On the Univalence Criterion of a General Integral Operator

Volume 2008, Article ID 702715,8pages doi:10.1155/2008/702715

Research Article

On the Univalence Criterion of a General

Integral Operator

Daniel Breaz1and H. ¨Ozlem G ¨uney2

1_{Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia 510009, Romania}

2_{Department of Mathematics, Faculty of Science and Arts, University of Dicle, Diyarbakir 21280, Turkey}

Correspondence should be addressed to Daniel Breaz,[email protected]

Received 12 March 2008; Accepted 16 May 2008

Recommended by Paolo Ricci

In this paper we considered an general integral operator and three classes of univalent functions for which the second order derivative is equal to zero. By imposing supplimentary conditions for these functions we proved some univalent conditions for the considered general operator. Also some interesting particullar results are presented.

Copyrightq2008 D. Breaz and H. ¨Ozlem G ¨uney. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let unit disk and letAdenote the class of functionsfof the form

fz za2z2a3z3· · · z∈ U, 1.1

which are analytic in the open diskUand satisfy the conditionsf0 f0−10. Consider

S {f ∈ A:fare univalent functions inU}.

LetA2be the subclass ofAconsisting of functions of the form

fz z

∞

k3

akzk. 1.2

LetTbe the univalent subclass ofAwhich satisfies

z

2_fz

fz2 −1

LetT2be the subclass ofTfor whichf0 0. LetT2,μbe the subclass ofT2consisting

of functions of the form1.2which satisfy

z2_fz

fz2 −1

≤μ z∈ U 1.4

for someμ0< μ≤1, and let us denoteT2,1≡ T2. Furthermore, for some realpwith 0< p≤2

we define a subclassSpofAconsisting of all functionsfzwhich satisfy

_fz_z≤p z∈ U. 1.5

In1, Singh has shown that iffz∈ Sp, thenfzsatisfies

z2_fz

fz2 −1

≤p|z|2 z∈ U. 1.6

Ahlfors2and Becker3had obtained the following univalence criterion.

Theorem 1.1. Let cbe a complex number,|c| ≤ 1_{, c /} −1. If fz z a2z2 · · · is a regular

function inUand

c|z|2

1− |z|2zf

_z

fz

≤1 1.7

for allz∈ U, then the functionfis regular and univalent inU.

In4, Pescar had obtained the following theorem.

Theorem 1.2see4. Letβbe a complex number,Reβ >0,ca complex number,|c| ≤ 1_{, c /} −1,

andhz za2z2· · · a regular function inU. If

c|z|2β

1− |z|2βzh

_z

βhz

≤1 1.8

for allz∈ U, then the function

Fβz

β z

0

tβ−1htdt 1/β

z· · · 1.9

is regular and univalent inU.

Lemma 1.3the general Schwarz lemma5. Let the functionfzbe regular in the diskUR

{z∈C;|z|< R},with|fz|< Mfor fixedM. Iffzhas one zero with multiplicity order bigger than

mforz0, then

_fz≤ M Rm|z|

m_{, z∈ UR. 1.10}

In6, Seenivasagan and Breaz consider forfi∈ A2i1,2, . . . , nandα1, α2, . . . , αn, β∈

C, the integral operator

Fα1,α2,...,αn,βz

β

z

0

tβ−1 n

i1

fit

t 1/αi

dt 1/β

. 1.11

When αi αfor all i 1,2, . . . , n, Fα1,α2,...,αn,βzbecomes the integral operator Fα,βz

considered in7.

2. Main results

Theorem 2.1. LetM ≥ 1,let the functionsfi ∈ Sp, fori ∈ {1, . . . , n},satisfy the condition1.6, letαi, βbe complex numbers, letReβ≥n_i11pM1/|αi|, and letcbe a complex number.

If

|c| ≤1−_Re1 β

n

i1

1pM1

|αi| ,

_fiz≤M, 2.1

for allz∈ U, then the functionsFα1,α2,...,αn,βdefined in1.11are in the classS.

Proof. Define a function

hz z

0

n

i1

fit

t 1/αi

dt, 2.2

then we haveh0 h0−10. Also, a simple computation yields

hz n

i1

fiz

z 1/αi

,

zhz hz

n

i1

1 αi

zf_iz

fiz −1

.

2.3

From2.3, we have

zh_hzz

≤n

i1

1

_αizfiz fiz

1

n

i1

1

|αi|

z2fiz f_i2z

fi_zz1

. 2.4

From the hypothesis, we have |fiz| ≤ Mz ∈ U, i 1,2, . . . , n, then by the general Schwarz lemma, we obtain that

We apply this result in inequality2.4, then we obtain

zh_hzz

≤n

i1

1 _αi

z2_f

iz

fiz2 M1

≤n

i1

1 _αi

z2_f

iz

fiz2 −1

MM1

n

i1

1

_αipM|z|2M1

< n

i1

1pM1 _αi .

2.6

We have

c|z|2β

1− |z|2βzh

_z

βhz

c|z|2β

1− |z|2β1 β

n

i1

1 αi

zf_iz

fiz −1

≤ |c|_|1 β|·

n

i1

1

|αi|

z2_f

iz f2

iz ·

|fiz|

|z| 1

.

2.7

We obtain

c|z|2β1− |z|2βzh

_z

βhz

<|c|_|1 β|

n

i1

1pM1 _αi

≤ |c|_Re1 β

n

i1

1pM1 _αi .

2.8

So, from2.1we have

c|z|2β1− |z|2βzh

_z

βhz

≤1. 2.9

ApplyingTheorem 1.2, we obtain thatFα1,α2,...,αn,βis univalent.

Theorem 2.2. LetM ≥ 1,let the functionsfi ∈ Sp, fori ∈ {1, . . . , n}, satisfy the condition1.6, letα, βbe complex numbers, letReβ≥n1pM1/|α|, and letcbe a complex number.

If

|c| ≤1− _Re1 β

np1M1

|α| ,

_fiz≤M, 2.10

for allz∈ U, then the function

Fα,βz

β z

0

tβ−1 n

i1

fit

t 1/α

dt 1/β

2.11

Proof. InTheorem 2.1, we considerα1α2· · ·αnα.

Corollary 2.3. Let the functionsfi∈ Sp, fori∈ {1, . . . , n}, satisfy the condition1.6, letαi, β be complex numbers, letReβ≥n_i1p2/|αi|, and letcbe a complex number.

If

|c| ≤1− 1

Reβ n

i1

p2 _αi ,

_fiz≤1, 2.12

for allz∈ U, then the functionFα1,α2,...,αn,βdefined in1.11is in the classS.

Proof. InTheorem 2.1, we considerM 1.

Corollary 2.4. LetM ≥ 1, let the functionf ∈ Spsatisfy the condition1.6, letα, βbe complex numbers, letReβ≥1pM1/|α|, and letcbe a complex number.

If

|c| ≤1− 1

Reβ

1pM1

|α| ,

_fz≤M 2.13

for allz∈ U, then the function

Gα,βz

β z

0

tβ−1

ft t

1/α dt

1/β

2.14

is in the classS.

Proof. InTheorem 2.1, we considern1.

Corollary 2.5. Let the functionf ∈ Spsatisfy the condition1.6, letα, βbe complex numbers, let Reβ≥p2/|α|, and letcbe a complex number.

If

|c| ≤1− 1

Reβ p2

|α| ,

|fz| ≤1

2.15

for allz∈ U, then the function

Gα,βz

β z

0

tβ−1

ft t

1/α dt

1/β

2.16

is in the classS.

Theorem 2.6. LetM≥1, let the functionsfi∈ T2,μi, fori∈ {1, . . . , n}, satisfy the condition1.4, let

αi, βbe complex numbers, letReβ≥n_i11μiM1/|αi|, and letcbe a complex number.

If

|c| ≤1− 1

Reβ n

i1

1μiM1 _αi ,

_fiz≤M 2.17

for allz∈ U, then the functionFα1,α2,...,αn,βdefined in1.11is in the classS.

Proof. Define a function

hz z

0

n

i1

fit

t 1/αi

dt, 2.18

then we haveh0 h0−10. Also a simple computation yields

hz n

i1

fiz

z 1/αi

,

zhz hz

n

i1

1 αi

_zf

iz fiz −1

.

2.19

From2.19, we have

zh_hzz

≤n

i1

1

_αizfiz fiz

1

n

i1

1

_αiz2fiz f2

iz

fi_zz1

. 2.20

From the hypothesis, we have |fiz| ≤ Mz ∈ U, i 1,2, . . . , n, then by the general Schwarz lemma, we obtain that

_{fiz≤M|z| z∈ U, i1,2, . . . , n. 2.21}

From2.20, we obtain zh_hzz

≤n

i1

1 _αi

z2_f

iz

fiz2 M1

≤n

i1

1 _αi

z2_f

iz

fiz2 −1

MM1

≤n

i1

1

_αiμiMM1

n

i1

1μiM1 _αi .

We have

c|z|2β1− |z|2βzh

_z

βhz

c|z|2β1− |z|2β1 β

n

i1

1 αi

_zf

iz fiz −

1

≤ |c|_|1 β|·

n

i1

1

_αiz2fiz f_i2z

·|fiz| |z| 1

.

2.23

We obtain

c|z|2β1− |z|2βzh

_z

βhz

≤ |c|_|1 β|

n

i1

1μiM1 _αi

≤ |c|_Re1 β

n

i1

1μiM1 _αi .

2.24

So, from2.17we have

c|z|2β1− |z|2βzh

_z

βhz

≤1. 2.25

ApplyingTheorem 1.2, we obtain thatFα1,α2,...,αn,βis univalent.

Corollary 2.7. LetM ≥1, let the functionsfi ∈ T2,μi, fori ∈ {1, . . . , n}, satisfy the condition1.4, letα, βbe complex numbers, letReβ≥n1μiM1/|α|, and letcbe a complex number.

If

|c| ≤1− 1

Reβ

n1μiM1

|α| ,

_fiz≤M 2.26

for allz∈ U, then the function

Fα,βz

β z

0

tβ−1 n

i1

fit

t 1/α

dt 1/β

2.27

is in the classS.

Proof. InTheorem 2.6, we considerα1α2· · ·αnα.

Corollary 2.8. Let the functionsfi ∈ T2,μi, fori ∈ {1, . . . , n}, satisfy the condition1.4, letαi, β be complex numbers, letReβ≥n_i1μi2/|αi|, and letcbe a complex number.

If

|c| ≤1− 1

Reβ n

i1

μi2 _αi ,

_fiz≤1 2.28

Proof. InTheorem 2.6, we considerM 1.

Corollary 2.9. LetM ≥ 1, let the function f ∈ T2,μ satisfy the condition1.4, letα, βbe complex

numbers, letReβ≥1μM1/|α|, and letcbe a complex number. If

|c| ≤1− 1

Reβ

1μM1

|α| ,

_fz≤M 2.29

for allz∈ U, then the function

Gα,βz

β z

0

tβ−1

ft t

1/α dt

1/β

2.30

is in the classS.

Proof. InTheorem 2.6, we considern1.

Corollary 2.10. Let the functionf ∈ T2,μsatisfy the condition1.4, letα, βbe complex numbers, let

Reβ≥μ2/|α|, and letcbe a complex number. If

|c| ≤1− 1

Reβ μ2

|α| ,

_fz≤1 2.31

for allz∈ U, then the function

Gα,βz

β z

0

tβ−1

ft t

1/α dt

1/β

2.32

is in the classS.

Proof. InCorollary 2.9, we considerM1.

Acknowledgment

D. Breaz was supported by the GAR 20/2007.

References

1 V. Singh, “On a class of univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 12, pp. 855–857, 2000.

2 L. V. Ahlfors, “Sufficient conditions for quasiconformal extension,”Annals of Mathematics Studies, vol. 79, pp. 23–29, 1973.

3 J. Becker, “L ¨ownersche Differentialgleichung und Schlichtheitskriterien,”Mathematische Annalen, vol. 202, no. 4, pp. 321–335, 1973.

4 V. Pescar, “A new generalization of Ahlfors’s and Becker’s criterion of univalence,” Bulletin of the Malaysian Mathematical Society. Second Series, vol. 19, no. 2, pp. 53–54, 1996.

5 Z. Nehari,Conformal Mapping, Dover, New York, NY, USA, 1975.

6 N. Seenivasagan and D. Breaz, “Certain sufficient conditions for univalence,”General Mathematics, vol. 15, no. 4, pp. 7–15, 2007.

7 D. Breaz and N. Breaz, “The univalent conditions for an integral operator on the classesSpandT2,”