AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation26797210.1155/2012/267972267972Research ArticleSome Convexity Properties of Certain General Integral OperatorsMacarieVasile Marius1BreazDaniel2NoorKhalida Inayat1Department of MathematicsUniversity of PiteştiTargul din Vale Street No. 1, 110040 PiteştiRomaniaupit.ro2Department of Mathematics“1 Decembrie 1918” University of Alba Iulia Nicolae Iorga Street No. 11-13, 510009 Alba IuliaRomaniauab.ro201223112011201220092011081120112012Copyright © 2012 Vasile Marius Macarie and Daniel Breaz.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:|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 Re(zf(z)f(z))>α,(zU) for some α  (0α<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 Re(zf′′(z)f(z)+1)>α,(zU) for some α (0α<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 Re(f(z))>α,(zU).

Frasin and Jahangiri introduced in  the family B(μ,α),  μ0,  0α<1 consisting of functions f𝒜 satisfying the condition |f(z)(zf(z))μ-1|<1-α,(zU). 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: Hn,γ(z)=0zi=1n(tefi(t))1/γdt,Gn(z)=0zi=1n(fi(t))βi-1dt,F(z)=0zi=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 [<xref ref-type="bibr" rid="B5">2</xref>, General Schwarz Lemma]).

Let the function f be regular in the disk UR={z:|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)|MRm|z|m(zUR). The equality can hold only if f(z)=eiθMRmzm, where θ is constant.

2. Main ResultsTheorem 2.1.

Let fi(z)𝒜 be in the class B(μi,αi),  μi0,  0αi<1 for all i=1,2,,n. If |fi(z)|Mi (Mi1,  zU) for all i=1,2,,n, then the integral operator Hn,γ(z)=0zi=1n(tefi(t))1/γdt is in K(δ), where δ=1-1|γ|[n+i=1n(2-αi)Miμi] and 1/|γ|<1/(n+i=1n(2-αi)Miμi),  γ{0}.

Proof.

Let fi𝒜 be in the class B(μi,αi),  μi0,  0αi<1. We have from (1.5) that Hn,γ(z)=0ztn/γe(1/γ)i=1nfi(t)dt,Hn,γ(z)=zn/γe(1/γ)i=1nfi(z). Also Hn,γ′′(z)=1γ(znei=1nfi(z))(1/γ)-1zn-1ei=1nfi(z)(n+zi=1nfi(z)). Then Hn,γ′′(z)Hn,γ(z)=1γ(nz+i=1nfi(z)) and, hence, |zHn,γ′′(z)Hn,γ(z)|=1|γ||n+zi=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 |zHn,γ′′(z)Hn,γ(z)|1|γ|i=1n[1+|fi(z)(zfi(z))μi|Miμi],(zU). From (1.4) and (2.7), we see that |zHn,γ′′(z)Hn,γ(z)|1|γ|[n+i=1n(2-αi)Miμi]=1-δ.

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

Corollary 2.2.

Let fi(z)𝒜 be in the class R(αi),  0αi<1 for all i=1,2,,n. Then the integral operator defined in (1.5) is in K(δ), where δ=1-1|γ|(3n-i=1nαi) and 1/|γ|<1/(3n-i=1nαi),  γ{0}.

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

Corollary 2.3.

Let fi𝒜 be in the class S*(αi),  0αi<1 for all i=1,2,,n. If |fi(z)|M (M1,  zU) for all i=1,2,,n, then the integral operator defined in (1.5) is in K(δ), where δ=1-1|γ|[n+M(2n-i=1nαi)] and 1/|γ|<1/(n+M(2n-i=1nαi)),γ{0}.

Letting αi=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.

Let fi(z)𝒜 be starlike functions in U for all i=1,2,,n. If |fi(z)|M (M1,  zU) 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{0}.

Theorem 2.5.

Let fi(z) be in the class B(μi,αi),  μi1,  0αi<1 for all i=1,2,,n. If |fi(z)|Mi (Mi1,  zU) for all i=1,2,,n, then the integral operator Gn(z)=0zi=1n(fi(t))βi-1dt is in K(δ), where δ=1-i=1n|βi-1|(2-αi)Miμi-1 and i=1n|βi-1|(2-αi)Miμi-1<1,  βi for all i=1,2,,n.

Proof.

Let fi(z) be in the class B(μi,αi),  μi1,  0αi<1. It follows from (1.6) that Gn′′(z)Gn(z)=i=1n(βi-1)fi(z)fi(z), and, hence, |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,  (zU) for all i=1,2,,n. Therefore, from (2.14), we obtain |zGn′′(z)Gn(z)|i=1n|βi-1||fi(z)(zfi(z))μi|Miμi-1,(zU). From (1.4) and (2.15), we see that |zGn′′(z)Gn(z)|i=1n|βi-1|(2-αi)Miμi-1=1-δ. This completes the proof.

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

Corollary 2.6.

Let fi(z) be in the class B(μi,αi),  μi1,  0αi<1 for all i=1,2,,n. If |fi(z)|M (M1,zU) for all i=1,2,,n, then the integral operator defined in (1.6) is convex function in U, where i=1n|βi-1|(2-αi)Mμi-1=1,  βiC  i=1,2,,n.

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

Corollary 2.7.

Let fi(z) be in the class S*(αi),  0αi<1 for all i=1,2,,n. If |fi(z)|M (M1,  zU) for all i=1,2,,n, then the integral operator defined in (1.6) is in K(δ), where δ=1-i=1n|βi-1|(2-αi) and i=1n|βi-1|·(2-αi)<1,  βi for all i=1,2,,n.

Letting n=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.

Let f(z) be a starlike function in U. If |f(z)|M (M1,  zU), then the integral operator 0zf(t)β-1dt is convex in U, where |β-1|=1/2,  β.

Theorem 2.9.

Let fi(z) be in the class B(μi,αi),  μi1,  0αi<1 for all i=1,2,,n. If |fi(z)|Mi (Mi1,  zU) for all i=1,2,,n, then the integral operator F(z)=0zi=1n(fi(t)t)βidt is in K(δ), where δ=1-i=1n|βi|[(2-αi)Miμi-1+1] and i=1n|βi|·[(2-αi)Miμi-1+1]<1,  βi  for  all  i=1,2,n.

Proof.

Let fi(z) be in the class B(μi,αi),  μi1,  0αi<1. It follows from (1.7) that zF′′(z)F(z)=i=1nβi(zfi(z)fi(z)-1). So, from (2.21), we have |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,  (zU) for all i=1,2,,n. Therefore, from (2.22), we obtain |zF′′(z)F(z)|i=1n|βi|(|fi(z)(zfi(z))μi|Miμi-1+1),(zU). From (1.4) and (2.23), we see that |zF′′(z)F(z)|i=1n|βi|[(2-αi)Miμi-1+1]=1-δ. This completes the proof.

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

Corollary 2.10.

Let fi(z) be in the class B(μi,αi),  μi1,  0αi<1 for all i=1,2,,n. If |fi(z)|M (M1,  zU) for all i=1,2,,n, then the integral operator defined in (1.7) is convex function in U, where i=1n|βi|[(2-αi)Mμi-1+1]=1,βiC  i=1,2,,n.

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

Corollary 2.11.

Let fi(z)𝒜 be in the class S*(αi),  0αi<1 for all i=1,2,,n. If |fi(z)|M (M1,  zU) for all i=1,2,,n, then the integral operator defined in (1.7) is in K(δ), where δ=1-i=1n|βi|(3-αi) and i=1n|βi|(3-αi)<1,  βi  for  all  i=1,2,,n.

Letting n=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.

Let f(z)𝒜 be a starlike function in U. If |f(z)|M (M1,  zU), then the integral operator 0z(f(t)/t)βdt is convex in U, where |β|=1/3,  β.

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