CJM Chinese Journal of Mathematics 2314-8071 Hindawi Publishing Corporation 923984 10.1155/2014/923984 923984 Research Article Univalency and Convexity Conditions for a General Integral Operator Bulut Serap 1 Breaz Daniel 2 Uhlig F. You H. Zhu C.-G. 1 Civil Aviation College Kocaeli University Arslanbey Campus İzmit, 41285 Kocaeli Turkey kocaeli.edu.tr 2 Department of Mathematics “1 Decembrie 1918” University of Alba Iulia Nicolae Iorga Street No. 11-13, 510000 Alba Iulia Romania uab.ro 2014 2022014 2014 24 10 2013 26 12 2013 20 2 2014 2014 Copyright © 2014 Serap Bulut and Daniel Breaz. 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.

For analytic functions f and g in the open unit disc 𝕌, a new general integral operator is introduced. The main objective of this paper is to obtain univalence condition and order of convexity for this general integral operator.

1. Introduction and Preliminaries

Let 𝒜 be the class of all functions of the form (1)f(z)=z+k=2akzk, which are analytic in the open unit disk (2)𝕌={z:|z|<1}. Also let 𝒮 denote the subclass of 𝒜 consisting of functions f which are univalent in 𝕌.

A function f𝒜 is said to be starlike of order α(0α<1) if it satisfies the inequality (3)Re{zf(z)f(z)}>α, for all z𝕌. We say that f is in the class 𝒮*(α) for such functions.

A function f𝒜 is said to be convex of order α(0α<1) if it satisfies the inequality (4)Re{1+zf′′(z)f(z)}>α, for all z𝕌. We say that f is in the class 𝒦(α) for such functions.

We note that f𝒦(α) if and only if zf𝒮*(α).

A function f𝒜 belongs to the class (α)(0α<1) if it satisfies the inequality (5)Re{f(z)}>α, for all z𝕌.

The family (μ,α)(μ0,0α<1) which contains the functions f that satisfy the condition (6)|f(z)(zf(z))μ-1|<1-α(z𝕌) was studied by Frasin and Jahangiri .

Remark 1.

This family is a comprehensive class of analytic functions that contains other new classes of analytic univalent functions as well as some very well-known ones. For example,

for μ=1, we have the class (7)(1,α)=𝒮*(α),

for μ=0, we have the class (8)(0,α)=(α),

for μ=2, the class (9)(α)={f𝒜:|z2f(z)f2(z)-1|<1-α;0α<1;z𝕌}

introduced by Frasin and Darus .

In this paper, we introduce a new general integral operator defined by (10)In,α(f,g)(z)=0zi=1n(fi(t)egi(t))αidt, where z𝕌, fi,gi𝒜, and αi for all i=1,,n.

Remark 2.

For n=1, f1=f, g1=g, and α1=α, we have the integral operator (11)I1(f,g)(z)=0z(f(t)eg(t))αdt introduced by Ularu and Breaz .

The following results will be required in our investigation.

General Schwarz Lemma (see ). Let the function f be regular in the disk 𝕌R={z:|z|<R}, with |f(z)|<M for fixed M. If f has one zero with multiplicity order bigger than m for z=0, then (12)|f(z)|MRm|z|m(z𝕌R).The equality can hold only if f(z)=eiθ(M/Rm)zm, where θ is constant.

Theorem A (see [<xref ref-type="bibr" rid="B1">5</xref>]).

If the function f is regular in the unit disk 𝕌, f(z)=z+a2z2+ and (13)(1-|z|2)|zf′′(z)f(z)|1, for all z𝕌, then the function f is univalent in 𝕌.

Theorem B (see [<xref ref-type="bibr" rid="B5">6</xref>]).

If f𝒜 satisfies the condition (14)|z2f(z)(f(z))2-1|1(z𝕌), then f is univalent in 𝕌.

2. Main Results Theorem 3.

Let fi,gi𝒜, where gi satisfies the condition (15)|z2gi(z)(gi(z))2-1|1,αi, and Mi>0 for all i=1,,n. If (16)|fi′′(z)fi(z)|1,(17)|gi(z)|<Mi,(18)|αi|92n3(1+2Mi2), for all i=1,,n, then the integral operator In,α(f,g) defined by (10) is in the univalent function class 𝒮.

Proof.

From (10), we obtain (19)In,α(f,g)(z)=(f1(z)eg1(z))α1(fn(z)egn(z))αn, for z𝕌. This equality implies that(20)lnIn,α(f,g)(z)=α1(lnf1(z)+g1(z))++αn(lnfn(z)+gn(z)). By differentiating the above equality, we get (21)In,α′′(f,g)(z)In,α(f,g)(z)=α1(f1′′(z)f1(z)+g1(z))++αn(fn′′(z)fn(z)+gn(z)), or equivalently (22)zIn,α′′(f,g)(z)In,α(f,g)(z)=α1(zf1′′(z)f1(z)+zg1(z))++αn(zfn′′(z)fn(z)+zgn(z)). So we find (23)(1-|z|2)|zIn,α′′(f,g)(z)In,α(f,g)(z)|=(1-|z|2)|i=1nαi(zfi′′(z)fi(z)+zgi(z))|(1-|z|2)i=1n|αi|(|z||fi′′(z)fi(z)|+|zgi(z)|)(1-|z|2)i=1n|αi|(|z|+|zgi(z)|)(1-|z|2)i=1n|αi|(|z|+|z2gi(z)gi2(z)||gi2(z)||z|). From the hypothesis, we have |gi(z)|<Mi(z𝕌;i{1,,n}); then by the general Schwarz lemma, we obtain that(24)|gi(z)|Mi|z|(z𝕌;i{1,,n}). Thus we have (25)(1-|z|2)|zIn,α′′(f,g)(z)In,α(f,g)(z)|(1-|z|2)i=1n|αi|(|z|+|z2gi(z)gi2(z)|Mi2|z|)(1-|z|2)i=1n|αi||z|(1+|z2gi(z)gi2(z)-1|Mi2+Mi2)|z|(1-|z|2)i=1n|αi|(1+2Mi2). Let us define the function F:[0,1], F(x)=x(1-x2), x=|z|. Then we obtain(26)F(x)239, for all x[0,1]. It follows from (26) that (27)|z|(1-|z|2)239.

Hence from (18), (25), and (27) we find (28)(1-|z|2)|zIn,α′′(f,g)(z)In,α(f,g)(z)|1. By applying Theorem A for the function In,α(f,g), we prove that In,α(f,g) is in the univalent function class 𝒮.

Setting n=1, f1=f, g1=g, and α1=α in Theorem 3, we obtain the following consequence of Theorem 3.

Corollary 4 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

Let f,g𝒜, where g satisfies the condition(29)|z2g(z)(g(z))2-1|1,α, and M>0. If (30)|f′′(z)f(z)|1,|g(z)|<M,|α|923(1+2M2), then the integral operator I1(f,g) defined by (11) is in the univalent function class 𝒮.

Theorem 5.

Let fi,gi𝒜, where gi(μi,βi)(μi0,0βi<1), αi, and Mi1 for all i=1,,n. If (31)|fi′′(z)fi(z)|1,|gi(z)|<Mi, for all i=1,,n, and if (32)0<i=1n|αi|(1+(2-βi)Miμi)1, then the integral operator In,α(f,g) defined by (10) is in the class 𝒦(δ), where (33)δ=1-i=1n|αi|(1+(2-βi)Miμi).

Proof.

From (22), we have (34)zIn,α′′(f,g)(z)In,α(f,g)(z)=α1(zf1′′(z)f1(z)+zg1(z))++αn(zfn′′(z)fn(z)+zgn(z)). It follows that (35)|zIn,α′′(f,g)(z)In,α(f,g)(z)|i=1n|αi||z|(|fi′′(z)fi(z)|+|gi(z)|)i=1n|αi||z|(1+|gi(z)(zgi(z))μi|gi(z)z|μi|). Since gi(μi,βi), |gi(z)|<Mi for all i=1,,n, applying general Schwarz lemma and using (35), we obtain (36)|zIn,α′′(f,g)(z)In,α(f,g)(z)|i=1n|αi||z|(1+|gi(z)(zgi(z))μiMiμi|)i=1n|αi||z|(1+|gi(z)(zgi(z))μi-1|Miμi+Miμi)<i=1n|αi|(1+(2-βi)Miμi)=1-δ. This implies that the integral operator In,α(f,g)𝒦(δ).

Setting μ1=μ2==μn=1 in Theorem 5, we have the following.

Corollary 6.

Let fi,gi𝒜, where gi𝒮*(βi)(0βi<1), αi, and Mi1 for all i=1,,n. If (37)|fi′′(z)fi(z)|1,|gi(z)|<Mi, for all i=1,,n, and if (38)0<i=1n|αi|(1+(2-βi)Mi)1, then the integral operator In,α(f,g) defined by (10) is in the class 𝒦(γ), where (39)γ=1-i=1n|αi|(1+(2-βi)Mi).

Setting μ1=μ2==μn=0 in Theorem 5, we have the following.

Corollary 7.

Let fi,gi𝒜, where gi(βi)(0βi<1), αi, and Mi1 for all i=1,,n. If (40)|fi′′(z)fi(z)|1,|gi(z)|<Mi, for all i=1,,n, and if (41)0<i=1n|αi|(3-βi)1, then the integral operator In,α(f,g) defined by (10) is in the class 𝒦(λ), where (42)λ=1-i=1n|αi|(3-βi).

Setting n=1, f1=f, g1=g, and α1=α in Theorem 5, we obtain the following consequence of Theorem 5.

Corollary 8 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

Let f,g𝒜, where g(μ,β)(μ0,0β<1), α, and M1. If (43)|f′′(z)f(z)|1,|g(z)|<M,0<|α|(1+(2-β)Mμ)1, then the integral operator I1(f,g) defined by (11) is in the class 𝒦(ρ), where (44)ρ=1-|α|(1+(2-β)Mμ).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Frasin B. A. Jahangiri J. M. A new and comprehensive class of analytic functions Analele Universităţii din Oradea 2008 15 59 62 Frasin B. A. Darus M. On certain analytic univalent functions International Journal of Mathematics and Mathematical Sciences 2001 25 5 305 310 10.1155/S0161171201004781 Ularu N. Breaz D. Univalence criterion and convexity for an integral operator Applied Mathematics Letters 2012 25 3 658 661 2-s2.0-80955151712 10.1016/j.aml.2011.10.011 Nehari Z. Conformal Mapping 1975 New York, NY, USA Dover Becker J. Löwnersche differentialgleichung und quasikonform fortsetzbare schlichte funktionen Journal für die Reine und Angewandte Mathematik 1972 255 23 43 10.1515/crll.1972.255.23 Ozaki S. Nunokawa M. The Schwarzian derivative and univalent functions Proceedings of the American Mathematical Society 1972 33 392 394 10.1090/S0002-9939-1972-0299773-3