Univalence Conditions Related to a General Integral Operator

Abstract

We consider a general integral operator based on two types of analytic functions, namely, regular functions and, respectively, functions having a positive real part. Some univalence conditions for this integral operator are obtained.

1. Introduction

Let 𝒜 be the class of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑖=2𝑎_𝑖𝑧^𝑖, (1.1) which are analytic in the open unit disk, 𝒰={𝑧∈ℂ∶|𝑧|<1} and normalized by 𝑓(0)=𝑓^′(0)−1=0. Also, let 𝒮 denote the subclass of 𝒜 consisting of univalent functions, in the open unit disk. We denote by 𝒫, the class of functions 𝑝 which are analytic in 𝒰, Re𝑝(𝑧)>0, for all 𝑧∈𝒰 and 𝑝(0)=1, the so-called Caratheodory functions.

In [1], Pescar introduced and studied the following integral operator: 𝐸_𝑛(𝑧)=^𝑧₀^𝑛_𝑗=1𝑓_𝑗(𝑢)𝑢^𝛼𝑗𝑔_𝑗(𝑢)^𝛽𝑗𝑑𝑢, (1.2) where 𝛼_𝑗,𝛽_𝑗 are complex numbers, 𝑓_𝑗∈𝒜, 𝑔_𝑗∈𝒫, 𝑗=1,𝑛.

In this paper, we generalize this integral operator, by considering the general integral operator defined as follows: 𝐾_𝑛(𝑧)=𝛿^𝑧₀𝑢^𝑛𝛿−1_𝑗=1𝑓_𝑗(𝑢)𝑢^𝛼𝑗𝑔_𝑗(𝑢)^𝛽𝑗𝑑𝑢^1/𝛿, (1.3) where 𝛿, 𝛼_𝑗,𝛽_𝑗 are complex numbers, 𝛿≠0, 𝑓_𝑗∈𝒜, 𝑔_𝑗∈𝒫, 𝑗=1,𝑛.

Remarks. This integral operator extends many other integral operators from related works on this field as those given by Srivastava, Mocanu, Owa, Pescar, Orhan, Breaz, and others. Using this new operator and the related univalence conditions that are going to be proved here, one can study some other already known operators in a unified perspective. Thus, for different particular cases of the parameters 𝛿, 𝛼_𝑗, 𝛽_𝑗, our integral operator 𝐾_𝑛 is shown to be an extension of the following integral operators.(i)For 𝛿=1, the integral operator 𝐾_𝑛 is the operator 𝐸_𝑛 from (1.2), introduced by Pescar in [1].(ii)For 𝑛=1, 𝛼_𝑗=𝛿, 𝛽_𝑗=0, the integral operator 𝐾_𝑛 was studied in [2], by Miller and Mocanu.(iii)For 𝛽_𝑗=0, 𝑗=1,𝑛, the integral operator 𝐾_𝑛 was studied in [3], by Pescar and Breaz.(iv)For 𝛽_𝑗=0, 𝑗=1,𝑛, 𝛼_𝑗=𝛼−1, and 𝛿=𝑛(𝛼−1)+1, the integral operator 𝐾_𝑛 was studied in [4], by Breaz et al. and also in [5], by Srivastava et al.(v)For 𝛿=1, 𝛽_𝑗=0, 𝑗=1,𝑛, we get the integral operator defined by D. Breaz and N. Breaz, in the paper [6].(vi)For 𝛿=1, 𝛼_𝑗=0, 𝛽_𝑗>0, and 𝑔_𝑗=𝑓^′_𝑗, the operator was studied by Breaz et al., in the paper [7].

More precisely, if we are interested to study various properties of two or more different operators reminded above (and other similar operators which are not mentioned here), we can do this in an integrated manner by simply allowing the parameters involved in the definition, to be more general and consequently by studying only one operator, having the form (1.3). In this paper, we will study some univalence criteria for this new operator.

The following known results will be used in order to prove our results.

Lemma 1.1 (see [8]). Let 𝛾 be a complex number, Re𝛾>0 and 𝑓∈𝒜. If 1−|𝑧|^2Re𝛾Re𝛾𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≤1, (1.4) for all 𝑧∈𝒰, then for any complex number 𝛿, Re𝛿≥Re𝛾, 𝐹_𝛿(𝑧)=𝛿^𝑧₀𝑢^𝛿−1𝑓^′(𝑢)𝑑𝑢^1/𝛿∈𝒮. (1.5)

Lemma 1.2 (see [9]). Let 𝑓 be a regular function in the open disk 𝒰_𝑅={𝑧∈ℂ∶|𝑧|<𝑅} with |𝑓(𝑧)|<𝑀, 𝑀 fixed. If 𝑓 has in 𝑧=0 one zero with multiplicity ≥𝑚, then 𝑓(𝑧≤𝑀)𝑅^𝑚|𝑧|^𝑚,𝑧∈𝒰_𝑅, (1.6) the equality for (𝑧≠0) can hold only if 𝑓(𝑧)=𝑒^𝑖𝜃𝑀𝑅^𝑚𝑧^𝑚, (1.7) where 𝜃 is constant.

Lemma 1.3 (see [10]). For each 𝑓∈𝒮, 1−𝑟≤1+𝑟𝑧𝑓^′(𝑧)𝑓(𝑧)≤1+𝑟,|𝑧|1−𝑟=𝑟<1. (1.8)

Lemma 1.4 (see [11]). If 𝑔(𝑧)=1+∑^∞_𝑛=1𝑏_𝑛𝑧^𝑛 is regular in |𝑧|<1 and Re(𝑔(𝑧))>0, then 𝑧𝑔^′(𝑧)𝑔(𝑧)≤2|𝑧|1−|𝑧|². (1.9)

Also, for the statement of our main results, we need to define the following classes: 𝒜_𝑀=𝑓∈𝒜∶𝑧𝑓^′(𝑧)𝑓(𝑧),𝒫−1≤𝑀,𝑀≥1_𝐿=𝑓∈𝒫∶𝑧𝑓^′(𝑧)𝑓(𝑧)≤𝐿,𝐿>0. (1.10)

2. Main Results

Theorem 2.1. Let 𝛿,𝛼_𝑗,𝛽_𝑗 be complex numbers, Re𝛿≥1, 𝑀_𝑗,𝐿_𝑗 positive real numbers, 𝑀_𝑗≥1, 𝑗=1,𝑛, and the functions 𝑓_𝑗∈𝒜_𝑀𝑗, 𝑔_𝑗∈𝒫_𝐿𝑗, 𝑗=1,𝑛.
If ^𝑛_𝑗=1𝛼_𝑗𝑀_𝑗+𝛽_𝑗𝐿_𝑗≤3√32, (2.1) then the general integral operator 𝐾_𝑛, defined by (1.3), is in the class 𝒮.

Proof. We consider the regular function ℎ_𝑛(𝑧)=^𝑧₀𝑓₁(𝑢)𝑢^𝛼1⋯𝑓_𝑛(𝑢)𝑢^𝛼𝑛⋅𝑔₁(𝑢)^𝛽1⋯𝑔_𝑛(𝑢)^𝛽𝑛𝑑𝑢. (2.2) After some calculus, we have 1−|𝑧|²𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1−|𝑧|^2𝑛_𝑗=1𝛼_𝑗𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)−1+𝛽_𝑗𝑧𝑔^′_𝑗(𝑧)𝑔_𝑗(𝑧), (2.3) for all 𝑧∈𝒰.
Since 𝑓_𝑗∈𝐴_𝑀𝑗, by applying Lemma 1.2 for 𝑚=1, we get 𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)−1≤𝑀_𝑗|𝑧|,𝑗=1,𝑛;𝑧∈𝒰. (2.4)
Also since 𝑔_𝑗∈𝑃_𝐿𝑗, from Lemma 1.2, we have 𝑧𝑔^′_𝑗(𝑧)𝑔_𝑗(𝑧)≤𝐿_𝑗|𝑧|,𝑗=1,𝑛;𝑧∈𝒰. (2.5)
If we put these last two inequalities in (2.3), together with the inequality from the hypothesis, we get 1−|𝑧|²𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1−|𝑧|²|𝑧|⋅3√32. (2.6)
Now we take into account the fact that max_|𝑧|<11−|𝑧|²|𝑧=2|3√3, (2.7) thus obtaining 1−|𝑧|²𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1,(𝑧∈𝒰). (2.8)
Further from Lemma 1.1, for Re𝛿≥Re𝛾=1, it is obvious that 𝐾_𝑛∈𝒮.

Corollary 2.2. Let 𝛿, 𝛼_𝑗 be complex numbers, Re𝛿≥1, 𝑀_𝑗 positive real numbers, 𝑀_𝑗≥1, 𝑓_𝑗∈𝒜_𝑀𝑗, 𝑗=1,𝑛. If ^𝑛_𝑗=1𝛼_𝑗𝑀_𝑗≤3√32 (2.9) then the function𝐾_𝑛(𝑧)=𝛿^𝑧₀𝑢^𝑛𝛿−1_𝑗=1𝑓_𝑗(𝑢)𝑢^𝛼𝑗𝑑𝑢^1/𝛿, (2.10) is in the class 𝒮.

Proof. In Theorem 2.1, we take 𝛽_𝑗=0,𝑗=1,𝑛.

Corollary 2.3. Let 𝛿, 𝛽_𝑗 be complex numbers, Re𝛿≥1, 𝐿_𝑗 positive real numbers and 𝑔_𝑗∈𝒫_𝐿𝑗, 𝑗=1,𝑛. If ^𝑛_𝑗=1𝛽_𝑗𝐿_𝑗≤3√32, (2.11) then the function 𝐾_𝑛(𝑧)=𝛿^𝑧₀𝑢^𝑛𝛿−1_𝑗=1𝑔_𝑗(𝑢)^𝛽𝑗𝑑𝑢^1/𝛿 (2.12) belongs to the class 𝒮.

Proof. In Theorem 2.1, we take 𝛼_𝑗=0,𝑗=1,𝑛.

Theorem 2.4. Let 𝛾,𝛿,𝛼_𝑗,𝛽_𝑗 be complex numbers, 𝑗=1,𝑛, Re𝛿≥Re𝛾>0, and the functions 𝑓_𝑗∈𝒜_𝜇, 𝑔_𝑗∈𝒫_𝜇, 𝜇≥1, with 𝜇=2Re𝛾+1^{1+(1/2Re𝛾)}2. (2.13)
If ^𝑛_𝑗=1𝛼_𝑗+𝛽_𝑗≤1, (2.14) then the general integral operator 𝐾_𝑛 belongs to the class 𝒮.

Proof. Let consider again the regular function: ℎ_𝑛(𝑧)=^𝑧₀𝑓₁(𝑢)𝑢^𝛼1⋯𝑓_𝑛(𝑢)𝑢^𝛼𝑛⋅𝑔₁(𝑢)^𝛽1⋯𝑔_𝑛(𝑢)^𝛽𝑛𝑑𝑢. (2.15)
After some calculus, we have 1−|𝑧|^2Re𝛾Re𝛾𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1−|𝑧|^2Re𝛾Re𝛾^𝑛_𝑗=1𝛼_𝑗𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)−1+𝛽_𝑗𝑧𝑔^′_𝑗(𝑧)𝑔_𝑗(𝑧),(𝑧∈𝒰). (2.16)
Since 𝑓_𝑗∈𝐴_𝜇 and 𝑔_𝑗∈𝑃_𝜇, for all 𝑗=1,𝑛, from Lemma 1.2, we obtain 𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)−1≤𝜇|𝑧|,𝑗=1,𝑛;𝑧∈𝒰,𝑧𝑔^′_𝑗(𝑧)𝑔_𝑗(𝑧)≤𝜇|𝑧|,𝑗=1,𝑛;𝑧∈𝒰. (2.17)
Using these last two inequalities, the inequality from the hypothesis, and the fact that max_|𝑧|<11−|𝑧|^2Re𝛾|𝑧|=1Re𝛾𝜇, (2.18) from (2.16) we get 1−|𝑧|^2Re𝛾Re𝛾𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1, (2.19) for all 𝑧∈𝒰.
Hence, by Lemma 1.1, we have that 𝐾_𝑛∈𝒮.

Corollary 2.5. Let 𝛾,𝛿,𝛼_𝑗 be complex numbers, Re𝛿≥Re𝛾>0, and 𝑓_𝑗∈𝒜_𝜇, 𝑗=1,𝑛, 𝜇≥1, with 𝜇=2Re𝛾+1^{1+(1/2Re𝛾)}2. (2.20) If ^𝑛_𝑗=1𝛼_𝑗≤1, (2.21) then the integral operator defined by 𝐾_𝑛(𝑧)=𝛿^𝑧₀𝑢^𝑛𝛿−1_𝑗=1𝑓_𝑗(𝑢)𝑢^𝛼𝑗𝑑𝑢^1/𝛿, (2.22) is in the class 𝒮.

Proof. In Theorem 2.4, we take 𝛽₁=⋯=𝛽_𝑛=0.

Corollary 2.6. Let 𝛾,𝛿,𝛽_𝑗 be complex numbers, 𝑗=1,𝑛, Re𝛿≥Re𝛾>0, and 𝑔_𝑗∈𝒫_𝜇, with 𝜇=2Re𝛾+1^{1+(1/2Re𝛾))}2. (2.23) If ^𝑛_𝑗=1𝛽_𝑗≤1, (2.24) then the integral operator defined by 𝐾_𝑛(𝑧)=𝛿^𝑧₀𝑢^𝑛𝛿−1_𝑗=1𝑔_𝑗(𝑢)^𝛽𝑗𝑑𝑢^1/𝛿, (2.25) is in the class 𝒮.

Proof. We put in Theorem 2.4, 𝛼₁=⋯=𝛼_𝑛=0.

Theorem 2.7. Let 𝛾,𝛿,𝛼_𝑗,𝛽_𝑗 be complex numbers, 𝑗=1,𝑛, Re𝛿≥Re𝛾>0 and 𝑓_𝑗∈𝒮, 𝑔_𝑗∈𝒫. If 2^𝑛_𝑗=1𝛼_𝑗+^𝑛_𝑗=1𝛽_𝑗≤minRe𝛾2,12, (2.26) then the general integral operator 𝐾_𝑛∈𝒮.

Proof. We consider the same regular function ℎ_𝑛 as in the proof of the previous theorems, and after some calculus we get 1−|𝑧|^2Re𝛾Re𝛾𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1−|𝑧|^2Re𝛾Re𝛾^𝑛_𝑗=1𝛼_𝑗𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)+1+𝛽_𝑗𝑧𝑔^′_𝑗(𝑧)𝑔_𝑗(𝑧), (2.27) for all 𝑧∈𝒰.
Since 𝑓_𝑗∈𝒮, from Lemma 1.3, we get 𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)≤1+|𝑧|1−|𝑧|, (2.28)𝑗=1,𝑛, 𝑧∈𝒰, and since 𝑔_𝑗∈𝑃, from Lemma 1.4, we get 𝑧𝑔^′_𝑗(𝑧)𝑔_𝑗(𝑧)≤2|𝑧|1−|𝑧|² (2.29)𝑗=1,𝑛, 𝑧∈𝒰.
Now if we use (2.28) and (2.29) in (2.27), we obtain 1−|𝑧|^2Re𝛾Re𝛾𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1−|𝑧|^2Re𝛾2Re𝛾1−|𝑧|^𝑛_𝑗=1𝛼_𝑗+2|𝑧|1−|𝑧|^2𝑛_𝑗=1𝛽_𝑗, (2.30) for all 𝑧∈𝒰.
We consider two cases.
(1) If min{Re𝛾/2,1/2}=Re𝛾/2, we have 1−|𝑧|^2Re𝛾≤1−|𝑧|² (2.31) and further if we use this inequality together with the inequality from the hypothesis, from (2.30), we get 1−|𝑧|^2Re𝛾Re𝛾𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤1, (2.32) for all 𝑧∈𝒰, and with Lemma 1.1 the proof is complete.
(2) If min{Re𝛾/2,1/2}=1/2, we obtain 1−|𝑧|^2Re𝛾|𝑧|Re𝛾≤1−², (2.33)𝑧∈𝒰 and further if we put this last inequality in (2.30) we get 1−|𝑧|^2Re𝛾Re𝛾𝑧ℎ_𝑛^′′(𝑧)ℎ^′_𝑛(𝑧)≤4^𝑛_𝑗=1𝛼_𝑗+2^𝑛_𝑗=1𝛽_𝑗, (2.34) for all 𝑧∈𝒰.
Now by applying first the inequality condition from the hypothesis and then Lemma 1.1 for ℎ_𝑛, the proof is complete.

Corollary 2.8. Let 𝛾,𝛿,𝛼_𝑗 be complex numbers, 𝑗=1,𝑛, Re𝛿≥Re𝛾>0 and 𝑓_𝑗∈𝒮. If ^𝑛_𝑗=1𝛼_𝑗≤minRe𝛾4,14, (2.35) then the integral operator 𝐾_𝑛(𝑧)=𝛿^𝑧₀𝑢^𝑛𝛿−1_𝑗=1𝑓_𝑗(𝑢)𝑢^𝛼𝑗𝑑𝑢^1/𝛿 (2.36) is in the class 𝒮.

Proof. In Theorem 2.7, we take 𝛽₁=⋯=𝛽_𝑛=0.

Corollary 2.9. Let 𝛾,𝛿,𝛽_𝑗 be complex numbers, 𝑗=1,𝑛, Re𝛿≥Re𝛾>0 and 𝑔_𝑗∈𝒫. If ^𝑛_𝑗=1𝛽_𝑗≤minRe𝛾2,12 (2.37) then the integral operator 𝐾_𝑛(𝑧)=𝛿^𝑧₀𝑢^𝑛𝛿−1_𝑗=1𝑔_𝑗(𝑢)^𝛽𝑗𝑑𝑢^1/𝛿 (2.38) is in the class 𝒮.

Proof. In Theorem 2.7 we take 𝛼₁=⋯=𝛼_𝑛=0.

Acknowledgment

The authors wish to thank the reviewers for their constructive comments which helped them to improve the quality of the paper.