Mathematical and Computer Modelling 55 (2012) 1466–1470

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Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

A study of Ahlfors’ univalence criteria for a space of analytic functions:Criteria IIImran Faisal, Maslina Darus ∗

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia

a r t i c l e i n f o

Article history:Received 17 June 2011Received in revised form 26 September2011Accepted 24 October 2011

Keywords:Analytic functionsIntegral operatorUnivalence criteria

a b s t r a c t

An attempt has been made to give some criteria (Criteria II) for a function defined in thespace of analytic functions to be univalent. Such criteria extend those obtained earlier fromAhlfors-type univalence criteria of analytic functions. We also discuss its application in thespace of analytic functions.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction and preliminaries

Let A denote the class of analytic functions of the form f (z) = z +∑

∞

k=2 akzk in the open unit disk U = {z : |z| < 1}

normalized by f (0) = f ′(0) − 1 = 0.We denote by S the subclass of A consisting of functions which are univalent in U.Next, we define some well known subclasses of A, denoted by A2, K, K2, K2,µ and S(p) respectively as follows:

A2 ⊂ A =

f ∈ A : f (z) = z +

∞−k=3

akzk, z ∈ U

;

K ⊂ A =

f ∈ A :

z2f ′(z)(f (z))2

− 1 < 1, z ∈ U

;

K2 ⊂ K =f ∈ K : f ′′(0) = 0

;

K2,µ ⊂ K2 =

f ∈ K2 :

z2f ′(z)(f (z))2

− 1 < µ, 0 < µ ≤ 1, z ∈ U

;

S(p) ⊂ A =

f ∈ A :

z(f (z))

′′ ≤ p, 0 < p ≤ 2, z ∈ U

.

∗ Corresponding author.E-mail addresses: faisalmath@gmail.com (I. Faisal), maslina@ukm.my, maslina@pkrisc.cc.ukm.my (M. Darus).

0895-7177/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2011.10.025

I. Faisal, M. Darus / Mathematical and Computer Modelling 55 (2012) 1466–1470 1467

Breaz (cf., [1,2]) introduced a family of integral operators for fi ∈ A univalent in U denoted by Gn,α such that

Gn,α(z) =

(n(α − 1) + 1)

∫ z

0[f1(t)]α−1

· · · [fn(t)]α−1dt

1n(α−1)+1

. (1)

In case of n = 1, the operator Gn,α becomes identical to the operator Gα given in Lemma 2 introduced by Pescar in 1996.

Lemma 1.1 ([3,4]). Let c be a complex number, |c| ≤ 1, c = −1. If f (z) = z + a2z2 + · · · is a regular function in U and|c|z |

2+(1 − |z |

2)zf ′′(z)f ′(z) | ≤ 1, ∀z ∈ U then the function f is regular and univalent in U.

Lemma 1.2 ([5, Schwarz Lemma]). Let the function f (z) be regular in the disk UR = {z ∈ C : |z| < R} with |f (z)| < M for fixedM. If f (z) has one zero with multiplicity order bigger than m for z = 0, then |f (z)| ≤

MRm |z|m,m,M ∈ ℜ, z ∈ UR. The equality

holds only if f (z) = eiθ MRm (z)m, where θ is constant.

Lemma 1.3 ([6,7]). Let δ be a complex number with ℜ(δ) > 0 such that c ∈ C, |c| ≤ 1, c = −1. If f ∈ A satisfies thecondition |c|z |

2δ+(1 − |z |

2δ)zf ′(z)δf (z) | ≤ 1, ∀z ∈ U, then the function (Fδ(z))δ = δ

z0 tδ−1f ′(t)dt is analytic and univalent

in U.

Lemma 1.4 ([8]). If a function f ∈ S(p), then |z2f ′(z)(f (z))2

− 1| ≤ p|z|2, ∀z ∈ U.

The aim of the present paper is to extend the idea of Ahlfors for the set of natural numbers. We have used a pair ofconsecutive natural numbers for this purpose and prove that their use gives us newunivalence criteria for analytic functions.

2. Main Univalence Criteria (II) for space of analytic functions

In this section, we introduce new univalence criteria for the space of analytic functions.

Theorem 2.1. Let c be a complex number and |fi(z)| ≤ Mi,Mi ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ K2,µi , ∀i and

ℜ(ρ) ≥

∞−i=1

(α − 1)3n(ui + 3) − 3.2n

3n

Mi, ρ ∈ C,

if |c| ≤ 1 −α − 1ℜ(ρ)

∞−i=1

3n(ui + 3) − 3.2n

3n

Mi, Mi ≥ 1, (2)

then Gn,α ∈ S.

Theorem 2.2. Let c be a complex number and |fi(z)| ≤ Mi,Mi ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ K2,µi , ∀i and

ℜ(ρ) ≥

∞−i=1

(α − 1)4n(ui + 4) − 4.3n

4n

Mi, ρ ∈ C,

if |c| ≤ 1 −α − 1ℜ(ρ)

∞−i=1

4n(ui + 4) − 4.3n

4n

Mi, Mi ≥ 1, (3)

then Gn,α ∈ S.

Theorem 2.3. Let c be a complex number and |fi(z)| ≤ Mi,Mi ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ K2,µi , ∀i and

ℜ(ρ) ≥

∞−i=1

(α − 1)5n(ui + 5) − 5.4n

5n

Mi, ρ ∈ C,

if |c| ≤ 1 −α − 1ℜ(ρ)

∞−i=1

5n(ui + 5) − 5.4n

5n

Mi, Mi ≥ 1, (4)

then Gn,α ∈ S.Proof of Theorem 2.1. Since each fi ∈ A implies

f1(z)(z)

=f2(z)(z)

= · · · =fn(z)(z)

= 1 at z = 0,

⇒

n∏i=1

fi(z)(z)

= 1 at z = 0.

1468 I. Faisal, M. Darus / Mathematical and Computer Modelling 55 (2012) 1466–1470

If

F(z) =

∫ z

0

f1(t)t

α−1

· · ·

fn(t)t

α−1

dt,

then, we get

F ′′(z) = (α − 1)∞−i=2

fi(z)z

α−2 zf ′

i (z) − fi(z)z2

n∏j=1

fj(z)z

α−1

,

andzF ′′(z)F ′(z)

= (α − 1)∞−i=2

zf ′

i (z)fi(z)

− 1

,

⇒

zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

z2f ′

i (z)(fi(z))2

fi(z)z

+ 1

.

Since fi(t) ∈ K2,µi ∀i, therefore |z2f ′i (z)(fi(z))2

−1| < ui. Also by using Schwarz Lemma 1.2, we get |fi(z)| ≤ Miz (∵ R = 1). Therefore zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

z2f ′

i (z)(fi(z))2

− 1 + 1

Mi + Mi + 1

, zF ′′(z)

F ′(z)

≤ (α − 1)∞−i=2

(uiMi + 2Mi), ∵ Mi ≥ 1.

After calculations, we have zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

uiMi +

3Mi + 2Mi

3

, ∵ Mi ≥ 1, zF ′′(z)

F ′(z)

≤ (α − 1)∞−i=2

uiMi + Mi +

3Mi + 2Mi

3

+

3Mi + Mi

9

+ · · · + nth term

, ∵ Mi ≥ 1.

Hence by doing calculation, we get zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

3n(ui + 3) − 3.2n

3n

Mi.

Now by using Lemma 1.3, we getc|z|2ρ + (1 − |z|2ρ)zF ′′(z)ρF ′(z)

≤ |c| +1

|ρ|

zF ′′(z)F ′(z)

≤ |c| +1

ℜ(ρ)

zF ′′(z)F ′(z)

,c|z|2ρ + (1 − |z|2ρ)zF ′′(z)ρF ′(z)

≤ |c| +α − 1ℜ(ρ)

∞−i=1

3n(ui + 3) − 3.2n

3n

Mi.

Finally by using the hypothesis, we getc|z|2ρ + (1 − |z|2ρ)zF ′′(z)ρF ′(z)

≤ 1.

Hence the theorem is proved. �

Similarly, we proved Theorems 2.2 and 2.3 as well.

Theorem 2.4. Let c be a complex number and |fi(z)| ≤ Mi,Mi ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ S(p), ∀i and

ℜ(ρ) ≥

∞−i=1

(α − 1)2n(pi − 2) + 2.3n

2n

Mi, ρ ∈ C,

if |c| ≤ 1 −α − 1ℜ(ρ)

∞−i=1

2n(pi − 2) + 2.3n

2n

Mi, Mi ≥ 1, (5)

then Gn,α ∈ S.

I. Faisal, M. Darus / Mathematical and Computer Modelling 55 (2012) 1466–1470 1469

Theorem 2.5. Let c be a complex number and |fi(z)| ≤ Mi,Mi ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ S(p), ∀i and

ℜ(ρ) ≥

∞−i=1

(α − 1)3n(pi − 3) + 3.4n

3n

Mi, ρ ∈ C,

if |c| ≤ 1 −α − 1ℜ(ρ)

∞−i=1

3n(pi − 3) + 3.4n

3n

Mi, Mi ≥ 1, (6)

then Gn,α ∈ S.

Theorem 2.6. Let c be a complex number and |fi| ≤ Mi,Mi ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ S(p), ∀i and

ℜ(ρ) ≥

∞−i=1

(α − 1)4n(pi − 4) + 4.5n

4n

Mi, ρ ∈ C,

if |c| ≤ 1 −α − 1ℜ(ρ)

∞−i=1

4n(pi − 4) + 4.5n

4n

Mi, Mi ≥ 1, (7)

then Gn,α ∈ S. Also

Proof of Theorem 2.5. Since each fi ∈ S(p), by Lemma 1.4, we have z2f ′

i (z)(fi(z))2

− 1 ≤ pi|z|2, ∀i.

Also, since |fi| ≤ Mi, by Schwarz Lemma 1.2, we have

|fi(z)| ≤ Mi|z|, ∀i.

Now by using the proof of Theorem 2.1, we have zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

z2f ′

i (z)(fi(z))2

fi(z)z

+ 1

.

Therefore zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

(piMi|z|2 + 2Mi), ∵ ∀Mi ≥ 1.

This implies that zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

piMi +

3Mi + 4Mi

3

, ∀Mi ≥ 1,

⇒

zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

piMi + Mi +

4Mi

3

, ∀Mi ≥ 1,

and zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

piMi +

3Mi + 4Mi

3

+

Mi

3+

13Mi

9

, ∀Mi ≥ 1,

therefore zF ′′(z)F ′(z)

≤ (α − 1)∞−i=2

piMi +

3Mi + 4Mi

3+

3Mi + 13Mi

9+ · · · + nth term

, ∀Mi ≥ 1.

After calculation, we get zF ′′(z)F ′(z)

≤ (α − 1)3n(pi − 3) + 3.4n

3n

Mi. (8)

Again by using Lemma 1.3, we getc|z|2ρ + (1 − |z|2ρ)zF ′′(z)ρF ′(z)

≤ |c| +1

|ρ|

zF ′′(z)F ′(z)

≤ |c| +1

ℜ(ρ)

zF ′′(z)F ′(z)

,c|z|2ρ + (1 − |z|2ρ)zF ′′(z)ρF ′(z)

≤ |c| +1

ℜ(ρ)

zF ′′(z)F ′(z)

,

1470 I. Faisal, M. Darus / Mathematical and Computer Modelling 55 (2012) 1466–1470

this implies thatc|z|2ρ + (1 − |z|2ρ)zF ′′(z)ρF ′(z)

≤ |c| +(α − 1)ℜ(ρ)

3n(pi − 3) + 3.4n

3n

Mi, (9)

as required.By using the same techniques for proving Theorem 2.5, we proved Theorem 2.4 and Theorem 2.6 as well. �

3. Applications of Main Univalence Criteria (II) in the space of analytic functions

ConsiderMi = M, ui = u, ∀i in Theorem 2.1, then, we have the following applications.

Corollary 3.1. Let c be a complex number and |fi(z)| ≤ M,M ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ K2,µi , ∀i and

ℜ(ρ) ≥ n3n(u + 3) − 3.2n

3n

M, ρ ∈ C,

if |c| ≤ 1 −n(α − 1)

ℜ(ρ)

3n(u + 3) − 3.2n

3n

M, M ≥ 1, (10)

then Gn,α ∈ S.

ConsiderMi = M, ui = u, ∀i andM = n = 1 in Theorem 2.1, then, we have:

Corollary 3.2. Let c be a complex number and |f | ≤ 1 and f (t) ∈ K2,µ, and

ℜ(ρ) ≥ (u + 1), ρ ∈ C,

if |c| ≤ 1 −n(α − 1)

ℜ(ρ)(u + 1), (11)

then Gn,α = f ∈ S.

Note that if we substitute Mi = M, ui = u, ∀i and M = n = 1 in any Theorems 2.2 and 2.3 its gives the same result ofCorollary 3.2.

ConsiderMi = M, pi = p, ∀i in Theorem 2.4. Then we have the following applications.

Corollary 3.3. Let c be a complex number and |fi(z)| ≤ M,M ≥ 1 for all by i = {1, 2, 3, . . . , n} and fi(t) ∈ S(p), ∀i and

ℜ(ρ) ≥ n(α − 1)2n(p − 2) + 2.3n

2n

M, ρ ∈ C,

if |c| ≤ 1 −n(α − 1)

ℜ(ρ)

2n(pi − 2) + 2.3n

2n

M,

then Gn,α ∈ S.

ConsiderMi = M, pi = p, ∀i and M = n = 1 in Theorem 2.4. Then we get the following:

Corollary 3.4. Let c be a complex number and |f (z)| ≤ 1 and f (t) ∈ S(p) and

ℜ(ρ) ≥ (α − 1)(p + 1), ρ ∈ C,

if |c| ≤ 1 −(α − 1)ℜ(ρ)

(p + 1),

then Gn,α = f ∈ S.

Acknowledgment

The work presented here was fully supported by UKM-ST-06-FRGS0244-2010.

References

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Park, Md., 1973, in: Ann. of Math. Studies, vol. 79, Princeton Univ. Press, Princeton, NJ, 1974, pp. 23–29.[4] J. Becker, Lownersche differentialgleichung und schlichtheitskriterien, Math. Ann. 202 (1973) 321–335.[5] Z. Nehari, Conformal Mapping, Mc Graw-Hill Book Company, New York, 1952. Reprinted by Dover, New York, 1975.[6] V. Pescar, On the univalence of some integral operators, J. Indian Acad. Math. 27 (2005) 239–243.[7] V. Pescar, A new generalization of Ahlfors and Becker’s criterion of univalence, Bull. Malaysian Math. Soc. (2) 19 (1996) 53–54.[8] V. Singh, On a class of univalent functions, Int. J. Math. Math. Sci. 23 (12) (2000) 855–857.