Univalence Criteria for Two Integral Operators

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.

1. Introduction and Preliminaries

Let 𝒜 denote the class of functions of the form:𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1.1) which are analytic in the open unit disk: 𝕌={𝑧∈ℂ∶|𝑧|<1}, (1.2) and satisfy the following usual normalization condition:𝑓(0)=𝑓′(0)−1=0. (1.3) 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:𝐺_𝑏=𝑓∈𝒜∶1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝑧𝑓^′(𝑧)𝑓(𝑧)<𝑏𝑧𝑓^′(𝑧)𝑓(𝑧),𝑧∈𝕌. (1.4)

Here, in our present investigation, we consider two general families of integral operators:𝐼𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛(𝑧)=𝛽^𝑧₀𝑡^𝑛𝛽−1_𝑖=1𝑓_𝑖(𝑡)𝑡^{(𝛼𝑖−1)/𝑀𝑖}𝑔^′_𝑖(𝑡)^𝛾𝑖𝑑𝑡^1/𝛽, (1.5)𝛼_𝑖,𝛾_𝑖∈ℂ; 𝛽∈ℂ⧵{0}; 𝑓_𝑖,𝑔_𝑖∈𝒜, 𝑀_𝑖≥1 for all 𝑖∈{1,2,…,𝑛};𝐽𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛(𝑧)=1+^𝑛_𝑖=1𝛼_𝑖^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)^𝛼𝑖𝑔^′_𝑖(𝑡)^𝛾𝑖𝑑𝑡^{1/(1+∑𝑛𝑖=1𝛼𝑖)}, (1.6)𝛼_𝑖,𝛾_𝑖∈ℂ; 𝑓_𝑖,𝑔_𝑖∈𝒜 for all 𝑖∈{1,2,…,𝑛}.

Many authors have studied the problem of integral operators which preserve the class 𝒮 (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 (Theorem 2.1 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]). Let 𝛽∈ℂ with Re𝛽>0, 𝑐∈ℂ with |𝑐|≤1, 𝑐≠−1. If 𝑓∈𝒜 satisfies 𝑐|𝑧|^2𝛽+1−|𝑧|^2𝛽𝑧𝑓^′′(𝑧)𝛽𝑓^′(𝑧)≤1, (1.7) for all 𝑧∈𝕌, then the integral operator, 𝐹_𝛽(𝑧)=𝛽^𝑧₀𝑡^𝛽−1𝑓^′(𝑡)𝑑𝑡^1/𝛽, (1.8) 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 function 𝑓 be regular in the disk 𝕌_𝑅={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 𝑓(𝑧≤𝑀)𝑅^𝑚|𝑧|^𝑚𝑧∈𝕌_𝑅. (1.9) The equality can hold only if 𝑓(𝑧)=𝑒^𝑖𝜃𝑀𝑅^𝑚𝑧^𝑚, (1.10) where 𝜃 is constant.

2. Univalence Conditions for 𝐼(𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛)(𝑧)

Theorem 2.1. Let 𝑀_𝑖≥1 (𝑖∈{1,2,…,𝑛}) and 𝛽, 𝛼_𝑖, 𝛾_𝑖 be complex numbers with Re𝛽≥0 and Re𝛽≥^𝑛_𝑖=13𝛼_𝑖−1+𝛾_𝑖2𝑏_𝑖+1, (2.1) and let 𝑐∈ℂ be such that |𝑐|1≤1−Re𝛽^𝑛_𝑖=13𝛼_𝑖−1+𝛾_𝑖2𝑏_𝑖+1. (2.2) If for all 𝑖∈{1,2,…,𝑛},𝑓_𝑖∈𝒜 satisfy the conditions: 𝑓_𝑖(𝑧)≤𝑀_𝑖(),𝑧𝑧∈𝕌²𝑓^′_𝑖(𝑧)𝑓²_𝑖(𝑧)−1≤2𝑀_𝑖−1𝑀_𝑖(𝑧∈𝕌), (2.3) and 𝑔_𝑖∈𝐺_𝑏𝑖, 0<𝑏_𝑖≤1 with 𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1<1(𝑧∈𝕌), (2.4) then the integral operator 𝐼(𝑓₁,…,𝑓_𝑛;g₁,…,𝑔_𝑛)(𝑧) defined by (1.5) is in the class 𝒮.

Proof. We begin by setting ℎ(𝑧)=^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)𝑡^{(𝛼𝑖−1)/𝑀𝑖}𝑔^′_𝑖(𝑡)^𝛾𝑖𝑑𝑡, (2.5) and then we calculate for ℎ(𝑧) the derivates of the first and second orders.
From (2.5), we obtain ℎ^′(𝑧)=^𝑛_𝑖=1𝑓_𝑖(𝑧)𝑧^{(𝛼𝑖−1)/𝑀𝑖}𝑔^′_𝑖(𝑧)^𝛾𝑖,𝑧ℎ^′′(𝑧)ℎ^′(𝑧)=^𝑛_𝑖=1𝛼_𝑖−1𝑀_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)−1+𝛾_𝑖𝑧𝑔_𝑖^′′(𝑧)𝑔^′_𝑖(𝑧)=^𝑛_𝑖=1𝛼_𝑖−1𝑀_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)−1+𝛾_𝑖𝑧𝑔_𝑖^′′(𝑧)𝑔^′_𝑖(𝑧)−𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)+1+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1. (2.6) Thus, we have 𝑧ℎ^′′(𝑧)ℎ^′(𝑧)≤^𝑛_𝑖=1𝛼_𝑖−1𝑀_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)+𝛾+1_𝑖𝑧𝑔_𝑖^′′(𝑧)𝑔^′_𝑖(𝑧)−𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)+1+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)≤−1_𝑛𝑖=1𝛼_𝑖−1𝑀_𝑖𝑧²𝑓^′_𝑖(𝑧)𝑓²_𝑖(𝑧)𝑓−1+1^𝑖(𝑧)𝑧+𝛾+1_𝑖𝑧𝑔_𝑖^′′(𝑧)𝑔^′_𝑖(𝑧)−𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)+1+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1. (2.7) From the hypothesis (2.3) of Theorem 2.1, we have 𝑓_𝑖(𝑧)≤𝑀_𝑖𝑧∈𝕌;𝑀_𝑖,𝑧≥1₂𝑓^′_𝑖(𝑧)𝑓²_𝑖(𝑧)−1≤2𝑀_𝑖−1𝑀_𝑖𝑧∈𝕌;𝑀_𝑖≥1 (2.8) for all 𝑖∈{1,2,…,𝑛}.
By applying the General Schwarz Lemma, we thus obtain 𝑓_𝑖(𝑧)≤𝑀_𝑖|𝑧|(𝑧∈𝕌;𝑖∈{1,2,…,𝑛}). (2.9) Since 𝑔_𝑖∈𝐺_𝑏𝑖, 0<𝑏_𝑖≤1 for all 𝑖∈{1,2,…,𝑛}, from (1.4), (2.4), we obtain 𝑧ℎ^′′(𝑧)ℎ^′(𝑧)≤^𝑛_𝑖=13𝛼_𝑖−1+𝛾_𝑖𝑏_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)≤−1_𝑛𝑖=13𝛼_𝑖−1+𝛾_𝑖𝑏_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1+1+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)≤−1_𝑛𝑖=13𝛼_𝑖−1+𝛾_𝑖𝑏_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1+𝛾_𝑖𝑏_𝑖+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)≤−1_𝑛𝑖=13𝛼_𝑖−1+𝛾_𝑖𝑏_𝑖+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1+𝛾_𝑖𝑏_𝑖≤∑^𝑛_𝑖=13𝛼_𝑖−1+𝛾_𝑖2𝑏_𝑖+1, (2.10) which readily shows that 𝑐|𝑧|^2𝛽+1−|𝑧|^2𝛽𝑧ℎ^′′(𝑧)𝛽ℎ^′(𝑧)≤|𝑐|+1𝛽^𝑛_𝑖=13𝛼_𝑖−1+𝛾_𝑖2𝑏_𝑖≤|𝑐|+1+1Re𝛽^𝑛_𝑖=13𝛼_𝑖−1+𝛾_𝑖2𝑏_𝑖+1≤1, (2.11) where we have also used the hypothesis (2.2) of Theorem 2.1.
Finally, by applying Theorem 1.1, we conclude that the integral operator 𝐼(𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛)(𝑧) defined by (1.5) is in the class 𝒮. This evidently completes the proof of Theorem 2.1.

Setting 𝛼₁=𝛼₂=⋯=𝛼_𝑛=1 in Theorem 2.1, we have the following result.

Corollary 2.2. Let 𝑀_𝑖≥1 (𝑖∈{1,2,…,𝑛}) and 𝛽, 𝛾_𝑖 be complex numbers with Re𝛽≥0 and Re𝛽≥^𝑛_𝑖=1𝛾_𝑖2𝑏_𝑖+1 (2.12) and let 𝑐∈ℂ be such that |𝑐|1≤1−Re𝛽^𝑛_𝑖=1𝛾_𝑖2𝑏_𝑖+1. (2.13) If for all 𝑖∈{1,2,…,𝑛},  𝑔_𝑖∈𝐺_𝑏𝑖, 0<𝑏_𝑖≤1 with 𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1<1(𝑧∈𝕌), (2.14) then the integral operator, 𝐼𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛(𝑧)=𝛽^𝑧₀𝑡^𝑛𝛽−1_𝑖=1𝑔^′_𝑖(𝑡)^𝛾𝑖𝑑𝑡^1/𝛽, (2.15) is in the class 𝒮.

Setting 𝛾_𝑖=1 for all 𝑖∈{1,2,…,𝑛} in Theorem 2.1, we have the following result.

Corollary 2.3. Let 𝑀_𝑖≥1 (𝑖∈{1,2,…,𝑛}) and 𝛽, 𝛼_𝑖 be complex numbers with Re𝛽≥0 and Re𝛽≥^𝑛_𝑖=13𝛼_𝑖−1+2𝑏_𝑖+1, (2.16) and let 𝑐∈ℂ be such that |𝑐|1≤1−Re𝛽^𝑛_𝑖=13𝛼_𝑖−1+2𝑏_𝑖+1. (2.17) If for all 𝑖∈{1,2,…,𝑛},𝑓_𝑖∈𝒜 satisfy the conditions: 𝑓_𝑖(𝑧)≤𝑀_𝑖(),𝑧𝑧∈𝕌²𝑓^′_𝑖(𝑧)𝑓²_𝑖(𝑧)−1≤2𝑀_𝑖−1𝑀_𝑖(𝑧∈𝕌) (2.18) and 𝑔_𝑖∈𝐺_𝑏𝑖,0<𝑏_𝑖≤1 with 𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1<1(𝑧∈𝕌), (2.19) then the integral operator 𝐼𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛(𝑧)=𝛽^𝑧₀𝑡^𝑛𝛽−1_𝑖=1𝑓_𝑖(𝑡)𝑡^{(𝛼𝑖−1)/𝑀𝑖}𝑔^′_𝑖(𝑡)𝑑𝑡^1/𝛽 (2.20) is in the class 𝒮.

Setting 𝑛=1 in Theorem 2.1, we have the following result.

Corollary 2.4. Let 𝑀≥1 and 𝛽, 𝛼, 𝛾 be complex numbers with Re𝛽≥0 and Re𝛽≥3|𝛼−1|+𝛾(2𝑏+1), (2.21) and let 𝑐∈ℂ be such that |𝑐|1≤1−3||+𝛾.Re𝛽𝛼−1(2𝑏+1) (2.22) If the function 𝑓∈𝒜 satisfies the conditions: 𝑓(𝑧(),𝑧)≤𝑀𝑧∈𝕌²𝑓^′(𝑧)𝑓²(𝑧)−1≤2𝑀−1𝑀(𝑧∈𝕌), (2.23) and 𝑔∈𝐺_𝑏, 0<𝑏≤1 with 𝑧𝑔^′(𝑧)𝑔(𝑧)−1<1(𝑧∈𝕌), (2.24) then the integral operator, 𝐼𝑓;𝑔(𝑧)=𝛽^𝑧₀𝑡^𝛽−1𝑓(𝑡)𝑡^{(𝛼−1)/𝑀}𝑔^′(𝑡)^𝛾𝑑𝑡^1/𝛽, (2.25) is in the class 𝒮.

3. Univalence Conditions for 𝐽(𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛)(𝑧)

Theorem 3.1. Let 𝑀_𝑖≥1 (𝑖∈{1,2,…,𝑛}) and 𝛽, 𝛼_𝑖, 𝛾_𝑖 be complex numbers, 𝛽=(1+∑^𝑛_𝑖=1𝛼_𝑖), Re𝛽≥0 and Re𝛽≥^𝑛_𝑖=1𝛼_𝑖+𝛾_𝑖𝑏_𝑖+12𝑀_𝑖+1+𝛾_𝑖𝑏_𝑖, (3.1) and let 𝑐∈ℂ be such that |𝑐|1≤1−Re𝛽^𝑛_𝑖=1𝛼_𝑖+𝛾_𝑖𝑏_𝑖+12𝑀_𝑖+1+𝛾_𝑖𝑏_𝑖. (3.2) If for all 𝑖∈{1,2,…,𝑛},𝑓_𝑖∈𝒜 satisfy the condition: 𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)−1<1(𝑧∈𝕌), (3.3) and 𝑔_𝑖∈𝐺_𝑏𝑖,0<𝑏_𝑖≤1 with 𝑧²𝑔^′_𝑖(𝑧)𝑔²_𝑖(𝑧)(),𝑔−1<1𝑧∈𝕌_𝑖(𝑧)≤𝑀_𝑖(𝑧∈𝕌;𝑖∈{1,2,…,𝑛}), (3.4)(3.5) then the integral operator 𝐽(𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛)(𝑧) defined by (1.6) is in the class 𝒮.

Proof. We begin by observing that the integral operator 𝐽(𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛)(𝑧) defined by (1.6) can be rewritten as follows: 𝐽𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛(𝑧)=1+^𝑛_𝑖=1𝛼_𝑖^𝑧₀𝑡^{∑𝑛𝑖=1𝛼𝑖𝑛}_𝑖=1𝑓_𝑖(𝑡)𝑡^𝛼𝑖𝑔^′_𝑖(𝑡)^𝛾𝑖𝑑𝑡^{1/(1+∑𝑛𝑖=1𝛼𝑖)}, (3.6) where 𝑓_𝑖∈𝒜 for all 𝑖∈{1,2,…,𝑛}.
Defining the function ℎ(𝑧) by ℎ(𝑧)=^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)𝑡^𝛼𝑖𝑔^′_𝑖(𝑡)^𝛾𝑖𝑑𝑡, (3.7) we take the same steps as in the proof of Theorem 2.1, and we have 𝑧ℎ^′′(𝑧)ℎ^′(𝑧)≤^𝑛_𝑖=1𝛼_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)−1+𝛾_𝑖𝑏_𝑖+𝛾_𝑖𝑧𝑔^′_𝑖(𝑧)𝑔_𝑖(𝑧)−1+𝛾_𝑖𝑏_𝑖. (3.8) Thus, we have 𝑐|𝑧|^2𝛽+1−|𝑧|^2𝛽𝑧ℎ^′′(𝑧)𝛽ℎ^′(𝑧)≤|𝑐|+1𝛽^𝑛_𝑖=1𝛼_𝑖+𝛾_𝑖𝑏_𝑖+𝛾_𝑖𝑧²𝑔^′_𝑖(𝑧)𝑔²_𝑖(𝑧)𝑔_𝑖(𝑧)𝑧+1+𝛾_𝑖𝑏_𝑖. (3.9) Furthermore, from the hypothesis (3.4) of Theorem 3.1, we have 𝑧²𝑔^′_𝑖(𝑧)𝑔²_𝑖(𝑧)(),𝑔−1<1𝑧∈𝕌_𝑖(𝑧)≤𝑀_𝑖(𝑧∈𝕌;𝑖∈{1,2,…,𝑛}). (3.10) By applying the General Schwarz Lemma, we obtain 𝑔_𝑖(𝑧)≤𝑀_𝑖|𝑧|(𝑧∈𝕌;𝑖∈{1,2,…,𝑛}). (3.11) So, we obtain 𝑐|𝑧|^2𝛽+1−|𝑧|^2𝛽𝑧ℎ^′′(𝑧)𝛽ℎ^′(𝑧)≤|𝑐|+1𝛽^𝑛_𝑖=1𝛼_𝑖+𝛾_𝑖𝑏_𝑖+𝛾_𝑖𝑧²𝑔^′_𝑖(𝑧)𝑔²_𝑖(𝑧)−1+1𝑀_𝑖+1+𝛾_𝑖𝑏_𝑖≤|𝑐|+1𝛽^𝑛_𝑖=1𝛼_𝑖+𝛾_𝑖𝑏_𝑖+𝛾_𝑖2𝑀_𝑖+1+𝛾_𝑖𝑏_𝑖≤|𝑐|+1Re𝛽^𝑛_𝑖=1𝛼_𝑖+𝛾_𝑖𝑏_𝑖+𝛾_𝑖2𝑀_𝑖+1+𝛾_𝑖𝑏_𝑖≤1. (3.12)
Finally, by applying Theorem 1.1, we conclude that the integral operator 𝐽(𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛)(𝑧) defined by (1.6) is in the class 𝒮. This evidently completes the proof of Theorem 3.1.

Setting 𝛼₁=𝛼₂=⋯=𝛼_𝑛=1 in Theorem 3.1, we have

Corollary 3.2. Let 𝑀_𝑖≥1 (𝑖∈{1,2,…,𝑛}) and 𝛽, 𝛾_𝑖 be complex numbers, 𝛽=(1+𝑛), Re𝛽≥0 and Re𝛽≥^𝑛_𝑖=11+𝛾_𝑖𝑏_𝑖+12𝑀_𝑖+1+𝛾_𝑖𝑏_𝑖, (3.13) and let 𝑐∈ℂ be such that |𝑐|1≤1−Re𝛽^𝑛_𝑖=11+𝛾_𝑖𝑏_𝑖+12𝑀_𝑖+1+𝛾_𝑖𝑏_𝑖. (3.14) If for all 𝑖∈{1,2,…,𝑛},𝑓_𝑖∈𝒜 satisfy the condition: 𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)−1<1(𝑧∈𝕌), (3.15) and 𝑔_𝑖∈𝐺_𝑏𝑖, 0<𝑏_𝑖≤1 with 𝑧²𝑔^′_𝑖(𝑧)𝑔²_𝑖(𝑧)−1<1(𝑧∈𝕌),𝑔_𝑖(𝑧)≤𝑀_𝑖(𝑧∈𝕌), (3.16) then the integral operator, 𝐽𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛(𝑧)=(1+𝑛)^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)𝑔^′_𝑖(𝑡)^𝛾𝑖𝑑𝑡^1/(1+𝑛), (3.17) is in the class 𝒮.

Setting 𝛾_𝑖=1 for all 𝑖∈{1,2,…,𝑛} in Theorem 3.1, we have the following result.

Corollary 3.3. Let 𝑀_𝑖≥1 (𝑖∈{1,2,…,𝑛}) and 𝛽, 𝛼_𝑖 be complex numbers, 𝛽=(1+∑^𝑛_𝑖=1𝛼_𝑖), Re𝛽≥0 and Re𝛽≥^𝑛_𝑖=1𝛼_𝑖+𝑏_𝑖+12𝑀_𝑖+1+𝑏_𝑖, (3.18) and let 𝑐∈ℂ be such that |𝑐|1≤1−Re𝛽^𝑛_𝑖=1𝛼_𝑖+𝑏_𝑖+12𝑀_𝑖+1+𝑏_𝑖. (3.19) If for all 𝑖∈{1,2,…,𝑛},𝑓_𝑖∈𝒜 satisfy the condition: 𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)−1<1(𝑧∈𝕌), (3.20) and 𝑔_𝑖∈𝐺_𝑏𝑖,0<𝑏_𝑖≤1 with 𝑧²𝑔^′_𝑖(𝑧)𝑔²_𝑖(𝑧)−1<1(𝑧∈𝕌),𝑔_𝑖(𝑧)≤𝑀_𝑖(𝑧∈𝕌), (3.21) then the integral operator, 𝐽𝑓₁,…,𝑓_𝑛;𝑔₁,…,𝑔_𝑛(𝑧)=1+^𝑛_𝑖=1𝛼_𝑖^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)^𝛼𝑖𝑔^′_𝑖(𝑡)𝑑𝑡^{1/(1+∑𝑛𝑖=1𝛼𝑖)}, (3.22) is in the class 𝒮.

Setting 𝑛=1 in Theorem 3.1, we have the following result.

Corollary 3.4. Let 𝑀≥1 and 𝛽, 𝛼, 𝛾 be complex numbers, 𝛽=(1+𝛼), Re𝛽≧0 and Re𝛽≥|𝛼|+𝛾(𝑏+1)(2𝑀+1)+𝛾𝑏, (3.23) and let 𝑐∈ℂ be such that |𝑐|1≤1−𝛼|+𝛾)+𝛾𝑏.Re𝛽|(𝑏+1)(2𝑀+1 (3.24) If one has that the function 𝑓∈𝒜 satisfies the condition: 𝑧𝑓^′(𝑧)𝑓(𝑧)−1≤1(𝑧∈𝕌), (3.25) and 𝑔∈𝐺_𝑏,0<𝑏≤1 with 𝑧²𝑔^′(𝑧)𝑔²(𝑧)−1<1(𝑧∈𝕌),𝑔(𝑧)≤𝑀(𝑧∈𝕌), (3.26) then the integral operator, 𝐽𝑓;𝑔(𝑧)=(1+𝛼)^𝑧₀𝑓(𝑡)^𝛼𝑔^′(𝑡)^𝛾𝑑𝑡^1/(1+𝛼), (3.27) is in the class 𝒮.

Acknowledgment

This work was partially supported by the strategic project POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007–2013 co-financed by the European Social Fund-Investing in People.