHOSE REAL PART IS BOUNDED
S.K. BAJPAI
Universidade de BrasiliaDepartamento de Matematica70.910- Brasilia- DF
Brasil
(Received October 22, 1979)
ABSTRACT. In this paper we investigate various subclasses of univalent analytic
functions. We find that many of the subclasses introduced in the recent years are
no more new and infact coincide with the class due to Jakubowski. We further
study the generalised Jakubowski class of univalent functions and obtain a
representational formula and use it in deriving the coefficient relations for
this class.
KEY WORDS AND PHRASES. Univent Stke, onvex, Sp like, Boanded fcurcio,Fanction positive rl pa.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. P OA92, Second OA94.
i. INTRODUCTION.,
The Class S (,v).m,M
Let M,m be arbitrary fixed real numbers satisfying the condition
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90 S.K. BAJPAI
D*(re,M) E Dlt D2 andI
Jakubowski [7] introduced the class S*,M(’)m of regular functions f(z)z + akzk defined in E {z Iz[ < I} and satisfying the cndition
eiUzf’(z)isinu Ucosf(z)- -ml
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CRITICAL INVESTIGATION OF SUBCLASSES OF FUNCTIONS 91
Izf’(z) zf’ (z) le-ilYet another class is investigated by Gopalakrishna nd Shetiya ]. They nned
this class by SA(e,O,) and f E SA(u,O,D) if and only if f E S with
eiAzf (z)’’f’(’’) isin
cos I - p < p’ (1.5)But we now show that these classes are not new and some of the results of the
above authors are also not new. In fact, we have
THEOREM I.,
(a) S (u,8) S M,M(e,O) M 112(1-8)
(e,); S 282/(1-82(b) Sl(e,8) SI+M,M(c) M(e 8) S*m,S(0,0) m (l+e82)/(l-e282) and M (I+e)8/(I-(282)
(d) SX(,O,) S,s(,) S p/(l-e).,
PROOF. Suppose X zf’(z)/f(z). Then f e S (e,8) satisfies (1.2). Hence,
we have
[(X-n/{2(X-l-(X-) } < . (.61The inequality (1.6) is true if and only if
slx-l 2 > Re{ (X-s) (-I) }.
This last inequality is equivalent to
[X12 {l+e-2eS}. e(l-8) < 0B {x} + (S)By simple manipulations, the last inequality holds if and only if
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92 S.K. BAJPAI
,:__(z) 1 1< --CT=" (’)
Hence, if M 2--2, uwith u 0 then f e S (a,8) implies f e SM,M(a,0) mdconversely. This proves (a).
For.proving (b), we let Y {Xeil isinl acosl}l(1-a)cosl and
X zf’ (z) S x(z) Then, as before f e (a,) if and rely if
(1.8)
(In (1.8), if k -0 c, then this class is due to Caplinger and Causey 3). But
simple calculations shoe tha (1.8) is equivalent to
il zf’ (z) isinl-acoslf (z) 282
282--(l---a) cosX 1 <
1-2 1-" (1.9)
Thus, a comparision of (1.9) to (1.1) implies (b) and conversely. The proof for
(c) is similar. For (d) we observe tha f e Sl(cz,p,q) if md only if
ejAzf ’(z) isinl acosX’(z)""’(:t’)"cos), < I--" (1.10)
Again a cmparision of (I.I0) to (I.I) gives (d). Hence the proof of the theorem
is complete.
We note as a consequence of theorem I, that the coefficients estimates
obtained in the Series of ppers (I]-6, 8]-Ol, [-133-27, 29], 30) arecontained in [7]. But some of the coefficients estimates obtained in [I, [6],
ZI2 are not contained in the theorem 1 of [7]. We also note that the classintroduced in 83, is a very general one but appears unnatural in thepresentation. Therefore, next section,we shall consider the natural generalization
of the class S*m,M(,v) by admitting m to be complex. This we do as follows.
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CRITICAL INVESTIGATION OF SUBCLASSES OF FUNCTIONS 93
,2. The class Sm,M(,,o,t)._
,We define he class Sm,M(,U,a,t)_ to denote the class of f E S with
elUzf’(z) isinu cosu(I-) cosv 0.
The following lenna shows that when u 0, the utmost class of Szynal et al
and he class introduced by Libera and Livingston 1 are obtained.
S*LEMMA I. f E (U,v,a t) if md only if here exists some w(z) regular inm ,ME, w(0) 0 and [0(z) < 1 such that
zf’(z) I + (De-iV B) w(z) (2.2)"-EG)-- -- s ,.o(z)where D (A+S)(1-U)cceV; A+B > 0and
A= _2_ m2+ m(l_2u)+a(l_u)+it_t2B=. 1 +m+ a-i
PROOF. Let us write
L(z) R m a i
g(z) eiVzf’(z) isinv f(z) Uccev f(z)md h(z) (I-) cosy (z).
ll-m-a-itJ. Hence, there exists a (z) which isThen, we have [L(z)[ < I, L(0) =regular in E and which cm be written in the form
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94 S.K. BAJPAI
(z) L(.Z,.) .L_(0)I-L(O)L(z)
Clearly, we have 00(0) 0 and l(z) < i. Solving the equation (2.3), we have
L(z) L(z) m-oiL (0) +.(z)_I+L(0) (z)
and this in turn on solving gives the lenma.
Using this lemma, we prove the following theorem which generalizes the
THEOREM 2. For Izl < I, let f(z) z +k=n+ I
kz and f E Sm,M(,v,,t).Also, let qo be a natural number such that qo E o-I, Ko where
(i) If, either Re{Beiv} > O and M(l-)cos[D + 2n Re{BeiV}] _< n2Re{Bei} < 0 md DM(l-)cosv < n2 then we have
or,
(p+ l)n(k-l) 2 ak 12
knp+ I< D2; p i, 2, 3,
2(ii) If Re{BexV} > 0 and (l-)Mcos + 2n Re{BeXV}] > n(q+l)n q-I 2
{ D+nmBe" }I (k-1)2lak 12 n II n
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CRITICAL INVESTIGATION OF SUBCLASSES OF FUNCTIONS 95
The estimates (2.4) are sharp for all k and the estimates in (2.5) are sharp for
k qn+l, q 1,2,... because equality holds for the functions
f(z)
DZe-kB for B 0l-Bzk)
De-izk/ke for B -0(A)
PROOF. Under the conditions of theorem 2, we have
k0J(Z) I bkZk=n
for which (2.2) is true. Then simple calculations lead to
[ ak(k-l)zk- z+ I {D+B(k-l)ei}akzk3 [ I bkzk]"k’+l k--n+l k-n
(2.7)
Equating the coefficients of zk frcm both sides of (2.7) for k n+l,...,2n,
we have
e (k-l) ak Dbk_I.
This immediately gives us,
2nD2 2 D2I (k-l)21ak 12 < . Ibkl <
k=n+ I k=n+I(2.9)
But, for p > n+l, (2.7) can be written in the form
G(z) H(z) 0(z) (2.10)
where
and
n+pG(z) e:l’V n (k-l)akzk + [k= + I k-n+p+ I dkzk
P kH(z) Dz + [ {D+B(k-l)eiV}akzk=n+ I
Then (2.10) gives us
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96 S.K. BAJPAI
n+p2 2, (k-l)[ak[ +
k=p+ 1 k=n+p+ 1
k=n+ 1(k-)
From the hypothesis of (i) of theorem 2, we have
and so
ID+(k-1) BeiVl2_
(k-l) 2
pn (k-1)21ak 12 < D2;k=p+l p >n+l.
Replacing p by pn in (2.12) then we have
(Pl)n(k-1)2]akl2 < D2k=pn+l
f or p-2,3,...
Combining (2.13) with (2.9), we get (i).
If Re{Bei9} > 0 then the functicx%
(k_ll) 2 [i D+ (k_l) Bei 12_ (k_l) 2]is always a decreasing function of k-l.
On using this fact, we get,.by (2.11) for p > n+l.
(k-l) akl < D2+ D+nBei 12 n2k--p+ 1 k=n+ I n (k- I)
Let p 2n. Then (2.14) gives us
(2+l)n(k-l)mlak 12 < D,-----6
2n+lby (2.9),
Under the conditions of (ii) of theorem 2, we intend to prove the following.(q+l)n, (k-l) 2 ak 12k=qn+ 1 < ’nl)’ qKl D+mnBezu 2.m=0 n
(2.zz)
(2. 2)
(2.13)
(2.14)
(2.15)
(2.16)
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CRITICAL INVESTIGATION OF SUBCLASSES OF FUNCTIONS 97
The relation (2.16) is true for q I and 2, so let us assume (2.16) to be true
for 1,2,...,q-I and let us prove it for q if q < qo + I. By (2.14) we have(q/l)n
2 i72[ lakl2(k-l) < + ID+(k-l)Be1912 (k-l) 2 II 2k=qn+ I k--n+I
D2 + m--q-ll k(m+l)n!n+I { ID+ (k- l) Beiv12(k-l)2 (k-l) 2 l(k-l)2 ak 12
+m:l m
2 2 "(m-l)’m=0
m=0 n
The last inequality also follows by induction. This proves (ii) of theorem 2.
Also, if k > (qo+2)n then+l)n(q+l)n 2 -’(qi qn 2 2} 2]+ [ l)n+l ID+Be19(k-I) -(k-l) akl
(qo+l)n< D2+ I ID+Bei(k_l) 12_(k_I)2> 2 2
k-n+l [ (k-l)2(k-l) ak
m--I k--mn+l (k-l) 2(k-l) 21ak 12
m=0
This proves (iii) of theorem 2. Hence the proof of the theorem is complete.
COROLLARY. From (2.11) we obtain
{(k_l) 2k--n+ I
[D + (k-l)Bei[2}[ak 12 < D2.
obtained in (13-73) are contained in theorem 2 for the different choices of theparameters, m,M,/,9,o and t.
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98 S.K. BAJPAI
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