Some Convexity Properties of Certain General Integral Operators

Abstract

The main object of the present paper is to discuss some extensions of certain integral operators and to obtain their order of convexity. Several other closely related results are also considered.

1. Introduction

Let 𝒜 be the class of analytic functions defined in the open unit disk of the complex plane 𝑈={𝑧∈ℂ∶|𝑧|<1}.

We denote by 𝑆 the subclass of 𝒜 consisting of all univalent functions in 𝑈. A function 𝑓(𝑧)∈𝑆 is starlike function of order 𝛼 if it satisfies Re𝑧𝑓^′(𝑧)𝑓(𝑧)>𝛼,(𝑧∈𝑈) (1.1) for some 𝛼(0≤𝛼<1). We denote by 𝑆^∗(𝛼) the subclass of 𝒜 consisting of the functions which are starlike of order 𝛼 in 𝑈. For 𝛼=0, we obtain the class of starlike functions, denoted by 𝑆^∗.

A function 𝑓(𝑧)∈𝑆 is convex of order 𝛼 if it satisfies Re𝑧𝑓^′′(𝑧)𝑓^′(𝑧)+1>𝛼,(𝑧∈𝑈) (1.2) for some 𝛼 (0≤𝛼<1). We denote by 𝐾(𝛼) the subclass of 𝒜 consisting of the functions which are convex of order 𝛼 in 𝑈. For 𝛼=0, we obtain the class of convex functions, denoted by 𝐾.

A function 𝑓(𝑧)∈𝒜 is in the class 𝑅(𝛼) if Re𝑓^′(𝑧)>𝛼,(𝑧∈𝑈). (1.3)

Frasin and Jahangiri introduced in [1] the family 𝐵(𝜇,𝛼),𝜇≥0,0≤𝛼<1 consisting of functions 𝑓∈𝒜 satisfying the condition 𝑓^′(𝑧)𝑧𝑓(𝑧)^𝜇−1<1−𝛼,(𝑧∈𝑈). (1.4) For 𝜇=0 we have 𝐵(0,𝛼)≡𝑅(𝛼), and for 𝜇=1 we have 𝐵(1,𝛼)≡𝑆^∗(𝛼).

In this paper, we will obtain the order of convexity of the following general integral operators: 𝐻_𝑛,𝛾(𝑧)=^𝑧₀^𝑛_𝑖=1𝑡𝑒^{𝑓𝑖(𝑡)1/𝛾}𝐺d𝑡,_𝑛(𝑧)=^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)^𝛽𝑖−1𝐹(𝑧)=d𝑡,_{𝑧0𝑛𝑖=1}𝑓_𝑖(𝑡)𝑡^𝛽𝑖d𝑡, (1.5)(1.6)(1.7) where the functions 𝑓_𝑖(𝑡) are in 𝐵(𝜇_𝑖,𝛼_𝑖) for all 𝑖=1,2,…,𝑛.

In order to prove our main results, we recall the following lemma.

Lemma 1.1 (see [2, General Schwarz Lemma]). 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. (1.8) The equality can hold only if 𝑓(𝑧)=𝑒^𝑖𝜃⋅𝑀𝑅^𝑚⋅𝑧^𝑚, (1.9) where 𝜃 is constant.

2. Main Results

Theorem 2.1. Let 𝑓_𝑖(𝑧)∈𝒜 be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥0,0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀_𝑖 (𝑀_𝑖≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator 𝐻_𝑛,𝛾(𝑧)=^𝑧₀^𝑛_𝑖=1𝑡𝑒^{𝑓𝑖(𝑡)1/𝛾}d𝑡 (2.1) is in 𝐾(𝛿), where 1𝛿=1−𝛾𝑛+^𝑛_𝑖=12−𝛼_𝑖𝑀^𝜇𝑖_𝑖 (2.2) and 1/|𝛾|<1/(𝑛+∑^𝑛_𝑖=1(2−𝛼_𝑖)𝑀^𝜇𝑖_𝑖),𝛾∈ℂ⧵{0}.

Proof. Let 𝑓_𝑖∈𝒜 be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥0,0≤𝛼_𝑖<1. We have from (1.5) that 𝐻_𝑛,𝛾(𝑧)=^𝑧₀𝑡^𝑛/𝛾𝑒^{(1/𝛾)∑𝑛𝑖=1𝑓𝑖(𝑡)}d𝑡,𝐻^′_𝑛,𝛾(𝑧)=𝑧^𝑛/𝛾𝑒^{(1/𝛾)∑𝑛𝑖=1𝑓𝑖(𝑧)}. (2.3) Also 𝐻^′′_𝑛,𝛾(𝑧)=1𝛾𝑧^𝑛𝑒^{∑𝑛𝑖=1𝑓𝑖(𝑧)(1/𝛾)−1}⋅𝑧^𝑛−1⋅𝑒^{∑𝑛𝑖=1𝑓𝑖(𝑧)}𝑛+𝑧^𝑛_𝑖=1𝑓^′_𝑖(𝑧). (2.4) Then 𝐻^′′_𝑛,𝛾(𝑧)𝐻^′_𝑛,𝛾(𝑧)=1𝛾𝑛𝑧+^𝑛_𝑖=1𝑓^′_𝑖(𝑧) (2.5) and, hence, 𝑧𝐻^′′_𝑛,𝛾(𝑧)𝐻^′_𝑛,𝛾(𝑧)=1𝛾𝑛+𝑧^𝑛_𝑖=1𝑓^′_𝑖(𝑧≤1)𝛾^𝑛_𝑖=11+𝑧𝑓^′_𝑖(𝑧≤1)𝛾^𝑛_𝑖=11+𝑓^′_𝑖(𝑧)𝑧𝑓_𝑖(𝑧)^𝜇𝑖⋅𝑓_𝑖(𝑧)𝑧^𝜇𝑖⋅|𝑧|. (2.6) Applying the General Schwarz lemma, we have |𝑓_𝑖(𝑧)/𝑧|≤𝑀_𝑖, for all 𝑖=1,2,…,𝑛. Therefore, from (2.6), we obtain 𝑧𝐻^′′_𝑛,𝛾(𝑧)𝐻^′_𝑛,𝛾(𝑧)≤1𝛾^𝑛_𝑖=11+𝑓^′_𝑖(𝑧)𝑧𝑓_𝑖(𝑧)^𝜇𝑖⋅𝑀^𝜇𝑖_𝑖,(𝑧∈𝑈). (2.7) From (1.4) and (2.7), we see that 𝑧𝐻^′′_𝑛,𝛾(𝑧)𝐻^′_𝑛,𝛾(𝑧)≤1𝛾𝑛+^𝑛_𝑖=12−𝛼_𝑖𝑀^𝜇𝑖_𝑖=1−𝛿. (2.8)

Letting 𝜇_𝑖=0 and 𝑀_𝑖=𝑀 for all 𝑖=1,2,…,𝑛 in Theorem 2.1, we have the following corollary.

Corollary 2.2. Let 𝑓_𝑖(𝑧)∈𝒜 be in the class 𝑅(𝛼_𝑖),0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. Then the integral operator defined in (1.5) is in 𝐾(𝛿), where 1𝛿=1−𝛾3𝑛−^𝑛_𝑖=1𝛼_𝑖 (2.9) and 1/|𝛾|<1/(3𝑛−∑^𝑛_𝑖=1𝛼_𝑖),𝛾∈ℂ⧵{0}.

Letting 𝜇_𝑖=1 and 𝑀_𝑖=𝑀 for all 𝑖=1,2,…,𝑛 in Theorem 2.1, we have the following corollary.

Corollary 2.3. Let 𝑓_𝑖∈𝒜 be in the class 𝑆^∗(𝛼_𝑖),0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀 (𝑀≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator defined in (1.5) is in 𝐾(𝛿), where 1𝛿=1−𝛾𝑛+𝑀2𝑛−^𝑛_𝑖=1𝛼_𝑖 (2.10) and 1/|𝛾|<1/(𝑛+𝑀(2𝑛−∑^𝑛_𝑖=1𝛼_𝑖)),𝛾∈ℂ⧵{0}.

Letting 𝛼_𝑖=𝛿=0, 𝜇_𝑖=1, and 𝑀_𝑖=𝑀 for all 𝑖=1,2,…,𝑛 in Theorem 2.1, we have the following corollary.

Corollary 2.4. Let 𝑓_𝑖(𝑧)∈𝒜 be starlike functions in 𝑈 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀 (𝑀≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator defined in (1.5) is convex in 𝑈, where 1/|𝛾|=1/(𝑛(2𝑀+1)),𝛾∈ℂ⧵{0}.

Theorem 2.5. Let 𝑓_𝑖(𝑧) be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥1,0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀_𝑖 (𝑀_𝑖≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator 𝐺_𝑛(𝑧)=^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)^𝛽𝑖−1d𝑡 (2.11) is in 𝐾(𝛿), where 𝛿=1−^𝑛_𝑖=1𝛽_𝑖−12−𝛼_𝑖𝑀^𝜇𝑖_𝑖⁻¹ (2.12) and ∑^𝑛_𝑖=1|𝛽_𝑖−1|(2−𝛼_𝑖)𝑀^𝜇𝑖_𝑖⁻¹<1,𝛽_𝑖∈ℂ for all 𝑖=1,2,…,𝑛.

Proof. Let 𝑓_𝑖(𝑧) be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥1,0≤𝛼_𝑖<1. It follows from (1.6) that 𝐺_𝑛^′′(𝑧)𝐺^′_𝑛(𝑧)=^𝑛_𝑖=1𝛽_𝑖−1𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧), (2.13) and, hence, 𝑧𝐺_𝑛^′′(𝑧)𝐺^′_𝑛(𝑧)≤^𝑛_𝑖=1𝛽_𝑖−1𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)≤^𝑛_𝑖=1𝛽_𝑖−1⋅𝑓^′_𝑖(𝑧)𝑧𝑓_𝑖(𝑧)^𝜇𝑖⋅𝑓_𝑖(𝑧)𝑧^𝜇𝑖−1. (2.14) Applying the General Schwarz lemma, we have |𝑓_𝑖(𝑧)/𝑧|≤𝑀_𝑖,(𝑧∈𝑈) for all 𝑖=1,2,…,𝑛. Therefore, from (2.14), we obtain 𝑧𝐺_𝑛^′′(𝑧)𝐺^′_𝑛(𝑧)≤^𝑛_𝑖=1𝛽_𝑖−1⋅𝑓^′_𝑖(𝑧𝑧)𝑓_𝑖(𝑧)^𝜇𝑖⋅𝑀^𝜇𝑖_𝑖⁻¹,(𝑧∈𝑈). (2.15) From (1.4) and (2.15), we see that 𝑧𝐺_𝑛^′′(𝑧)𝐺^′_𝑛(𝑧)≤^𝑛_𝑖=1𝛽_𝑖−1⋅2−𝛼_𝑖⋅𝑀^𝜇𝑖_𝑖⁻¹=1−𝛿. (2.16) This completes the proof.

Letting 𝛿=0 and 𝑀_𝑖=𝑀 for all 𝑖=1,2,…,𝑛 in Theorem 2.5, we have the following corollary.

Corollary 2.6. Let 𝑓_𝑖(𝑧) be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥1,0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀 (𝑀≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator defined in (1.6) is convex function in 𝑈, where ^𝑛_𝑖=1𝛽_𝑖−12−𝛼_𝑖𝑀^𝜇𝑖−1=1,𝛽_𝑖∈ℂ∀𝑖=1,2,…,𝑛. (2.17)

Letting 𝜇_𝑖=1 and 𝑀_𝑖=𝑀 for all 𝑖=1,2,…,𝑛 in Theorem 2.5, we have the following corollary.

Corollary 2.7. Let 𝑓_𝑖(𝑧) be in the class 𝑆^∗(𝛼_𝑖),0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(z)|≤𝑀 (𝑀≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator defined in (1.6) is in 𝐾(𝛿), where 𝛿=1−^𝑛_𝑖=1𝛽_𝑖−1⋅2−𝛼_𝑖 (2.18) and ∑^𝑛_𝑖=1|𝛽_𝑖−1|⋅(2−𝛼_𝑖)<1,𝛽_𝑖∈ℂ for all 𝑖=1,2,…,𝑛.

Letting 𝑛=1,𝜇_𝑖=1,𝑀_𝑖=𝑀, and 𝛼_𝑖=𝛿=0 for all 𝑖=1,2,…,𝑛 in Theorem 2.5, we have the following corollary.

Corollary 2.8. Let 𝑓(𝑧) be a starlike function in 𝑈. If |𝑓(𝑧)|≤𝑀 (𝑀≥1,𝑧∈𝑈), then the integral operator ∫^𝑧₀𝑓(𝑡)^𝛽−1d𝑡 is convex in 𝑈, where |𝛽−1|=1/2,𝛽∈ℂ.

Theorem 2.9. Let 𝑓_𝑖(𝑧) be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥1,0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀_𝑖 (𝑀_𝑖≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator 𝐹(𝑧)=^𝑧₀^𝑛_𝑖=1𝑓_𝑖(𝑡)𝑡^𝛽𝑖d𝑡 (2.19) is in 𝐾(𝛿), where 𝛿=1−^𝑛_𝑖=1𝛽_𝑖⋅2−𝛼_𝑖𝑀^𝜇𝑖_𝑖⁻¹+1 (2.20) and ∑^𝑛_𝑖=1|𝛽_𝑖|⋅[(2−𝛼_𝑖)𝑀^𝜇𝑖_𝑖⁻¹+1]<1,𝛽_𝑖∈ℂforall𝑖=1,2,…𝑛.

Proof. Let 𝑓_𝑖(𝑧) be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥1,0≤𝛼_𝑖<1. It follows from (1.7) that 𝑧𝐹^′′(𝑧)𝐹^′(𝑧)=^𝑛_𝑖=1𝛽_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)−1. (2.21) So, from (2.21), we have 𝑧𝐹^′′(𝑧)𝐹^′(𝑧)≤^𝑛_𝑖=1𝛽_𝑖𝑧𝑓^′_𝑖(𝑧)𝑓_𝑖(𝑧)≤+1_𝑛𝑖=1𝛽_𝑖𝑓^′_𝑖(𝑧)𝑧𝑓_𝑖(𝑧)^𝜇𝑖⋅𝑓_𝑖(𝑧)𝑧^𝜇𝑖−1+1. (2.22) Applying the General Schwarz lemma, we have |𝑓_𝑖(𝑧)/𝑧|≤𝑀_𝑖,(𝑧∈𝑈) for all 𝑖=1,2,…,𝑛. Therefore, from (2.22), we obtain 𝑧𝐹^′′(𝑧)𝐹^′(𝑧)≤^𝑛_𝑖=1𝛽_𝑖𝑓^′_𝑖(𝑧)𝑧𝑓_𝑖(𝑧)^𝜇𝑖⋅𝑀^𝜇𝑖_𝑖⁻¹+1,(𝑧∈𝑈). (2.23) From (1.4) and (2.23), we see that 𝑧𝐹^′′(𝑧)𝐹^′(𝑧)≤^𝑛_𝑖=1𝛽_𝑖2−𝛼_𝑖𝑀^𝜇𝑖_𝑖⁻¹+1=1−𝛿. (2.24) This completes the proof.

Letting 𝛿=0 and 𝑀_𝑖=𝑀 for all 𝑖=1,2,…,𝑛 in Theorem 2.9, we have the following corollary.

Corollary 2.10. Let 𝑓_𝑖(𝑧) be in the class 𝐵(𝜇_𝑖,𝛼_𝑖),𝜇_𝑖≥1,0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀 (𝑀≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator defined in (1.7) is convex function in 𝑈, where ^𝑛_𝑖=1𝛽_𝑖2−𝛼_𝑖𝑀^𝜇𝑖−1+1=1,𝛽_𝑖∈ℂ∀𝑖=1,2,…,𝑛. (2.25)

Letting 𝜇_𝑖=1 and 𝑀_𝑖=𝑀 for all 𝑖=1,2,…,𝑛 in Theorem 2.9, we have the following corollary.

Corollary 2.11. Let 𝑓_𝑖(𝑧)∈𝒜 be in the class 𝑆^∗(𝛼_𝑖),0≤𝛼_𝑖<1 for all 𝑖=1,2,…,𝑛. If |𝑓_𝑖(𝑧)|≤𝑀 (𝑀≥1,𝑧∈𝑈) for all 𝑖=1,2,…,𝑛, then the integral operator defined in (1.7) is in 𝐾(𝛿), where 𝛿=1−^𝑛_𝑖=1𝛽_𝑖3−𝛼_𝑖 (2.26) and ∑^𝑛_𝑖=1|𝛽_𝑖|(3−𝛼_𝑖)<1,𝛽_𝑖∈ℂforall𝑖=1,2,…,𝑛.

Letting 𝑛=1, 𝜇_𝑖=1,𝑀_𝑖=𝑀, and 𝛼_𝑖=𝛿=0 for all 𝑖=1,2,…,𝑛 in Theorem 2.9, we have the following corollary.

Corollary 2.12. Let 𝑓(𝑧)∈𝒜 be a starlike function in 𝑈. If |𝑓(𝑧)|≤𝑀 (𝑀≥1,𝑧∈𝑈), then the integral operator ∫^𝑧₀(𝑓(𝑡)/𝑡)^𝛽𝑑𝑡 is convex in 𝑈, where |𝛽|=1/3,𝛽∈ℂ.