# Some Convexity Properties of Certain General Integral Operators

Authors: Vasile Marius Macarie; Daniel Breaz
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction

Let𝒜 be the class of analytic functions defined in the open unit disk of the complex plane U={z∈ℂ:|z|<1}.We denote byS the subclass of 𝒜 consisting of all univalent functions in U. A function f(z)∈S is starlike function of orderα if it satisfies
(1.1)Re(zf′(z)f(z))>α,(z∈U)
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 functionf(z)∈S is convex of orderα if it satisfies
(1.2)Re(zf′′(z)f′(z)+1)>α,(z∈U)
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 functionf(z)∈𝒜 is in the class R(α) if
(1.3)Re(f′(z))>α,(z∈U).Frasin and Jahangiri introduced in [1] the family B(μ,α),μ≥0,0≤α<1 consisting of functions f∈𝒜 satisfying the condition
(1.4)|f′(z)(zf(z))μ-1|<1-α,(z∈U).
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:(1.5)Hn,γ(z)=∫0z∏i=1n(tefi(t))1/γdt,(1.6)Gn(z)=∫0z∏i=1n(fi(t))βi-1dt,(1.7)F(z)=∫0z∏i=1n(fi(t)t)βidt,
where the functions fi(t) are in B(μi,αi) 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 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
(1.8)|f(z)|≤MRm⋅|z|m(z∈UR).
The equality can hold only if
(1.9)f(z)=eiθ⋅MRm⋅zm,
where θ is constant.


## 2. Main Results

Theorem 2.1.
Letfi(z)∈𝒜 be in the class B(μi,αi),μi≥0,0≤αi<1 for all i=1,2,…,n. If |fi(z)|≤Mi (Mi≥1,z∈U) for all i=1,2,…,n, then the integral operator
(2.1)Hn,γ(z)=∫0z∏i=1n(tefi(t))1/γdt
is in K(δ), where
(2.2)δ=1-1|γ|[n+∑i=1n(2-αi)Miμi]
and 1/|γ|<1/(n+∑i=1n(2-αi)Miμi),γ∈ℂ∖{0}.Proof.
Letfi∈𝒜 be in the class B(μi,αi),μi≥0,0≤αi<1. We have from (1.5) that
(2.3)Hn,γ(z)=∫0ztn/γe(1/γ)∑i=1nfi(t)dt,Hn,γ′(z)=zn/γe(1/γ)∑i=1nfi(z).
Also
(2.4)Hn,γ′′(z)=1γ(zne∑i=1nfi(z))(1/γ)-1⋅zn-1⋅e∑i=1nfi(z)(n+z∑i=1nfi′(z)).
Then
(2.5)Hn,γ′′(z)Hn,γ′(z)=1γ(nz+∑i=1nfi′(z))
and, hence,
(2.6)|zHn,γ′′(z)Hn,γ′(z)|=1|γ||n+z∑i=1nfi′(z)|≤1|γ|(∑i=1n|1+zfi′(z)|)≤1|γ|∑i=1n[1+|fi′(z)(zfi(z))μi|⋅|(fi(z)z)μi|⋅|z|].
Applying the General Schwarz lemma, we have |fi(z)/z|≤Mi, for all i=1,2,…,n. Therefore, from (2.6), we obtain
(2.7)|zHn,γ′′(z)Hn,γ′(z)|≤1|γ|∑i=1n[1+|fi′(z)(zfi(z))μi|⋅Miμi],(z∈U).
From (1.4) and (2.7), we see that
(2.8)|zHn,γ′′(z)Hn,γ′(z)|≤1|γ|[n+∑i=1n(2-αi)Miμi]=1-δ.Lettingμi=0 and Mi=M for all i=1,2,…,n in Theorem 2.1, we have the following corollary.Corollary 2.2.
Letfi(z)∈𝒜 be in the class R(αi),0≤αi<1 for all i=1,2,…,n. Then the integral operator defined in (1.5) is in K(δ), where
(2.9)δ=1-1|γ|(3n-∑i=1nαi)
and 1/|γ|<1/(3n-∑i=1nαi),γ∈ℂ∖{0}.Lettingμi=1 and Mi=M for all i=1,2,…,n in Theorem 2.1, we have the following corollary.Corollary 2.3.
Letfi∈𝒜 be in the class S*(αi),0≤αi<1 for all i=1,2,…,n. If |fi(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
(2.10)δ=1-1|γ|[n+M(2n-∑i=1nαi)]
and 1/|γ|<1/(n+M(2n-∑i=1nαi)),γ∈ℂ∖{0}.Lettingαi=δ=0, μi=1, and Mi=M for all i=1,2,…,n in Theorem 2.1, we have the following corollary.Corollary 2.4.
Letfi(z)∈𝒜 be starlike functions in U for all i=1,2,…,n. If |fi(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.
Letfi(z) be in the class B(μi,αi),μi≥1,0≤αi<1 for all i=1,2,…,n. If |fi(z)|≤Mi (Mi≥1,z∈U) for all i=1,2,…,n, then the integral operator
(2.11)Gn(z)=∫0z∏i=1n(fi(t))βi-1dt
is in K(δ), where
(2.12)δ=1-∑i=1n|βi-1|(2-αi)Miμi-1
and ∑i=1n|βi-1|(2-αi)Miμi-1<1,βi∈ℂ for all i=1,2,…,n.Proof.
Letfi(z) be in the class B(μi,αi),μi≥1,0≤αi<1. It follows from (1.6) that
(2.13)Gn′′(z)Gn′(z)=∑i=1n(βi-1)fi′(z)fi(z),
and, hence,
(2.14)|zGn′′(z)Gn′(z)|≤∑i=1n|βi-1||zfi′(z)fi(z)|≤∑i=1n|βi-1|⋅|fi′(z)(zfi(z))μi|⋅|(fi(z)z)μi-1|.
Applying the General Schwarz lemma, we have |fi(z)/z|≤Mi,(z∈U) for all i=1,2,…,n. Therefore, from (2.14), we obtain
(2.15)|zGn′′(z)Gn′(z)|≤∑i=1n|βi-1|⋅|fi′(z)(zfi(z))μi|⋅Miμi-1,(z∈U).
From (1.4) and (2.15), we see that
(2.16)|zGn′′(z)Gn′(z)|≤∑i=1n|βi-1|⋅(2-αi)⋅Miμi-1=1-δ.
This completes the proof.Lettingδ=0 and Mi=M for all i=1,2,…,n in Theorem 2.5, we have the following corollary.Corollary 2.6.
Letfi(z) be in the class B(μi,αi),μi≥1,0≤αi<1 for all i=1,2,…,n. If |fi(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
(2.17)∑i=1n|βi-1|(2-αi)Mμi-1=1,βi∈C∀i=1,2,…,n.Lettingμi=1 and Mi=M for all i=1,2,…,n in Theorem 2.5, we have the following corollary.Corollary 2.7.
Letfi(z) be in the class S*(αi),0≤αi<1 for all i=1,2,…,n. If |fi(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
(2.18)δ=1-∑i=1n|βi-1|⋅(2-αi)
and ∑i=1n|βi-1|·(2-αi)<1,βi∈ℂ for all i=1,2,…,n.Lettingn=1,μi=1,Mi=M, and αi=δ=0 for all i=1,2,…,n in Theorem 2.5, we have the following corollary.Corollary 2.8.
Letf(z) be a starlike function in U. If |f(z)|≤M (M≥1,z∈U), then the integral operator ∫0zf(t)β-1dt is convex in U, where |β-1|=1/2,β∈ℂ.Theorem 2.9.
Letfi(z) be in the class B(μi,αi),μi≥1,0≤αi<1 for all i=1,2,…,n. If |fi(z)|≤Mi (Mi≥1,z∈U) for all i=1,2,…,n, then the integral operator
(2.19)F(z)=∫0z∏i=1n(fi(t)t)βidt
is in K(δ), where
(2.20)δ=1-∑i=1n|βi|⋅[(2-αi)Miμi-1+1]
and ∑i=1n|βi|·[(2-αi)Miμi-1+1]<1,βi∈ℂforalli=1,2,…n.Proof.
Letfi(z) be in the class B(μi,αi),μi≥1,0≤αi<1. It follows from (1.7) that
(2.21)zF′′(z)F′(z)=∑i=1nβi(zfi′(z)fi(z)-1).
So, from (2.21), we have
(2.22)|zF′′(z)F′(z)|≤∑i=1n|βi|(|zfi′(z)fi(z)|+1)≤∑i=1n|βi|(|fi′(z)(zfi(z))μi|⋅|(fi(z)z)μi-1|+1).
Applying the General Schwarz lemma, we have |fi(z)/z|≤Mi,(z∈U) for all i=1,2,…,n. Therefore, from (2.22), we obtain
(2.23)|zF′′(z)F′(z)|≤∑i=1n|βi|(|fi′(z)(zfi(z))μi|⋅Miμi-1+1),(z∈U).
From (1.4) and (2.23), we see that
(2.24)|zF′′(z)F′(z)|≤∑i=1n|βi|[(2-αi)Miμi-1+1]=1-δ.
This completes the proof.Lettingδ=0 and Mi=M for all i=1,2,…,n in Theorem 2.9, we have the following corollary.Corollary 2.10.
Letfi(z) be in the class B(μi,αi),μi≥1,0≤αi<1 for all i=1,2,…,n. If |fi(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
(2.25)∑i=1n|βi|[(2-αi)Mμi-1+1]=1,βi∈C∀i=1,2,…,n.Lettingμi=1 and Mi=M for all i=1,2,…,n in Theorem 2.9, we have the following corollary.Corollary 2.11.
Letfi(z)∈𝒜 be in the class S*(αi),0≤αi<1 for all i=1,2,…,n. If |fi(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
(2.26)δ=1-∑i=1n|βi|(3-αi)
and ∑i=1n|βi|(3-αi)<1,βi∈ℂforalli=1,2,…,n.Lettingn=1, μi=1,Mi=M, and αi=δ=0 for all i=1,2,…,n in Theorem 2.9, we have the following corollary.Corollary 2.12.
Letf(z)∈𝒜 be a starlike function in U. If |f(z)|≤M (M≥1,z∈U), then the integral operator ∫0z(f(t)/t)βdt is convex in U, where |β|=1/3,β∈ℂ.



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