# Univalence Criteria for Two Integral Operators

Authors: Laura Filofteia Stanciu; Daniel Breaz
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the univalence conditions for two integral operators to be univalent in the open unit disk. Many known univalence conditions are written to prove our main results.


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## Body

## 1. Introduction and Preliminaries

Let𝒜 denote the class of functions of the form:(1.1)f(z)=z+∑n=2∞anzn,
which are analytic in the open unit disk: (1.2)U={z∈C:|z|<1}, and satisfy the following usual normalization condition:(1.3)f(0)=f′(0)-1=0.
We denote by 𝒮 the subclass of 𝒜 consisting of functions f(z) which are univalent in 𝕌.In [1], for 0<b≤1, Silverman considered the class:(1.4)Gb={f∈A:|1+zf′′(z)f′(z)-zf′(z)f(z)|<b|zf′(z)f(z)|,z∈U}.Here, in our present investigation, we consider two general families of integral operators:(1.5)I(f1,…,fn;g1,…,gn)(z)=(β∫0ztβ-1∏i=1n(fi(t)t)(αi-1)/Mi(gi′(t))γidt)1/β,αi,γi∈ℂ; β∈ℂ∖{0}; fi,gi∈𝒜, Mi≥1 for all i∈{1,2,…,n};(1.6)J(f1,…,fn;g1,…,gn)(z)=((1+∑i=1nαi)∫0z∏i=1n(fi(t))αi(gi′(t))γidt)1/(1+∑i=1nαi),αi,γi∈ℂ; fi,gi∈𝒜 for all i∈{1,2,…,n}.Many authors have studied the problem of integral operators which preserve the class𝒮 (see, e.g., [2–5]).In the present paper, we study the univalence conditions involving the general families of integral operators defined by (1.5) and (1.6).In the proof of our main results (Theorem2.1 and Theorem 3.1), we need the following univalence criterion. The univalence criterion, asserted by Theorem 1.1, is a generalization of Ahlfor’s and Becker’s univalence criterion; it was proven by Pescar [6].Theorem 1.1 (see Pescar [6]).
Letβ∈ℂ with Reβ>0, c∈ℂ with |c|≤1, c≠-1. If f∈𝒜 satisfies
(1.7)|c|z|2β+(1-|z|2β)zf′′(z)βf′(z)|≤1,
for all z∈𝕌, then the integral operator,
(1.8)Fβ(z)=(β∫0ztβ-1f′(t)dt)1/β,
is in the class 𝒮.Finally, in the present investigation, one also needs the familiar Schwarz Lemma (see, for details, [7]).Lemma 1.2 ((General Schwarz Lemma) (see [7])).
Let the functionf be regular in the disk 𝕌R={z∈ℂ:|z|<R}, with |f(z)|<M for fixed M. If f has one zero with multiplicity order bigger than m for z=0, then
(1.9)|f(z)|≤MRm|z|m(z∈UR).
The equality can hold only if
(1.10)f(z)=eiθMRmzm,
where θ is constant.


## 2. Univalence Conditions forI(f1,…,fn;g1,…,gn)(z)

Theorem 2.1.
LetMi≥1 (i∈{1,2,…,n}) and β, αi, γi be complex numbers with Reβ≥0 and
(2.1)Reβ≥∑i=1n[3|αi-1|+|γi|(2bi+1)],
and let c∈ℂ be such that
(2.2)|c|≤1-1Reβ∑i=1n[3|αi-1|+|γi|(2bi+1)].
If for all i∈{1,2,…,n},fi∈𝒜 satisfy the conditions:
(2.3)|fi(z)|≤Mi(z∈U),|z2fi′(z)fi2(z)-1|≤2Mi-1Mi(z∈U),
and gi∈Gbi,   0<bi≤1 with
(2.4)|zgi′(z)gi(z)-1|<1(z∈U),
then the integral operator I(f1,…,fn;g1,…,gn)(z) defined by (1.5) is in the class 𝒮.Proof.
We begin by setting(2.5)h(z)=∫0z∏i=1n(fi(t)t)(αi-1)/Mi(gi′(t))γidt,
and then we calculate for h(z) the derivates of the first and second orders.
From (2.5), we obtain
(2.6)h′(z)=∏i=1n(fi(z)z)(αi-1)/Mi(gi′(z))γi,zh′′(z)h′(z)=∑i=1n[αi-1Mi(zfi′(z)fi(z)-1)+γizgi′′(z)gi′(z)]=∑i=1n[αi-1Mi(zfi′(z)fi(z)-1)+γi(zgi′′(z)gi′(z)-zgi′(z)gi(z)+1)+γi(zgi′(z)gi(z)-1)].
Thus, we have
(2.7)|zh′′(z)h′(z)|≤∑i=1n[|αi-1|Mi(|zfi′(z)fi(z)|+1)+|γi||zgi′′(z)gi′(z)-zgi′(z)gi(z)+1|+|γi||zg′i(z)gi(z)-1|]≤∑i=1n[|αi-1|Mi((|z2fi′(z)fi2(z)-1|+1)|fi(z)z|+1)+|γi||zgi′′(z)gi′(z)-zgi′(z)gi(z)+1|+|γi||zgi′(z)gi(z)-1|].
From the hypothesis (2.3) of Theorem 2.1, we have
(2.8)|fi(z)|≤Mi(z∈U;Mi≥1),|z2fi′(z)fi2(z)-1|≤2Mi-1Mi(z∈U;Mi≥1)
for all i∈{1,2,…,n}.
By applying the General Schwarz Lemma, we thus obtain(2.9)|fi(z)|≤Mi|z|(z∈U;i∈{1,2,…,n}).
Since gi∈Gbi, 0<bi≤1 for all i∈{1,2,…,n}, from (1.4), (2.4), we obtain
(2.10)|zh′′(z)h′(z)|≤∑i=1n[3|αi-1|+|γi|bi|zgi′(z)gi(z)|+|γi||zgi′(z)gi(z)-1|]≤∑i=1n[3|αi-1|+|γi|bi(|zgi′(z)gi(z)-1|+1)+|γi||zgi′(z)gi(z)-1|]≤∑i=1n[3|αi-1|+|γi|bi|zgi′(z)gi(z)-1|+|γi|bi+|γi||zgi′(z)gi(z)-1|]≤∑i=1n[3|αi-1|+(|γi|bi+|γi|)|zgi′(z)gi(z)-1|+|γi|bi]≤∑i=1n[3|αi-1|+|γi|(2bi+1)],
which readily shows that
(2.11)|c|z|2β+(1-|z|2β)zh′′(z)βh′(z)|≤|c|+1|β|∑i=1n[3|αi-1|+|γi|(2bi+1)]≤|c|+1Reβ∑i=1n[3|αi-1|+|γi|(2bi+1)]≤1,
where we have also used the hypothesis (2.2) of Theorem 2.1.
Finally, by applying Theorem1.1, we conclude that the integral operator I(f1,…,fn;g1,…,gn)(z) defined by (1.5) is in the class 𝒮. This evidently completes the proof of Theorem 2.1.Settingα1=α2=⋯=αn=1 in Theorem 2.1, we have the following result.Corollary 2.2.
LetMi≥1 (i∈{1,2,…,n}) and β, γi be complex numbers with Reβ≥0 and
(2.12)Reβ≥∑i=1n[|γi|(2bi+1)]
and let c∈ℂ be such that
(2.13)|c|≤1-1Reβ∑i=1n[|γi|(2bi+1)].
If for all i∈{1,2,…,n},  gi∈Gbi, 0<bi≤1 with
(2.14)|zgi′(z)gi(z)-1|<1(z∈U),
then the integral operator,
(2.15)I(f1,…,fn;g1,…,gn)(z)=(β∫0ztβ-1∏i=1n(gi′(t))γidt)1/β,
is in the class 𝒮.Settingγi=1 for all i∈{1,2,…,n} in Theorem 2.1, we have the following result.Corollary 2.3.
LetMi≥1 (i∈{1,2,…,n}) and β, αi be complex numbers with Reβ≥0 and
(2.16)Reβ≥∑i=1n[3|αi-1|+(2bi+1)],
and let c∈ℂ be such that
(2.17)|c|≤1-1Reβ∑i=1n[3|αi-1|+(2bi+1)].
If for all i∈{1,2,…,n},fi∈𝒜 satisfy the conditions:
(2.18)|fi(z)|≤Mi(z∈U),|z2fi′(z)fi2(z)-1|≤2Mi-1Mi(z∈U)
and gi∈Gbi,0<bi≤1 with
(2.19)|zgi′(z)gi(z)-1|<1(z∈U),
then the integral operator
(2.20)I(f1,…,fn;g1,…,gn)(z)=(β∫0ztβ-1∏i=1n(fi(t)t)(αi-1)/Mi(gi′(t))dt)1/β
is in the class 𝒮.Settingn=1 in Theorem 2.1, we have the following result.Corollary 2.4.
LetM≥1 and β, α, γ be complex numbers with Reβ≥0 and
(2.21)Reβ≥[3|α-1|+|γ|(2b+1)],
and let c∈ℂ be such that
(2.22)|c|≤1-1Reβ[3|α-1|+|γ|(2b+1)].
If the function f∈𝒜 satisfies the conditions:
(2.23)|f(z)|≤M(z∈U),|z2f′(z)f2(z)-1|≤2M-1M(z∈U),
and g∈Gb, 0<b≤1 with
(2.24)|zg′(z)g(z)-1|<1(z∈U),
then the integral operator,
(2.25)I(f;g)(z)=(β∫0ztβ-1(f(t)t)(α-1)/M(g′(t))γdt)1/β,
is in the class 𝒮.


## 3. Univalence Conditions forJ(f1,…,fn;g1,…,gn)(z)

Theorem 3.1.
LetMi≥1 (i∈{1,2,…,n}) and β, αi, γi be complex numbers, β=(1+∑i=1nαi),  Reβ≥0 and
(3.1)Reβ≥∑i=1n[|αi|+|γi|(bi+1)(2Mi+1)+|γi|bi],
and let c∈ℂ be such that
(3.2)|c|≤1-1Reβ∑i=1n[|αi|+|γi|(bi+1)(2Mi+1)+|γi|bi].
If for all i∈{1,2,…,n},fi∈𝒜 satisfy the condition:
(3.3)|zfi′(z)fi(z)-1|<1(z∈U),
and gi∈Gbi,0<bi≤1 with
(3.4)|z2gi′(z)gi2(z)-1|<1(z∈U),(3.5)|gi(z)|≤Mi(z∈U;i∈{1,2,…,n}),
then the integral operator J(f1,…,fn;g1,…,gn)(z) defined by (1.6) is in the class 𝒮.Proof.
We begin by observing that the integral operatorJ(f1,…,fn;g1,…,gn)(z) defined by (1.6) can be rewritten as follows:
(3.6)J(f1,…,fn;g1,…,gn)(z)=((1+∑i=1nαi)∫0zt∑i=1nαi∏i=1n(fi(t)t)αi(gi′(t))γidt)1/(1+∑i=1nαi),
where fi∈𝒜 for all i∈{1,2,…,n}.
Defining the functionh(z) by
(3.7)h(z)=∫0z∏i=1n(fi(t)t)αi(gi′(t))γidt,
we take the same steps as in the proof of Theorem 2.1, and we have
(3.8)|zh′′(z)h′(z)|≤∑i=1n[|αi||zfi′(z)fi(z)-1|+(|γi|bi+|γi|)|zgi′(z)gi(z)-1|+|γi|bi].
Thus, we have
(3.9)|c|z|2β+(1-|z|2β)zh′′(z)βh′(z)|≤|c|+1|β|∑i=1n[|αi|+(|γi|bi+|γi|)(|z2gi′(z)gi2(z)||gi(z)z|+1)+|γi|bi].
Furthermore, from the hypothesis (3.4) of Theorem 3.1, we have
(3.10)|z2gi′(z)gi2(z)-1|<1(z∈U),|gi(z)|≤Mi(z∈U;i∈{1,2,…,n}).
By applying the General Schwarz Lemma, we obtain
(3.11)|gi(z)|≤Mi|z|(z∈U;i∈{1,2,…,n}).
So, we obtain
(3.12)|c|z|2β+(1-|z|2β)zh′′(z)βh′(z)|≤|c|+1|β|∑i=1n[|αi|+(|γi|bi+|γi|)((|z2gi′(z)gi2(z)-1|+1)Mi+1)+|γi|bi]≤|c|+1|β|∑i=1n[|αi|+(|γi|bi+|γi|)(2Mi+1)+|γi|bi]≤|c|+1Reβ∑i=1n[|αi|+(|γi|bi+|γi|)(2Mi+1)+|γi|bi]≤1.
Finally, by applying Theorem1.1, we conclude that the integral operator J(f1,…,fn;g1,…,gn)(z) defined by (1.6) is in the class 𝒮. This evidently completes the proof of Theorem 3.1.Settingα1=α2=⋯=αn=1 in Theorem 3.1, we haveCorollary 3.2.
LetMi≥1 (i∈{1,2,…,n}) and β, γi be complex numbers, β=(1+n), Reβ≥0 and
(3.13)Reβ≥∑i=1n[1+|γi|(bi+1)(2Mi+1)+|γi|bi],
and let c∈ℂ be such that
(3.14)|c|≤1-1Reβ∑i=1n[1+|γi|(bi+1)(2Mi+1)+|γi|bi].
If for all i∈{1,2,…,n},fi∈𝒜 satisfy the condition:
(3.15)|zfi′(z)fi(z)-1|<1(z∈U),
and gi∈Gbi, 0<bi≤1 with
(3.16)|z2gi′(z)gi2(z)-1|<1(z∈U),|gi(z)|≤Mi(z∈U),
then the integral operator,
(3.17)J(f1,…,fn;g1,…,gn)(z)=((1+n)∫0z∏i=1n(fi(t))(gi′(t))γidt)1/(1+n),
is in the class 𝒮.Settingγi=1 for all i∈{1,2,…,n} in Theorem 3.1, we have the following result.Corollary 3.3.
LetMi≥1 (i∈{1,2,…,n}) and β, αi be complex numbers, β=(1+∑i=1nαi), Reβ≥0 and
(3.18)Reβ≥∑i=1n[|αi|+(bi+1)(2Mi+1)+bi],
and let c∈ℂ be such that
(3.19)|c|≤1-1Reβ∑i=1n[|αi|+(bi+1)(2Mi+1)+bi].
If for all i∈{1,2,…,n},fi∈𝒜 satisfy the condition:
(3.20)|zfi′(z)fi(z)-1|<1(z∈U),
and gi∈Gbi,0<bi≤1 with
(3.21)|z2gi′(z)gi2(z)-1|<1(z∈U),|gi(z)|≤Mi(z∈U),
then the integral operator,
(3.22)J(f1,…,fn;g1,…,gn)(z)=((1+∑i=1nαi)∫0z∏i=1n(fi(t))αi(gi′(t))dt)1/(1+∑i=1nαi),
is in the class 𝒮.Settingn=1 in Theorem 3.1, we have the following result.Corollary 3.4.
LetM≥1 and β, α, γ be complex numbers, β=(1+α), Reβ≧0 and
(3.23)Reβ≥[|α|+|γ|(b+1)(2M+1)+|γ|b],
and let c∈ℂ be such that
(3.24)|c|≤1-1Reβ[|α|+|γ|(b+1)(2M+1)+|γ|b].
If one has that the function f∈𝒜 satisfies the condition:
(3.25)|zf′(z)f(z)-1|≤1(z∈U),
and g∈Gb,0<b≤1 with
(3.26)|z2g′(z)g2(z)-1|<1(z∈U),|g(z)|≤M(z∈U),
then the integral operator,
(3.27)J(f;g)(z)=((1+α)∫0z(f(t))α(g′(t))γdt)1/(1+α),
is in the class 𝒮.



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