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

Abstract

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

1. Introduction

Let 𝕌={𝑧∈ℂ∶|𝑧|<1} be the open unit disc in the complex plane ℂ, 𝕌^∗=𝕌⧵{0}, the punctured open unit disk. Let Σ_𝑝 denote the class of meromorphic functions of the form 𝑓(𝑧)=1𝑧^𝑝+^∞_𝑛=0𝑎_𝑛𝑧^𝑛,𝑝∈ℕ, (1.1) which are analytic and 𝑝-valent in 𝕌^∗.

For 𝑝=1, we obtain the class of meromorphic functions Σ.

We say that a function 𝑓∈Σ_𝑝 is the meromorphic 𝑝-valent starlike of order 𝛼(0≤𝛼<𝑝) and belongs to the class 𝑓∈Σ^⋆_𝑝(𝛼), if it satisfies the inequality −ℜ𝑧𝑓^′(𝑧)𝑓(𝑧)>𝛼. (1.2) A function 𝑓∈Σ_𝑝 is the meromorphic 𝑝-valent convex function of order 𝛼(0≤𝛼<𝑝), if 𝑓 satisfies the following inequality: −ℜ1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝛼, (1.3) and we denote this class by Σ𝐾_𝑝(𝛼).

Many important properties and characteristics of various interesting subclasses of the class Σ_𝑝 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, for 𝑓∈Σ_𝑝, we define the following new subclasses.

Definition 1.1. Let a function 𝑓∈Σ_𝑝 be analytic in 𝕌^∗. Then 𝑓 is in the class Σ_𝑝ℳ(𝛽,𝑏) if it satisfies the inequality ℜ1𝑝−𝑏𝑧𝑓^′(𝑧)𝑓(𝑧)+𝑝<𝛽, (1.4) where 𝑝∈ℕ,𝑏∈ℂ⧵{0},𝛽>𝑝. (1.5)

Definition 1.2. Let a function 𝑓∈Σ_𝑝 be analytic in 𝕌^∗. Then 𝑓 is in the class Σ_𝑝𝒩(𝛽,𝑏) if it satisfies the inequality ℜ1𝑝−𝑏𝑧𝑓^′′(𝑧)𝑓^′(𝑧)+𝑝+1<𝛽, (1.6) where 𝑝∈ℕ,𝑏∈ℂ⧵{0},𝛽>𝑝. (1.7)

Definition 1.3. Let a function 𝑓∈Σ_𝑝 be analytic in 𝕌^∗. Then 𝑓 is in the class Σ𝐹_𝑝(𝛽,𝑏) if it satisfies the inequality ℜ1𝑝−𝑏𝑧𝑧𝑓^′′(𝑧)+𝑝+2𝑓^′(𝑧)𝑧𝑓^′(𝑧)+𝑝+1𝑓(𝑧)+𝑝<𝛽, (1.8) where 𝑝∈ℕ,𝑏∈ℂ⧵{0},𝛽>𝑝. (1.9)

For 𝑝=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 function 𝑓∈Σ be analytic in 𝕌^∗. Then 𝑓 is in the class Σ_ℳ(𝛽,𝑏) if it satisfies the inequality ℜ11−𝑏𝑧𝑓^′(𝑧)𝑓(𝑧)+1<𝛽, (1.10) where 𝑏∈ℂ⧵{0},𝛽>1. (1.11)

Definition 1.5. Let a function 𝑓∈Σ  be analytic in 𝕌^∗. Then 𝑓 is in the class Σ_𝒩(𝛽,𝑏) if it satisfies the inequality ℜ11−𝑏𝑧𝑓^′′(𝑧)𝑓^′(𝑧)+2<𝛽, (1.12) where 𝑏∈ℂ⧵{0},𝛽>1. (1.13)For 𝑏=1 in Definition 1.5, we obtain Σ_𝑁(𝛽), the class of meromorphic function, introduced and studied by Wang et al. [13–15] (see [16–18]).

Definition 1.6. Let a function 𝑓∈Σ  be analytic in 𝕌^∗. Then 𝑓 is in the class Σ𝐹(𝛽,𝑏) if it satisfies the inequality ℜ11−𝑏𝑧𝑧𝑓^′′(𝑧)+3𝑓^′(𝑧)𝑧𝑓^′(𝑧)+2𝑓(𝑧)+1<𝛽, (1.14) where 𝑏∈ℂ⧵{0},𝛽>1. (1.15) Most recently, Mohammed and Darus [19] introduced the following two general integral operators of 𝑝-valent meromorphic functions Σ_𝑝: ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=1𝑧^𝑝+1𝑧₀𝑢^𝑝𝑓₁(𝑢)^𝛾1⋯𝑢^𝑝𝑓_𝑛(𝑢)^𝛾𝑛𝒥𝑑𝑢;_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=1𝑧^𝑝+1𝑧₀−𝑢^𝑝+1𝑝𝑓^′₁(𝑢)^𝛾1⋯−𝑢^𝑝+1𝑝𝑓^′_𝑛(𝑢)^𝛾𝑛𝑑𝑢, (1.16)(1.17) where 𝑛,𝑝∈ℕ,𝑗∈{1,2,3,…,𝑛},𝛾_𝑗>0. (1.18)For 𝑝=1 in (1.16) and (1.17), respectively, we obtain the general integral operators ℱ_{1,𝛾1,…,𝛾𝑛}(𝑧)=ℋ(𝑧) and 𝒥_{1,𝛾1,…,𝛾𝑛}(𝑧)=ℋ_{𝛾1,…,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 operators 𝐹_{1,𝛾1,…,𝛾𝑛}(𝑧) and 𝒥_{1,𝛾1,…,𝛾𝑛}(𝑧) some sufficient conditions for a family of functions 𝑓_𝑖 to be in the the above new subclasses.

Theorem 2.1. Let 𝑓_𝑗∈Σ_𝑝. If 𝑓_𝑗∈Σ_𝑝ℳ(𝛽_𝑗,𝑏), then ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)∈Σ𝐹_𝑝𝜇,𝑏, (2.1) where 𝑛,𝑝∈ℕ,𝑗∈{1,...,𝑛},𝛾_𝑗>0,𝑏∈ℂ⧵{0},𝛽_𝑗>𝑝,𝜇=𝑝+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−𝑝. (2.2)

Proof. A differentiation of ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧) which is defined in (1.16), we get 𝑧^𝑝+1ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝑧^𝑝ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=𝑧^𝑝𝑓₁(𝑧)^𝛾1⋯𝑧^𝑝𝑓_𝑛(𝑧)^𝛾𝑛,𝑧^𝑝+1ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+2𝑝+1𝑧^𝑝ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝𝑝+1𝑧^𝑝−1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=^𝑛_𝑗=1𝛾_𝑗𝑧^𝑝𝑓^′_𝑗(𝑧)+𝑝𝑧^𝑝−1𝑓_𝑗(𝑧)𝑧^𝑝𝑓_𝑗(𝑧)𝑧^𝑝𝑓₁(𝑧)^𝛾1⋯𝑧^𝑝𝑓_𝑛(𝑧)^𝛾𝑛, (2.3) Then from , we obtain 𝑧^𝑝+1ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+2𝑝+1𝑧^𝑝ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝𝑝+1𝑧^𝑝−1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧^𝑝+1ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝑧^𝑝ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=^𝑛_𝑗=1𝛾_𝑗𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)+𝑝𝑧. (2.4) By multiplying (2.4) with 𝑧 yield, 𝑧^𝑝+1ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+2𝑝+1𝑧^𝑝ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝𝑝+1𝑧^𝑝−1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧^𝑝ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝑧^𝑝−1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=^𝑛_𝑗=1𝛾_𝑗𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)+𝑝. (2.5) That is equivalent to 𝑧^𝑝+1ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2𝑧^𝑝ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧^𝑝ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝑧^𝑝−1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝=^𝑛_𝑗=1𝛾_𝑗𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧);𝑧+𝑝𝑧ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝=^𝑛_𝑗=1𝛾_𝑗𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)+𝑝. (2.6) Equivalently, the above can be written as 1𝑝−𝑏𝑧𝑧ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=+𝑝_𝑛𝑗=1𝛾_𝑗1𝑝−𝑏𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)+𝑝+𝑝−𝑝^𝑛_𝑗=1𝛾_𝑗. (2.7) Taking the real part of both terms of (2.7), we have ℜ1𝑝−𝑏𝑧𝑧ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=+𝑝_𝑛𝑗=1𝛾_𝑗ℜ1𝑝−𝑏𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)+𝑝+𝑝−𝑝^𝑛_𝑗=1𝛾_𝑗. (2.8) Sine 𝑓_𝑗∈Σ_𝑝ℳ(𝛽_𝑗,𝑏), we get ℜ1𝑝−𝑏𝑧𝑧ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝<^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗+𝑝−𝑝^𝑛_𝑗=1𝛾_𝑗. (2.9) That is, ℜ1𝑝−𝑏𝑧𝑧ℱ^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧ℱ^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝<𝑝+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−𝑝. (2.10) Then ℱ_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)∈Σ𝐹_𝑝𝜇,𝑏,𝜇=𝑝+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−𝑝. (2.11) This completes the proof.

Theorem 2.2. Let 𝑓_𝑗∈Σ_𝑝. If 𝑓_𝑗∈Σ_𝑝𝒩(𝛽_𝑗,𝑏), then 𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)∈Σ𝐹_𝑝𝜇,𝑏, (2.12) where 𝑛,𝑝∈ℕ,𝑗∈{1,...,𝑛},𝛾_𝑗>0,𝑏∈ℂ⧵{0},𝛽_𝑗>𝑝,𝜇=𝑝+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−𝑝. (2.13)

Proof. A differentiation of 𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧), which is defined in (1.17), we get 𝑧^𝑝+1𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝑧^𝑝𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=−𝑧^𝑝+1𝑝𝑓^′₁(𝑧)^𝛾1⋯−𝑧^𝑝+1𝑝𝑓^′_𝑛(𝑧)^𝛾𝑛,𝑧^𝑝+1𝒥^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+2𝑝+1𝑧^𝑝𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝𝑝+1𝑧^𝑝−1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=^𝑛_𝑗=1𝛾_𝑗𝑧^𝑝+1𝑓_𝑗^′′(𝑧)+𝑝+1𝑧^𝑝𝑓^′_𝑗(𝑧)𝑧^𝑝+1𝑓^′_𝑗(𝑧)−𝑧^𝑝+1𝑝𝑓^′₁(𝑧)^𝛾1⋯−𝑧^𝑝+1𝑝𝑓^′_𝑛(𝑧)^𝛾𝑛. (2.14) Then from , we obtain 𝑧^𝑝+1𝒥^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+2𝑝+1𝑧^𝑝𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝𝑝+1𝑧^𝑝−1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧^𝑝𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝑧^𝑝−1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=^𝑛_𝑗=1𝛾_𝑗𝑧𝑓_𝑗^′′(𝑧)𝑓^′_𝑗(𝑧)+𝑝+1. (2.15) That is equivalent to 𝑧𝑧𝒥^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝=^𝑛_𝑗=1𝛾_𝑗𝑧𝑓_𝑗^′′(𝑧)𝑓^′_𝑗(𝑧)+𝑝+1. (2.16) Equivalently, the above can be written as 1𝑝−𝑏𝑧𝑧𝒥^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=+𝑝_𝑛𝑗=1𝛾_𝑗1𝑝−𝑏𝑧𝑓_𝑗^′′(𝑧)𝑓^′_𝑗(𝑧)+𝑝+1+𝑝−𝑝^𝑛_𝑗=1𝛾_𝑗. (2.17) Taking the real part of both terms of (2.17), we have ℜ1𝑝−𝑏𝑧𝑧𝒥^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)=+𝑝_𝑛𝑗=1𝛾_𝑗ℜ1𝑝−𝑏𝑧𝑓_𝑗^′′(𝑧)𝑓^′_𝑗(𝑧)+𝑝+1+𝑝−𝑝^𝑛_𝑗=1𝛾_𝑗. (2.18) Sine 𝑓_𝑗∈Σ_𝑝𝒩(𝛽_𝑗,𝑏), we get ℜ1𝑝−𝑏𝑧𝑧𝒥^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝<^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗+𝑝−𝑝^𝑛_𝑗=1𝛾_𝑗. (2.19) That is, ℜ1𝑝−𝑏𝑧𝑧𝒥^′′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+2𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)𝑧𝒥^′_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝+1𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)+𝑝<𝑝+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−𝑝. (2.20) Then 𝒥_{𝑝,𝛾1,…,𝛾𝑛}(𝑧)∈Σ𝐹_𝑝𝜇,𝑏,𝜇=𝑝+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−𝑝. (2.21) This completes the proof.

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

Theorem 2.3. Let 𝑓_𝑗∈Σ. If 𝑓_𝑗∈Σ_ℳ(𝛽_𝑗,𝑏), then ℋ_𝑛(𝑧)∈Σ𝐹𝜇,𝑏, (2.22) where 𝑛∈ℕ,𝑗∈{1,…,𝑛},𝛾_𝑗>0,𝑏∈ℂ⧵{0},𝛽_𝑗>1,𝜇=1+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−1. (2.23)

Putting 𝑝=1 in Theorem 2.2, we have the following.

Theorem 2.4. Let 𝑓_𝑗∈Σ. If 𝑓_𝑗∈Σ_𝒩(𝛽_𝑗,𝑏), then ℋ_{𝛾1,…,𝛾𝑛}(𝑧)∈Σ𝐹𝜇,𝑏, (2.24) where 𝑛∈ℕ,𝑗∈{1,…,𝑛},𝛾_𝑗>0,𝑏∈ℂ⧵{0},𝛽_𝑗>1,𝜇=1+^𝑛_𝑗=1𝛾_𝑗𝛽_𝑗−1. (2.25)

for other work that we can look at regarding integral operators see [17, 21, 22].

Acknowledgment

This work was supported by UKM-ST-06-FRGS0244-2010 and LRGS/TD/2011/UKM/ICT/03/02.