Báo cáo hóa học: " A study of Pescar’s univalence criteria for space of analytic functions" doc

Ehsan, Malaysia Abstract An attempt has been made to give a criteria to a family of functions defined in the space of analytic functions to be univalent.. Such criteria extended earlier

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R E S E A R C H Open Access

of analytic functions

Imran Faisal and Maslina Darus*

* Correspondence: maslina@ukm.

my

School of Mathematical Sciences,

Faculty of Science and Technology,

Universiti Kebangsaan Malaysia,

Bangi 43600 Selangor D Ehsan,

Malaysia

Abstract

An attempt has been made to give a criteria to a family of functions defined in the space of analytic functions to be univalent Such criteria extended earlier univalence criteria of Pescar’s-type of analytic functions

2000 MSC: 30C45

Keywords: analytic functions, univalent functions, integral operator

1 Introduction and preliminaries Let A denote the class of analytic functions of the form f (z) = z +∞

k=2 a k z kin the open unit diskU = {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 inU The results in this communication are motivated by Pescar [1] In [1], a new criteria for an analytic function to be univalent is introduced which is true only for two fixed natural numbers Then, Breaz and Breaz [2] introduced a new integral operator using product n-multiply analytic functions and gave another univalence criteria for such analytic integral operators Using such integral operator, we extend the criteria given

by Pescar in 2005 and prove that it is true for any two consecutive natural numbers First, we recall the main results of Pescar introduced in 1996 and later 2005 as follow:

Lemma 1.1 [1,3] Let a be a complex number with Re a > 0 such that c Î ℂ,

|c| ≤ 1, c = −1 If fÎ A satisfies the condition

c |z|2α+ (1− |z|2α

zf(z)

αf(z)

≤ 1, ∀z∈U,

then the function(F α (z)) α =αz

0t α−1 f(t)dt is analytic and univalent inU Lemma 1.2 [1] Let the function f Î A satisfies

z f22f (z)(z)− 1

≤ 1, ∀z ∈ U Also, let

α ∈ R(α ∈ [1,3

2])and cÎ ℂ If|c| ≤ 3− 2α

α (c= −1)and |g(z)|≤ 1, then the function

Ga(z) defined by(G α (z)) α =αz

0[f (t)] α−1is in the univalent function class S

Lemma 1.3 [4] If f Î A satisfies the condition

z f22f (z)(z)− 1

≤ 1, ∀z ∈ U, then the function f is univalent inU

© 2011 Faisal and Darus; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in

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Lemma 1.4 (Schwarz Lemma) Let the analytic function f be regular in the open unit diskUand let f(0) = 0 Iff (z) ≤1, (z∈U)then |f(z)| ≤ |z| where the equality holds

true only if f(z) = kz and |k| = 1

Breaz (cf., [2,5]) introduced a family of integral operators for fi Î A univalent inU

denoted by Gn,asuch that

G n, α (z) =

z

0

[f1(t)] α−1 · · · [f n (t)] α−1 dt

In the case of n = 1, the operator Gn, abecomes identical to the operator Gagiven in Lemma 1.2 which was introduced by Pescar in 1996

2 Main univalence criteria for analytic function

In this section, we make a criteria for space of analytic functions to be univalent We

give proof and applications only for the first theorem and for the remaining theorems

we use the same techniques

Theorem 2.1 Let fiÎ A,

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

4M(3 n− 2n)

2n

n, α ∈ R, c ∈ C

and

f i (z) ≤ M, ∀i and M ≥ 1.

Then, the family of functions f denoted by Gn,abelong to the class S

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

6M(4 n− 3n)

3n

n, α ∈ R, c ∈ C

and

f i (z) ≤ M, ∀i and M ≥ 1.

Then, the family of functions f denoted by Gn, abelong to the class S

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

8M(5 n− 4n)

4n

n, α ∈ R, c ∈ C

and

f i (z) ≤ M, ∀i and M ≥ 1.

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Proof of Theorem 2.1 Since for each fiÎ A implies

f i (z)

z = 1 +

∞ 2

a n z n−1, ∀i

and

f i (z)

z = 1 at z = 0, ∀i.

We can write

n

i=1

f i (z)

z = 1 at z = 0.

Now suppose that

F(z) =

z

0

f1(t)

t

α−1

· · ·

f n (t)

t

α−1

dt

and taking logarithmic derivative and doing some mathematics we get

F(z) = ( α − 1)

∞

i=2

f i (z)

z

α−2⎛

⎝zfi (z) − f i (z)

z2

n

j=1

f j (z) (z)

α−1⎞

⎠ ,

⇒ zF(z)

F(z) = (α − 1)

∞

i=2

zfi (z)

f i (z) − 1

,

⇒

zF F(z)(z) ≤ (α − 1) ∞

i=2

z2fi (z)

f i (z))2

f i (z)

z

+ 1 Using hypothesis of Theorem 2.1 such as

z2fi (z) (f i (z))2

≤ 2, f i (z) ≤ M, ∀ifor M ≥ 1 and after doing calculation we get

zF F(z) (z)

≤ (α − 1) m

i=1

(2M + (2M + 1)), ∵ 2M ≥ 1,

⇒

zF F(z) (z)

≤ (α − 1) m

i=1

2M +

2M + 2M

2

+

3M + 3M

2

, ∵ 3M/2 ≥ 1

⇒

zF F(z) (z)

≤ (α − 1) m

i=1

2M + 3M + 9M

2 M + · · · + nth term

Therefore, by Lemma 1.1, we get

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

1

α

zF(z)

F(z)

≤ |c| +

1

α

zF(z)

F(z)

,

⇒

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

1

α

zF(z)

F(z)

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c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

≤ |c|+

α − 1

α

n i=1

2M + 3M +9

2M + · · · + nth term .

Hence, after calculation, we have

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

α − 1 α

i=1

4M(3 n− 2n)

2n

or

c |z|2α+ (1− |z|2α

zF(z)

αF(z)

α − 1 α

n

4M(3 n− 2n)

2n

,

and again using the hypothesis of Theorem 2.1 we get

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

≤ 1, and hence proved

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

6M(3 n− 2n)

3n

n, α ∈ R, c ∈ C

and

f i (z) ≤ M, ∀i and M ≥ 1.

Proof Using the proof of Theorem 2.1, we have

zF F(z)(z) ≤ (α − 1) ∞

i−2

z2fi (z) (f i (z))2

f i (z)

z

+ 1

Again, using the hypothesis, we get

zF F(z) (z)

≤ (α − 1) m

i=1

(2M + (2M + 1)), ∵ 2M ≥ 1,

⇒

zF F(z) (z)

≤ (α − 1) m

i−1

2M +

M + M

2

+

M + 2M

9

,

⇒

zF F(z) (z)

≤ (α − 1) m

i=1

2M +4

3M +

8

9M + · · · + nth term

Thus, we have

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

≤ |c| +

1

α

zF(z)

F(z)

≤ |c| +

1

α

zF(z)

F(z)

,

⇒

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

≤ |c| +

1

α

zF(z)

F(z)

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c |z|2α+ (1− |z|2α

(z)

≤ |c| +

α − 1 α

n i=1

2M +4

3M +

8

9M + · · · + nth term

,

which implies that

c |z|2α+ (1− |z|2α

zF(z)

αF(z)

α − 1

α ) n

i=1

6M(3 n− 2n)

3n

or

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

α − 1

α

n

6M(3 n− 2n)

3n

Again, using the hypothesis of Theorem 2.1, we get

c |z|2α+ (1− |z|2α

zF(z)

αF

(z)

and hence proved

Similarly, we proved the following theorems:

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

8M(4 n− 3n)

4n

n, α ∈ R, c ∈ C

and

f i (z) ≤M, ∀i and M ≥ 1.

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

10M(5 n− 4n)

5n

n, α ∈ R, c ∈ C

and

f i (z) ≤ M, ∀i and M ≥ 1.

3 Applications of univalence criteria

Considering n = 1 in Theorem 2.1, we obtain the following application:

Corollary 3.1 Let fiÎ A,

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

(2M), α ∈ R, c ∈ C

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f i (z) ≤1, ∀i.

Considering M = n = 11 in Theorem 2.1, we obtain second application as follow:

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤

3α − 2 α

, α ∈ R, c ∈ C

and

f i (z) ≤1, ∀i.

Considering M = 1 in Theorem 2.1, we obtain third application such as:

z2fi (z) (f i (z))2− 1

≤ 1, z ∈ Ufor all i = {1, 2, , n}

If

|c| ≤ 1 +

α − 1

α

(3n− 2n)

2n−2

n, α ∈ R, c ∈ C

and

f i (z) ≤1, ∀i.

If we substitute n = 1 and M = n = 1 in Theorem 2.4, we get the results of Corol-laries 3.1 and 3.2, respectively

Other work related to integral operators concerning on univalence criteria and prop-erties can be found in [6,7]

Acknowledgements

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

Authors ’ contributions

The first author is currently a PhD student under supervision of the second author and jointly worked on the results.

All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 26 June 2011 Accepted: 10 November 2011 Published: 10 November 2011

References

1 Pescar, V: On the univalence of some integral operators J Indian Acad Math 27, 239 –243 (2005)

2 Breaz, D, Breaz, N: Univalence of an integral operator Mathematica (Cluj) 47(70), 35 –38 (2005)

3 Pescar, V: A new generalization of Ahlfors ’s and Becker’s criterion of univalence Bull Malaysian Math Soc 19, 53–54

(1996)

4 Ozaki, S, Nunokawa, M: The Schwarzian derivative and univalent func-tions Proc Am Math Soc 33, 392 –394 (1972).

doi:10.1090/S0002-9939-1972-0299773-3

5 Breaz, D: Integral Operators on Spaces of Univalent Functions Publishing House of the Romanian Academy of Sciences,

Bucharest (in Romanian) (2004)

6 Darus, M, Faisal, I: A study on Becker ’s univalence criteria Abstr Appl Anal 2011, 13 (2011) Article ID759175,

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7 Mohammed, A, Darus, M: Starlikeness properties for a new integral operator for meromorphic functions J Appl Math

2011, 8 (2011) Article ID 804150,

doi:10.1186/1029-242X-2011-109 Cite this article as: Faisal and Darus: A study of Pescar’s univalence criteria for space of analytic functions Journal

of Inequalities and Applications 2011 2011:109.

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