On Univalence Criteria for a General Integral Operator

Abstract

We consider a new general integral operator, and we give sufficient conditions for the univalence of this integral operator in the open unit disk of the complex plane. Several consequences of the main results are also shown.

1. Introduction

Let 𝑈={𝑧∈ℂ∶|𝑧|<1} be the unit disk of the complex plane and let 𝒜 be the class of functions 𝑓 of the form 𝑓(𝑧)=𝑧+^∞_𝑚₌₂𝑎_𝑚𝑧^𝑚,(1.1) that are analytic in 𝑈 and satisfy the usual normalization conditions 𝑓(0)=𝑓′(0)−1=0. We denote by 𝑆 the subclass of 𝒜 consisting of all univalent functions 𝑓 in 𝑈 and consider the class 𝑃 of functions ℎ that are analytic in 𝑈 and that satisfy ℎ(0)=1 and Reℎ(𝑧)>0 for all 𝑧∈𝑈.

In the present paper we obtain sufficient conditions for the following general integral operator to be in the class 𝑆 (The univalent functions are of importance in geometric functions theory and may have some applications in fluid mechanics and physics): 𝐻_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ1,…,ℎ𝑝}(𝑧)=𝛽^𝑧0𝑢^𝑛𝛽−1_𝑖=1𝑔^𝑖(𝑢)^𝛼𝑖𝑝_𝑗=1ℎ_𝑗(𝑢)𝑑𝑢^1/𝛽,(1.2) where 𝛼_𝑖,𝛽 are complex numbers, 𝛽≠0, the functions 𝑔_𝑖(𝑢)∈𝒜 for all 𝑖=1,2,…,𝑛 and ℎ_𝑗(𝑢)∈𝑃 for all 𝑗=1,2,…,𝑝, where 𝑛,𝑝 are positive integers.

For proving our main results we need the following theorems.

Theorem 1.1 (see [1]). Let 𝛼 be a complex number, let Re𝛼>0; and let 𝑓∈𝒜. If 1−|𝑧|^2Re𝛼Re𝛼||||𝑧𝑓^(𝑧)𝑓^(𝑧)||||≤1,(1.3) for all 𝑧∈𝑈, then for any complex number 𝛽 with Re𝛽≥Re𝛼, the function 𝐹_𝛽(𝑧)=𝛽^𝑧0𝑢^𝛽−1𝑓^(𝑢)𝑑𝑢^1/𝛽(1.4) is in the class S.

Theorem 1.2 (see [2]). If the function 𝑔(𝑧) is regular in 𝑈 and |𝑔(𝑧)|<1 in 𝑈, then for all 𝜉∈𝑈 and 𝑧∈𝑈, the following inequalities hold: ||||𝑔(𝜉)−𝑔(𝑧)1−𝑔||𝑔(𝑧)⋅𝑔(𝜉)||||≤||||1𝜉−−𝑧𝑧⋅𝜉||||,(1.5)_(𝑧)||≤1−||𝑔(𝑧)||²1−|𝑧|².(1.6) These are equalities if and only if 𝑔(𝑧)=𝜀(𝑧+𝑢)/(1+𝑢̄𝑧), where |𝜀|=1 and |𝑢|<1.

Remark 1.3 (see [2]). For 𝑧=0, from inequality (1.5) we have ||||𝑔(𝜉)−𝑔(0)1−𝑔(0)⋅𝑔(𝜉)||||≤||𝜉||,(1.7) and hence, ||𝑔(𝜉)||≤||𝜉||+||𝑔(0)||1+||𝑔(0)||⋅||𝜉||.(1.8) Writing 𝑔(0)=𝑎 and letting 𝜉=𝑧, we get ||𝑔(𝑧)||≤|𝑧|+|𝑎|1+|𝑎|⋅|𝑧|,(1.9) for all 𝑧∈𝑈.

2. Main Results

Theorem 2.1. Let 𝛿>0. For 𝑖=1,2,…,𝑛, let 𝛼_𝑖 be a complex number, let 𝑀_𝑖>0, and let 𝑔_𝑖∈𝒜. For 𝑗=1,2,…,𝑝, let 𝑁_𝑗>0 and ℎ_𝑗∈𝑃. If ||||𝑧𝑔^𝑖(𝑧)𝑔^𝑖(𝑧)||||≤𝑀^𝑖||||𝑧ℎ_𝑗,∀𝑖=1,2,…,𝑛(𝑧∈𝑈),(2.1)(𝑧)ℎ_𝑗(𝑧)||||≤𝑁^𝑗,∀𝑗=1,2,…,𝑝(𝑧∈𝑈),(2.2)_𝑛_𝑖=1||𝛼_𝑖||⋅𝑀_𝑖+^𝑝_𝑗=1𝑁_𝑗≤𝛿,(2.3) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐻_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ1,…,ℎ𝑝}(𝑧) given by (1.2) is in the class 𝑆.

Proof. Let us define the function: 𝑓(𝑧)=^𝑧0𝑛_𝑖=1𝑔^𝑖(𝑢)^𝛼𝑖𝑝_𝑗=1ℎ_𝑗(𝑢)𝑑𝑢,(2.4) with 𝑔_𝑖∈𝒜, for all 𝑖=1,2,…,𝑛 and ℎ_𝑗∈𝑃, for all 𝑗=1,2,…,𝑝, and, thus, we obtain 𝑓^(𝑧)=^𝑛_𝑖=1𝑔^𝑖(𝑧)^𝛼𝑖𝑝_𝑗=1ℎ_𝑗(𝑧).(2.5) The function 𝑓 is regular in 𝑈 and 𝑓(0)=𝑓′(0)−1=0. We have 𝑧𝑓^(𝑧)𝑓^(𝑧)=^𝑛_𝑖=1𝛼_𝑖𝑧𝑔^𝑖(𝑧)𝑔^𝑖(𝑧)+^𝑝_𝑗=1𝑧ℎ^𝑗(𝑧)ℎ_𝑗(𝑧)(𝑧∈𝑈).(2.6) From (2.1), (2.2), and (2.6) we obtain 1−|𝑧|^2𝛿𝛿⋅||||𝑧𝑓^(𝑧)𝑓^(𝑧)||||≤1−𝛿|𝑧|^2𝛿^𝑛_𝑖=1||𝛼_𝑖||⋅𝑀_𝑖+^𝑝_𝑗=1𝑁_𝑗(𝑧∈𝑈),(2.7) and by (2.3), we have 1−|𝑧|^2𝛿𝛿⋅||||𝑧𝑓^(𝑧)𝑓^(𝑧)||||≤1,∀𝑧∈𝑈.(2.8) Using (2.8), by Theorem 1.1, it results that the integral operator 𝐻_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ1,…,ℎ𝑝}(𝑧) given by (1.2) is in the class 𝑆.

Letting 𝑝=1 in Theorem 2.1, we have the following.

Corollary 2.2. Let 𝛿>0,  let 𝑁>0, and let ℎ∈𝑃. For 𝑖=1,2,…,𝑛, let 𝛼_𝑖 be a complex number, let 𝑀_𝑖>0, and let 𝑔_𝑖∈𝒜. If ||||𝑧𝑔^𝑖(𝑧)𝑔^𝑖(𝑧)||||≤𝑀^𝑖||||𝑧ℎ_,∀𝑖=1,2,…,𝑛(𝑧∈𝑈),(𝑧)ℎ(𝑧)||||≤𝑁(𝑧∈𝑈)_𝑛_𝑖=1||𝛼_𝑖||⋅𝑀_𝑖+𝑁≤𝛿,(2.9) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐹_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ}(𝑧)=𝛽^𝑧0𝑢^𝑛𝛽−1_𝑖=1𝑔^𝑖(𝑢)^𝛼𝑖ℎ(𝑢)𝑑𝑢^1/𝛽(2.10) is in the class 𝑆.

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

Corollary 2.3. Let 𝛿>0, let 𝑀>0, let 𝑔∈𝒜, and let 𝛼 be a complex number. For 𝑗=1,2,…,𝑝, let 𝑁_𝑗>0 and let ℎ_𝑗∈𝑃. If ||||𝑧𝑔^(𝑧)||||𝑧ℎ_𝑗𝑔^(𝑧)||||≤𝑀(𝑧∈𝑈),(𝑧)ℎ_𝑗(𝑧)||||≤𝑁^𝑗|𝛼|⋅,𝑀∀+𝑗=_𝑝1,2,…,𝑝(𝑧∈𝑈),_𝑗=1𝑁_𝑗≤𝛿,(2.11) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐺_{𝛼,𝛽,𝑔,ℎ1,…,ℎ𝑝}(𝑧)=𝛽^𝑧0𝑢^𝛽−1𝑔′(𝑢)^𝛼𝑝_𝑗=1ℎ_𝑗(𝑢)𝑑𝑢^1/𝛽(2.12) is in the class 𝑆.

For 𝑀₁=𝑀₂=⋯=𝑀_𝑛=𝑀 and 𝑁₁=𝑁₂=⋯=𝑁_𝑝=𝑁 in Theorem 2.1, we have the following.

Corollary 2.4. Let 𝛿>0, let 𝑀>0, and let 𝑁>0. For 𝑖=1,2,…,𝑛, let 𝛼_𝑖 be a complex number and let 𝑔_𝑖∈𝒜. For 𝑗=1,2,…,𝑝, let ℎ_𝑗∈𝑃. If ||||𝑧𝑔^𝑖(𝑧)||||𝑧𝑔ℎ^𝑖_𝑗(𝑧)||||≤𝑀,∀𝑖=1,2,…,𝑛(𝑧∈𝑈),(𝑧)ℎ_𝑗(𝑧)||||≤𝑀𝑁,∀𝑗=1,2,…,𝑝(𝑧∈𝑈),_𝑛_𝑖=1||𝛼_𝑖||+𝑝𝑁≤𝛿,(2.13) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐻_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ1,…,ℎ𝑝}(𝑧) given by (1.2) is in the class 𝑆.

Theorem 2.5. Let 𝛿>0. For 𝑖=1,2,…,𝑛, let 𝛼_𝑖 be a complex number, let 𝑀_𝑖>0, and let 𝑔_𝑖∈𝒜,𝑔_𝑖(𝑧)=𝑧+𝑎^𝑖2𝑧²+𝑎^𝑖3𝑧³+⋯. For 𝑗=1,2,…,𝑝, let 𝑁_𝑗>0 and ℎ_𝑗∈𝑃 with ℎ^𝑗(0)=0, and 𝑐=∑^𝑛𝑖=1𝛼_𝑖𝑎^𝑖2/|∏^𝑛𝑖=1𝛼_𝑖|. If ||||𝑔^𝑖(𝑧)𝑔^𝑖(𝑧)||||<𝑀^𝑖||||ℎ,∀𝑖=1,2,…,𝑛(𝑧∈𝑈),(2.14)_𝑗(𝑧)ℎ_𝑗(𝑧)||||<𝑁^𝑗∑,∀𝑗=1,2,…,𝑝(𝑧∈𝑈),(2.15)_𝑛𝑖=1||𝛼_𝑖||𝑀_𝑖+∑^𝑝𝑗=1𝑁_𝑗|||||||∏^𝑛𝑖=1𝛼_𝑖||<1,(2.16)_𝑛_𝑖=1𝛼_𝑖|||||≤1max_|𝑧|<11−|𝑧|^2𝛿/𝛿⋅|𝑧|⋅((|𝑧|+2|𝑐|)/(1+2|𝑐|⋅|𝑧|)),(2.17) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐻_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ1,…,ℎ𝑝}(𝑧) given by (1.2) is in the class 𝑆.

Proof. We define the function: 𝑓(𝑧)=^𝑧0𝑛_𝑖=1𝑔^𝑖(𝑢)^𝛼𝑖𝑝_𝑗=1ℎ_𝑗(𝑢)𝑑𝑢(𝑧∈𝑈),(2.18) with 𝑔_𝑖∈𝒜, for all 𝑖=1,2,…,𝑛 and ℎ_𝑗∈𝑃, for all 𝑗=1,2,…,𝑝.
We consider the function 𝐾(𝑧)=1||∏^𝑛𝑖=1𝛼_𝑖||⋅𝑓^(𝑧)𝑓^(𝑧),∀𝑧∈𝑈.(2.19) We have 𝑓^(𝑧)𝑓^(𝑧)=^𝑛_𝑖=1𝛼_𝑖𝑔^𝑖(𝑧)𝑔^𝑖(𝑧)+^𝑝_𝑗=1ℎ^𝑗(𝑧)ℎ_𝑗(𝑧)(𝑧∈𝑈).(2.20) From (2.14), (2.15), and (2.20) we obtain 1||∏^𝑛𝑖=1𝛼_𝑖||⋅||||𝑓^(𝑧)𝑓^(𝑧)||||≤∑^𝑛𝑖=1||𝛼_𝑖||𝑀_𝑖+∑^𝑝𝑗=1𝑁_𝑗||∏^𝑛𝑖=1𝛼_𝑖||(𝑧∈𝑈).(2.21) From (2.16), (2.19), and (2.21) we obtain |𝐾(𝑧)|<1 for all 𝑧∈𝑈.
We have 𝐾(0)=2∑^𝑛𝑖=1𝛼_𝑖𝑎^𝑖2/|∏^𝑛𝑖=1𝛼_𝑖|=2𝑐, and, using Remark 1.3 we get ||𝐾(𝑧)||≤|𝑧|+2|𝑐|1+2|𝑐|⋅|𝑧|(𝑧∈𝑈).(2.22) From (2.19) and (2.22), we obtain 1−|𝑧|^2𝛿𝛿⋅||||𝑧𝑓^(𝑧)𝑓^(𝑧)||||≤|||||^𝑛_𝑖=1𝛼_𝑖|||||⋅max_|𝑧|<11−𝛿|𝑧|^2𝛿⋅|𝑧|⋅1|+𝑧|2+|𝑐2||⋅𝑐||𝑧|,(2.23) for all 𝑧∈𝑈.
From (2.17) and (2.23), we have 1−|𝑧|^2𝛿𝛿⋅||||𝑧𝑓^(𝑧)𝑓^(𝑧)||||≤1(𝑧∈𝑈).(2.24) So, applying Theorem 1.1, we obtain that the integral operator 𝐻_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ1,…,ℎ𝑝}(𝑧) given by (1.2) is in the class 𝑆.

Letting 𝑝=1 in Theorem 2.5, we have the following.

Corollary 2.6. Let 𝛿>0, let 𝑁>0, and let ℎ∈𝑃 with ℎ′(0)=0. For 𝑖=1,2,…,𝑛, let 𝛼_𝑖 be a complex number, 𝑀_𝑖>0 and 𝑔_𝑖∈𝒜,𝑔_𝑖(𝑧)=𝑧+𝑎^𝑖2𝑧²+𝑎^𝑖3𝑧³+⋯, and 𝑐=∑^𝑛𝑖=1𝛼_𝑖𝑎^𝑖2/|∏^𝑛𝑖=1𝛼_𝑖|. If ||||𝑔^𝑖(𝑧)𝑔^𝑖(𝑧)||||<𝑀^𝑖||||ℎ,∀𝑖=1,2,…,𝑛(𝑧∈𝑈),_(𝑧)ℎ(𝑧)||||<𝑁(𝑧∈𝑈),∑^𝑛𝑖=1||𝛼_𝑖||𝑀_𝑖+𝑁|||||||∏^𝑛𝑖=1𝛼_𝑖||<1,_𝑛_𝑖=1𝛼_𝑖|||||≤1max_|𝑧|<11−|𝑧|^2𝛿/𝛿⋅|𝑧|⋅((|𝑧|+2|𝑐|)/(1+2|𝑐|⋅|𝑧|)),(2.25) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐹_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ}(𝑧)=𝛽^𝑧0𝑢^𝑛𝛽−1_𝑖=1𝑔^𝑖(𝑢)^𝛼𝑖ℎ(𝑢)𝑑𝑢^1/𝛽(2.26) is in the class 𝑆.

Letting 𝑛=1 in Theorem 2.5, we have the following.

Corollary 2.7. Let 𝛿>0, 𝑀>0,  𝑔∈𝒜,𝑔(𝑧)=𝑧+𝑎₂𝑧²+𝑎₃𝑧³+⋯ and let 𝛼 be a complex number. For 𝑗=1,2,…,𝑝, let 𝑁_𝑗>0 and ℎ_𝑗∈𝑃 with ℎ^𝑗(0)=0. If ||||𝑔^(𝑧)||||ℎ𝑔^(𝑧)||||<𝑀(𝑧∈𝑈),_𝑗(𝑧)ℎ_𝑗(𝑧)||||<𝑁^𝑗|𝛼|⋅M+_𝑝,∀𝑗=1,2,…,𝑝(𝑧∈𝑈),_𝑗∑₌₁𝑁_𝑗<|𝛼|≤1max_|𝑧|<11−|𝑧|^2𝛿/𝛿⋅|𝑧|⋅|𝑧|+2||𝑎₂||/1+2||𝑎₂||⋅|𝑧|,(2.27) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐺_{𝛼,𝛽,𝑔,ℎ1,…,ℎ𝑝}(𝑧)=𝛽^𝑧0𝑢^𝛽−1𝑔′(𝑢)^𝛼𝑝_𝑗=1ℎ_𝑗(𝑢)𝑑𝑢^1/𝛽(2.28) is in the class 𝑆.

For 𝑀₁=𝑀₂=⋯=𝑀_𝑛=𝑀 and 𝑁₁=𝑁₂=⋯=𝑁_𝑝=𝑁 in Theorem 2.5, we have the following.

Corollary 2.8. Let 𝛿>0, 𝑀>0, 𝑁>0. For 𝑖=1,2,…,𝑛, let 𝛼_𝑖 be a complex number and 𝑔_𝑖∈𝒜,𝑔_𝑖(𝑧)=𝑧+𝑎^𝑖2𝑧²+𝑎^𝑖3𝑧³+⋯. For 𝑗=1,2,…,𝑝, let ℎ_𝑗∈𝑃 with ℎ^𝑗(0)=0, and 𝑐=∑^𝑛𝑖=1𝛼_𝑖𝑎^𝑖2/|∏^𝑛𝑖=1𝛼_𝑖|.
If ||||𝑔^𝑖(𝑧)||||ℎ𝑔^𝑖(𝑧)||||<𝑀∀𝑖=1,2,…,𝑛(𝑧∈𝑈),_𝑗(𝑧)ℎ_𝑗(𝑧)||||<𝑀𝑁∑∀𝑗=1,2,…,𝑝(𝑧∈𝑈),_𝑛𝑖=1||𝛼_𝑖||+𝑝𝑁|||||||∏^𝑛𝑖=1𝛼_𝑖||<1,_𝑛_𝑖=1𝛼_𝑖|||||≤1max_|𝑧|<11−|𝑧|^2𝛿/𝛿⋅|𝑧|⋅((|𝑧|+2|𝑐|)/(1+2|𝑐|⋅|𝑧|)),(2.29) then for every complex number 𝛽 with Re𝛽≥𝛿>0, the integral operator 𝐻_{𝛼1,…,𝛼𝑛,𝛽,𝑔1,…,𝑔𝑛,ℎ1,…,ℎ𝑝}(𝑧) given by (1.2) is in the class 𝑆.