# Some Properties of Certain Integral Operators on New Subclasses of Analytic Functions with Complex Order

Authors: Aabed Mohammed; Maslina Darus
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We define new subclasses ofp-valent meromorphic functions with complex order. We prove some properties for certain integral operators on
these subclasses.


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## Body

## 1. Introduction

Let𝕌={z∈ℂ:|z|<1} be the open unit disc in the complex plane ℂ, 𝕌*=𝕌∖{0}, the punctured open unit disk. Let Σp denote the class of meromorphic functions of the form
(1.1)f(z)=1zp+∑n=0∞‍anzn,(p∈ℕ),
which are analytic and p-valent in 𝕌*.Forp=1, we obtain the class of meromorphic functions Σ.We say that a functionf∈Σp is the meromorphic p-valent starlike of order α(0≤α<p) and belongs to the class f∈Σp⋆(α), if it satisfies the inequality
(1.2)-ℜ(zf′(z)f(z))>α.
A function f∈Σp is the meromorphic p-valent convex function of order α(0≤α<p), if f satisfies the following inequality:
(1.3)-ℜ(1+zf′′(z)f′(z))>α,
and we denote this class by ΣKp(α).Many important properties and characteristics of various interesting subclasses of the classΣp of meromorphically p-valent functions were investigated extensively by (among others) Uralegaddi and Somanatha [1, 2], Liu and Srivastava [3, 4], Mogra [5, 6], Srivastava et a1. [7], Aouf et al. [8, 9], Joshi and Srivastava [10], Owa et a1. [11], and Kulkarni et al. [12].Now, forf∈Σp, we define the following new subclasses.Definition 1.1.
Let a functionf∈Σp be analytic in 𝕌*. Then f is in the class Σpℳ(β,b) if it satisfies the inequality
(1.4)ℜ{p-1b(zf′(z)f(z)+p)}<β,
where
(1.5)p∈ℕ,b∈ℂ∖{0},β>p.Definition 1.2.
Let a functionf∈Σp be analytic in 𝕌*. Then f is in the class Σp𝒩(β,b) if it satisfies the inequality
(1.6)ℜ{p-1b(zf′′(z)f′(z)+p+1)}<β,
where
(1.7)p∈ℕ,b∈ℂ∖{0},β>p.Definition 1.3.
Let a functionf∈Σp be analytic in 𝕌*. Then f is in the class ΣFp(β,b) if it satisfies the inequality
(1.8)ℜ{p-1b(z(zf′′(z)+(p+2)f′(z))zf′(z)+(p+1)f(z)+p)}<β,
where
(1.9)p∈ℕ,b∈ℂ∖{0},β>p.Forp=1 in Definitions 1.1, 1.2, and 1.3, we obtain the following new subclasses of meromorphic functions Σ.Definition 1.4.
Let a functionf∈Σ be analytic in 𝕌*. Then f is in the class Σℳ(β,b) if it satisfies the inequality
(1.10)ℜ{1-1b(zf′(z)f(z)+1)}<β,
where
(1.11)b∈ℂ∖{0},β>1.Definition 1.5.
Let a functionf∈Σ  be analytic in 𝕌*. Then f is in the class Σ𝒩(β,b) if it satisfies the inequality
(1.12)ℜ{1-1b(zf′′(z)f′(z)+2)}<β,
where
(1.13)b∈ℂ∖{0},β>1.For b=1 in Definition 1.5, we obtain ΣN(β), the class of meromorphic function, introduced and studied by Wang et al. [13–15] (see [16–18]).Definition 1.6.
Let a functionf∈Σ  be analytic in 𝕌*. Then f is in the class ΣF(β,b) if it satisfies the inequality
(1.14)ℜ{1-1b(z(zf′′(z)+3f′(z))zf′(z)+2f(z)+1)}<β,
where
(1.15)b∈ℂ∖{0},β>1.
Most recently, Mohammed and Darus [19] introduced the following two general integral operators of p-valent meromorphic functions Σp:
(1.16)ℱp,γ1,…,γn(z)=1zp+1∫0z(upf1(u))γ1⋯(upfn(u))γndu;(1.17)𝒥p,γ1,…,γn(z)=1zp+1∫0z(-up+1pf1′(u))γ1⋯(-up+1pfn′(u))γndu,
where
(1.18)n,p∈ℕ,j∈{1,2,3,…,n},γj>0.For p=1 in (1.16) and (1.17), respectively, we obtain the general integral operators ℱ1,γ1,…,γn(z)=ℋ(z) and 𝒥1,γ1,…,γn(z)=ℋγ1,…,γn(z), introduced by the authors in [16, 20].


## 2. Main Results

In this section, considering the above new subclasses we obtain for the integral operatorsF1,γ1,…,γn(z) and 𝒥1,γ1,…,γn(z) some sufficient conditions for a family of functions fi to be in the the above new subclasses.Theorem 2.1.
Letfj∈Σp. If fj∈Σpℳ(βj,b), then
(2.1)ℱp,γ1,…,γn(z)∈ΣFp(μ,b),
where
(2.2)n,p∈ℕ,j∈{1,...,n},γj>0,b∈ℂ∖{0},βj>p,μ=p+∑j=1n‍γj(βj-p).Proof.
A differentiation ofℱp,γ1,…,γn(z) which is defined in (1.16), we get
(2.3)zp+1ℱp,γ1,…,γn′(z)+(p+1)zpℱp,γ1,…,γn(z)=(zpf1(z))γ1⋯(zpfn(z))γn,zp+1ℱp,γ1,…,γn′′(z)+2(p+1)zpℱp,γ1,…,γn′(z)+p(p+1)zp-1ℱp,γ1,…,γn(z)=∑j=1n‍γj(zpfj'(z)+pzp-1fj(z)zpfj(z))[(zpf1(z))γ1⋯(zpfn(z))γn],
Then from , we obtain
(2.4)zp+1ℱp,γ1,…,γn′′(z)+2(p+1)zpℱp,γ1,…,γn′(z)+p(p+1)zp-1ℱp,γ1,…,γn(z)zp+1ℱp,γ1,…,γn′(z)+(p+1)zpℱp,γ1,…,γn(z)=∑j=1n‍γj(fj′(z)fj(z)+pz).
By multiplying (2.4) with z yield,
(2.5)zp+1ℱp,γ1,…,γn′′(z)+2(p+1)zpℱp,γ1,…,γn′(z)+p(p+1)zp-1ℱp,γ1,…,γn(z)zpℱp,γ1,…,γn′(z)+(p+1)zp-1ℱp,γ1,…,γn(z)=∑j=1n‍γj(zfj′(z)fj(z)+p).
That is equivalent to
(2.6)zp+1ℱp,γ1,…,γn′′(z)+(p+2)zpℱp,γ1,…,γn′(z)zpℱp,γ1,…,γn′(z)+(p+1)zp-1ℱp,γ1,…,γn(z)+p=∑j=1n‍γj(zfj′(z)fj(z)+p);z(zℱp,γ1,…,γn′′(z)+(p+2)ℱp,γ1,…,γn′(z))zℱp,γ1,…,γn′(z)+(p+1)ℱp,γ1,…,γn(z)+p=∑j=1n‍γj(zfj′(z)fj(z)+p).
Equivalently, the above can be written as
(2.7)p-1b(z(zℱp,γ1,…,γn′′(z)+(p+2)ℱp,γ1,…,γn′(z))zℱp,γ1,…,γn′(z)+(p+1)ℱp,γ1,…,γn(z)+p)=∑j=1n‍γj{p-1b(zfj′(z)fj(z)+p)}+p-p∑j=1n‍γj.
Taking the real part of both terms of (2.7), we have
(2.8)ℜ{p-1b(z(zℱp,γ1,…,γn′′(z)+(p+2)ℱp,γ1,…,γn′(z))zℱp,γ1,…,γn′(z)+(p+1)ℱp,γ1,…,γn(z)+p)}=∑j=1n‍γjℜ{p-1b(zfj′(z)fj(z)+p)}+p-p∑j=1n‍γj.
Sine fj∈Σpℳ(βj,b), we get
(2.9)ℜ{p-1b(z(zℱp,γ1,…,γn′′(z)+(p+2)ℱp,γ1,…,γn′(z))zℱp,γ1,…,γn′(z)+(p+1)ℱp,γ1,…,γn(z)+p)}<∑j=1n‍γjβj+p-p∑j=1n‍γj.
That is,
(2.10)ℜ{p-1b(z(zℱp,γ1,…,γn′′(z)+(p+2)ℱp,γ1,…,γn′(z))zℱp,γ1,…,γn′(z)+(p+1)ℱp,γ1,…,γn(z)+p)}<p+∑j=1n‍γj(βj-p).
Then
(2.11)ℱp,γ1,…,γn(z)∈ΣFp(μ,b),μ=p+∑j=1n‍γj(βj-p).
This completes the proof.Theorem 2.2.
Letfj∈Σp. If fj∈Σp𝒩(βj,b), then
(2.12)𝒥p,γ1,…,γn(z)∈ΣFp(μ,b),
where
(2.13)n,p∈ℕ,j∈{1,...,n},γj>0,b∈ℂ∖{0},βj>p,μ=p+∑j=1n‍γj(βj-p).Proof.
A differentiation of𝒥p,γ1,…,γn(z), which is defined in (1.17), we get
(2.14)zp+1𝒥p,γ1,…,γn′(z)+(p+1)zp𝒥p,γ1,…,γn(z)=(-zp+1pf1′(z))γ1⋯(-zp+1pfn′(z))γn,zp+1𝒥p,γ1,…,γn′′(z)+2(p+1)zp𝒥p,γ1,…,γn′(z)+p(p+1)zp-1𝒥p,γ1,…,γn(z)=∑j=1n‍γj(zp+1fj′′(z)+(p+1)zpfj′(z)zp+1fj′(z))[(-zp+1pf1′(z))γ1⋯(-zp+1pfn′(z))γn].
Then from , we obtain
(2.15)zp+1𝒥p,γ1,…,γn′′(z)+2(p+1)zp𝒥p,γ1,…,γn′(z)+p(p+1)zp-1𝒥p,γ1,…,γn(z)zp𝒥p,γ1,…,γn′(z)+(p+1)zp-1𝒥p,γ1,…,γn(z)=∑j=1n‍γj(zfj′′(z)fj′(z)+p+1).
That is equivalent to
(2.16)z(z𝒥p,γ1,…,γn′′(z)+(p+2)𝒥p,γ1,…,γn′(z))z𝒥p,γ1,…,γn′(z)+(p+1)𝒥p,γ1,…,γn(z)+p=∑j=1n‍γj(zfj′′(z)fj′(z)+p+1).
Equivalently, the above can be written as
(2.17)p-1b(z(z𝒥p,γ1,…,γn′′(z)+(p+2)𝒥p,γ1,…,γn′(z))z𝒥p,γ1,…,γn′(z)+(p+1)𝒥p,γ1,…,γn(z)+p)=∑j=1n‍γj{p-1b(zfj′′(z)fj′(z)+p+1)}+p-p∑j=1n‍γj.
Taking the real part of both terms of (2.17), we have
(2.18)ℜ{p-1b(z(z𝒥p,γ1,…,γn′′(z)+(p+2)𝒥p,γ1,…,γn′(z))z𝒥p,γ1,…,γn′(z)+(p+1)𝒥p,γ1,…,γn(z)+p)}=∑j=1n‍γjℜ{p-1b(zfj′′(z)fj′(z)+p+1)}+p-p∑j=1n‍γj.
Sine fj∈Σp𝒩(βj,b), we get
(2.19)ℜ{p-1b(z(z𝒥p,γ1,…,γn′′(z)+(p+2)𝒥p,γ1,…,γn′(z))z𝒥p,γ1,…,γn′(z)+(p+1)𝒥p,γ1,…,γn(z)+p)}<∑j=1n‍γjβj+p-p∑j=1n‍γj.
That is,
(2.20)ℜ{p-1b(z(z𝒥p,γ1,…,γn′′(z)+(p+2)𝒥p,γ1,…,γn′(z))z𝒥p,γ1,…,γn′(z)+(p+1)𝒥p,γ1,…,γn(z)+p)}<p+∑j=1n‍γj(βj-p).
Then
(2.21)𝒥p,γ1,…,γn(z)∈ΣFp(μ,b),μ=p+∑j=1n‍γj(βj-p).
This completes the proof.Puttingp=1 in Theorem 2.1, we have the following.Theorem 2.3.
Letfj∈Σ. If fj∈Σℳ(βj,b), then
(2.22)ℋn(z)∈ΣF(μ,b),
where
(2.23)n∈ℕ,j∈{1,…,n},γj>0,b∈ℂ∖{0},βj>1,μ=1+∑j=1n‍γj(βj-1).Puttingp=1 in Theorem 2.2, we have the following.Theorem 2.4.
Letfj∈Σ. If fj∈Σ𝒩(βj,b), then
(2.24)ℋγ1,…,γn(z)∈ΣF(μ,b),
where
(2.25)n∈ℕ,j∈{1,…,n},γj>0,b∈ℂ∖{0},βj>1,μ=1+∑j=1n‍γj(βj-1).for other work that we can look at regarding integral operators see [17, 21, 22].



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