# Univalency and Convexity Conditions for a General Integral Operator

Authors: Serap Bulut; Daniel Breaz
Journal: Chinese Journal of Mathematics
 (2014)

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## Abstract

For analytic functionsf 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.


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## Body

## 1. Introduction and Preliminaries

Let𝒜 be the class of all functions of the form
(1)f(z)=z+∑k=2∞‍akzk,
which are analytic in the open unit disk
(2)𝕌={z∈ℂ:|z|<1}.
Also let 𝒮 denote the subclass of 𝒜 consisting of functions f which are univalent in 𝕌.A functionf∈𝒜 is said to be starlike of order α(0≤α<1) if it satisfies the inequality
(3)Re{zf′(z)f(z)}>α,
for all z∈𝕌. We say that f is in the class 𝒮*(α) for such functions.A functionf∈𝒜 is said to be convex of order α(0≤α<1) if it satisfies the inequality
(4)Re{1+zf′′(z)f′(z)}>α,
for all z∈𝕌. We say that f is in the class 𝒦(α) for such functions.We note thatf∈𝒦(α) if and only if zf′∈𝒮*(α).A functionf∈𝒜 belongs to the class ℛ(α)(0≤α<1) if it satisfies the inequality
(5)Re{f′(z)}>α,
for all z∈𝕌.The familyℬ(μ,α)(μ≥0,0≤α<1) which contains the functions f that satisfy the condition
(6)|f′(z)(zf(z))μ-1|<1-α(z∈𝕌)
was studied by Frasin and Jahangiri [1].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,(i)
forμ=1, we have the class
(7)ℬ(1,α)=𝒮*(α),(ii)
forμ=0, we have the class
(8)ℬ(0,α)=ℛ(α),(iii)
forμ=2, the class
(9)ℬ(α)={f∈𝒜:|z2f′(z)f2(z)-1|<1-α;0≤α<1;z∈𝕌}introduced by Frasin and Darus [2].In this paper, we introduce a new general integral operator defined by(10)In,α(f,g)(z)=∫0z‍∏i=1n(fi′(t)egi(t))αidt,
where z∈𝕌, fi,gi∈𝒜, and αi∈ℂ for all i=1,…,n.Remark 2.
Forn=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 [3].The following results will be required in our investigation.General Schwarz Lemma (see [4]). Let the functionfbe regular in the disk𝕌R={z∈ℂ:|z|<R}, with|f(z)|<Mfor fixedM. Iffhas one zero with multiplicity order bigger thanmforz=0, then(12)|f(z)|≤MRm|z|m(z∈𝕌R).The equality can hold only iff(z)=eiθ(M/Rm)zm, whereθis constant.Theorem A (see [5]).
If the functionf is regular in the unit disk 𝕌, f(z)=z+a2z2+⋯ and
(13)(1-|z|2)|zf′′(z)f′(z)|≤1,
for all z∈𝕌, then the function f is univalent in 𝕌.Theorem B (see [6]).
Iff∈𝒜 satisfies the condition
(14)|z2f′(z)(f(z))2-1|≤1(z∈𝕌),
then f is univalent in 𝕌.


## 2. Main Results

Theorem 3.
Letfi,gi∈𝒜, where gi satisfies the condition
(15)|z2gi′(z)(gi(z))2-1|≤1,αi∈ℂ, 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 z∈𝕌. This equality implies that(20)ln⁡In,α′(f,g)(z)=α1(ln⁡f1′(z)+g1(z))+⋯+αn(ln⁡fn′(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(z∈𝕌;i∈{1,…,n}); then by the general Schwarz lemma, we obtain that(24)|gi(z)|≤Mi|z|(z∈𝕌;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]→ℝ, 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.Settingn=1, f1=f, g1=g, and α1=α in Theorem 3, we obtain the following consequence of Theorem 3.Corollary 4 (see [3]).
Letf,g∈𝒜, where g satisfies the condition(29)|z2g′(z)(g(z))2-1|≤1,α∈ℂ, 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.
Letfi,gi∈𝒜, where gi∈ℬ(μi,βi)(μi≥0,0≤βi<1), αi∈ℂ, and Mi≥1 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∈ℬ(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).Settingμ1=μ2=⋯=μn=1 in Theorem 5, we have the following.Corollary 6.
Letfi,gi∈𝒜, where gi∈𝒮*(βi)(0≤βi<1), αi∈ℂ, and Mi≥1 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).Settingμ1=μ2=⋯=μn=0 in Theorem 5, we have the following.Corollary 7.
Letfi,gi∈𝒜, where gi∈ℛ(βi)(0≤βi<1), αi∈ℂ, and Mi≥1 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).Settingn=1, f1=f, g1=g, and α1=α in Theorem 5, we obtain the following consequence of Theorem 5.Corollary 8 (see [3]).
Letf,g∈𝒜, where g∈ℬ(μ,β)(μ≥0,0≤β<1), α∈ℂ, and M≥1. 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).



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