Some Properties of Fractional Calculus and Linear Operators Associated with Certain Subclass of Multivalent Functions

Abstract

We investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivatives, and some well-known integral operators. In special cases, the results presented in this paper provide new approaches to several previously known results.

1. Introduction

Let 𝒜(𝑝) denote the class of functions 𝑓 of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=1𝑎_𝑝+𝑘𝑧^𝑝+𝑘(𝑝∈ℕ={1,2,3,…}), (1.1) which are analytic in the open unit disc 𝒰={𝑧∈ℂ∶|𝑧|<1}. Also, let 𝒦(𝑝) denote the subclass of 𝒜(𝑝) consisting of all functions 𝑓 of the form 𝑓(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑎_𝑝+𝑘𝑧^𝑝+𝑘(𝑝∈ℕ={1,2,3,…},𝑎_𝑝+𝑘≥0), (1.2) For functions 𝑓,𝑔∈𝒜(𝑝), given by 𝑓(𝑧)∶=𝑧^𝑝+^∞_𝑘=1𝑎_𝑝+𝑘𝑧^𝑝+𝑘,𝑔(𝑧)∶=𝑧^𝑝+^∞_𝑘=1𝑏_𝑝+𝑘𝑧^𝑝+𝑘, (1.3) the Hadamard product (or convolution) of 𝑓 and 𝑔 is defined by (𝑓∗𝑔)(𝑧)∶=𝑧^𝑝+^∞_𝑘=1𝑎_𝑝+𝑘𝑏_𝑝+𝑘𝑧^𝑝+𝑘. (1.4) The Ruscheweyh derivative of 𝑓 of order 𝛿+𝑝−1 is defined by 𝐷^{𝛿+𝑝−1}𝑧𝑓(𝑧)∶=^𝑝(1−𝑧)^𝛿+𝑝∗𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=1(𝛿+𝑝)_𝑘(1)_𝑘𝑎_𝑝+𝑘𝑧^𝑝+𝑘, (1.5) where 𝑓 is given by (1.1) and 𝛿>−𝑝. The Ruscheweyh derivative 𝐷^{𝛿+𝑝−1} has been studied by several authors; for example, see [1, 2].

For 𝛽<1, 𝛾≥0, 𝑝∈ℕ, and 𝛿>−𝑝, let 𝒯_{𝛿,𝑝,𝛾}(𝛽) consist of functions 𝑓∈𝒦(𝑝) so that ℜ{𝑒^𝑖𝜂𝐷((1−𝛾)^{𝛿+𝑝−1}𝑓(𝑧)𝑧^𝑝+𝛾(𝐷^{𝛿+𝑝−1}𝑓(𝑧))^′𝑝𝑧^𝑝−1−𝛽)}>0,𝑧∈𝒰, (1.6) for some 𝜂∈ℝ. In [3], the authors obtained four containment results for the class 𝒯_{𝛿,𝑝,𝛾}(𝛽). We denote 𝒯_0,1,𝛾(𝛽)=𝒯_𝛾(𝛽). The class 𝒯_𝛾(𝛽) was studied by Swaminathan [4–6], Barnard et al. [7], Kim and Rønning [8], and others.

In the present paper, we investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivative, and some well-known integral operators defined on the class 𝒯_{𝛿,𝑝,𝛾}(𝛽). In special cases, the results presented here provide new approaches to some previously known results.

Remark 1.1. Throughout this section, we assume that 𝛿+𝑝≥1.

2. Definitions and Lemmas

For the function 𝑓 given by (1.2), we define 𝑞_{𝛿,𝑝,𝛾}𝐷(𝑧)=(1−𝛾)^{𝛿+𝑝−1}𝑓(𝑧)𝑧^𝑝+𝛾(𝐷^{𝛿+𝑝−1}𝑓(𝑧))^′𝑝𝑧^𝑝−1. (2.1) It is easy to verify that 𝑞_{𝛿,𝑝,𝛾}(𝑧)=1−^∞_𝑘=1(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾)𝑝(1)_𝑘𝑎_𝑝+𝑘𝑧^𝑘,𝑎_𝑝+𝑘≥0. (2.2)

Lemma 2.1. Let the function 𝑓 be given by (1.2). Then 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽) if and only if ^∞_𝑘=1(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾)𝑝(1)_𝑘𝑎_𝑝+𝑘≤12(|1+𝑒^𝑖𝜂(1−𝛽)|−|1−𝑒^𝑖𝜂(1−𝛽)|) (2.3) for some 𝜂∈ℝ.

Proof. Using the fact that Re(𝜔)≥0 if and only if |1+𝜔|≥|1−𝜔|, it suffices to show that |1+𝑒^𝑖𝜂(𝑞_{𝛿,𝑝,𝛾}(𝑧)−𝛽)|−|1−𝑒^𝑖𝜂(𝑞_{𝛿,𝑝,𝛾}(𝑧)−𝛽)|≥0, (2.4) where 𝑞_{𝛿,𝑝,𝛾}(𝑧) is defined by (2.1). Letting 𝒜_𝑘=(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾)𝑝(1)_𝑘, (2.5) and assuming (2.3), we obtain |1+𝑒^𝑖𝜂(𝑞_{𝛿,𝑝,𝛾}(𝑧)−𝛽)|−|1−𝑒^𝑖𝜂(𝑞_{𝛿,𝑝,𝛾}(𝑧)−𝛽)|=|1+𝑒^𝑖𝜂(1−𝛽)−𝑒^∞𝑖𝜂_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|−|1−𝑒^𝑖𝜂(1−𝛽)+𝑒^∞𝑖𝜂_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|≥|1+𝑒^𝑖𝜂(1−𝛽)|−|𝑒^𝑖𝜂||^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|−|1+𝑒^𝑖𝜂(1−𝛽)|−|𝑒^𝑖𝜂||^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|≥|1+𝑒^𝑖𝜂(1−𝛽)|−|1−𝑒^𝑖𝜂(1−𝛽)|−2^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘≥0, (2.6) where 𝑧∈𝜕𝒰={𝑧;𝑧∈ℂand|𝑧|=1}. By (2.3), the desired inequality (2.4) follows at once. Conversely, if 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then Re{𝑒^𝑖𝜂(𝑞_{𝛿,𝑝,𝛾}(𝑧)−𝛽)}>0, (2.7) or, equivalently (2.4). This yields |1+𝑒^𝑖𝜂(1−^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘−𝛽)|≥|1−𝑒^𝑖𝜂(1−^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘−𝛽)|, (2.8) which implies that |1+𝑒^𝑖𝜂(1−𝛽)−𝑒^∞𝑖𝜂_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|≥|1−𝑒^𝑖𝜂(1−𝛽)+𝑒^∞𝑖𝜂_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|. (2.9) Squaring the above inequality, choosing the value of 𝑧 on the half line 𝑧=𝑟𝑒^−𝑖𝜃(0≤𝑟<1), and letting 𝑟→1⁻ through this line, we obtain |1+𝑒^𝑖𝜂(1−𝛽)|²+|𝑒^𝑖𝜂|²|^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|²−2|1+𝑒^𝑖𝜂(1−𝛽)||^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|≥|1−𝑒^𝑖𝜂(1−𝛽)|²+|𝑒^𝑖𝜂|²|^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|²+2|1−𝑒^𝑖𝜂(1−𝛽)||^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|. (2.10) Hence we get 2|^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘𝑧^𝑘|(|1−𝑒^𝑖𝜂(1−𝛽)|+|1+𝑒^𝑖𝜂(1−𝛽)|)≤|1+𝑒^𝑖𝜂(1−𝛽)|²−|1−𝑒^𝑖𝜂(1−𝛽)|²=(|1+𝑒^𝑖𝜂(1−𝛽)|−|1−𝑒^𝑖𝜂(1−𝛽)|)(|1+𝑒^𝑖𝜂(1−𝛽)|+|1−𝑒^𝑖𝜂(1−𝛽)|), (2.11) which reduces to 2^∞_𝑘=1𝐴_𝑘𝑎_𝑝+𝑘≤|1+𝑒^𝑖𝜂(1−𝛽)|−|1−𝑒^𝑖𝜂(1−𝛽)|. (2.12) So the desired inequality (2.3) follows upon using (2.5).

Setting 𝜂=𝛿=0 and 𝑝=1 in Lemma 2.1, we get the following result.

Corollary 2.2 ([5, Theorem  2.4]). Let 𝑓(𝑧) be of the form (1.2). Then necessary and sufficient condition for 𝑓 to be in 𝒯_𝛾(𝛽) is ^∞_𝑘=2[1+𝛾(𝑘−1)]𝑎_𝑘≤1−𝛽,𝑎_𝑘≥0. (2.13)

Throughout this paper, we define 𝐸_𝜂,𝛽∶=|1+𝑒^𝑖𝜂(1−𝛽)|−|1−𝑒^𝑖𝜂(1−𝛽)|. (2.14)

As an immediate consequence of Lemma 2.1, we have the following corollary.

Corollary 2.3. Let the function 𝑓 be defined by (1.2). If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then 𝑎_𝑝+𝑘≤𝑝(1)_𝑘𝐸_𝜂,𝛽2(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾) (2.15) for some 𝜂∈ℝ.

Let 𝛼₁,𝛼₂,…,𝛼_𝑝 and 𝛽₁,𝛽₂,…,𝛽_𝑞 (𝑝,𝑞∈ℕ∪{0}, 𝑝≤𝑞+1) be complex numbers such that 𝛽_𝑘≠0,−1,−2,…for 𝑘∈{1,2,…,𝑞}. The generalized hypergeometric function _𝑝𝐹_𝑞 is given by _𝑝𝐹_𝑞(𝛼₁,…,𝛼_𝑝;𝛽₁,…,𝛽_𝑞;𝑧)∶=^∞_𝑛=0(𝛼₁)_𝑛(𝛼₂)_𝑛⋯(𝛼_𝑝)_𝑛(𝛽₁)_𝑛(𝛽₂)_𝑛⋯(𝛽_𝑞)_𝑛𝑧^𝑛𝑛!,𝑧∈𝒰, (2.16) where (𝑥)_𝑛 denotes the Pochhammer symbol defined by (𝑥)_𝑛=Γ(𝑥+𝑛)Γ(𝑥)=𝑥(𝑥+1)(𝑥+2)⋯(𝑥+𝑛−1),for𝑛∈ℕ,(𝑥)₀=1. (2.17) The operator _𝑝𝐹_𝑞 has recently been studied by several authors; for example, [3, 5]. For 𝑝=𝑞+1=2, the above series give rise to the Gaussian hypergeometric series 𝐹(𝑎,𝑏;𝑐;𝑧).

In [9], Hohlov introduced the convolution operator 𝐻_{𝑎,𝑏;𝑐} by 𝐻_{𝑎,𝑏;𝑐}(𝑓)(𝑧)∶=𝑧𝐹(𝑎,𝑏;𝑐;𝑧)∗𝑓(𝑧),𝑓∈𝔸(1). (2.18) Motivated by the operator 𝐻_{𝑎,𝑏;𝑐}, the authors in [3] defined the convolution operators 𝐺^𝑝_𝑓 and 𝐻^𝑝,𝑑_{𝑎,𝑏;𝑐} as follows: 𝐺^𝑝_𝑓(𝑎,𝑏;𝑧)∶=𝐺(𝑧)∶=(^∞_𝑘=1(1+𝑎)(1+𝑏)𝑧(𝑘+𝑎)(𝑘+𝑏)^{𝑘+𝑝−1}𝐻)∗𝑓(𝑧),(𝑎>−1,𝑏>−1),_{𝑝,𝑑𝑎,𝑏;𝑐}(𝑓)(𝑧)=𝐻(𝑧)=𝑧^𝑝₃𝐹₂𝑝(𝑎,𝑏,1+𝑑;𝑝𝑑,𝑐;𝑧)∗𝑓(𝑧), (2.19)(2.20) where 𝑓∈𝒜(𝑝), 𝑝∈ℕ, and 𝑑≥0. For 𝑝=1, the operator 𝐺¹_𝑓(𝑎,𝑏;𝑧) was introduced in [7].

In Section 3, we will make use of the following well-known fractional calculus operators 𝐷_𝑧^−𝜇, 𝐷^𝜇_𝑧, and 𝐷_𝑧^𝑛+𝜇. For an analytic function 𝑓 defined in a simply connected region of the complex 𝑧-plane containing the origin, these operators are defined as follows (See [1, 10]): 𝐷_𝑧^−𝜇1𝑓(𝑧)=Γ(𝜇)^𝑧₀𝑓(𝑡)(𝑧−𝑡)^1−𝜇𝑑𝑡(𝜇>0), (2.21) where multiplicity of (𝑧−𝑡)^𝜇−1 is removed by requiring log(𝑧−𝑡) to be real when 𝑧−𝑡>0; 𝐷^𝜇_𝑧1𝑓(𝑧)=Γ(1−𝜇)^𝑧₀𝑓(𝑡)(𝑧−𝑡)^𝜇𝑑𝑡(0≤𝜇<1), (2.22) where the multiplicity of (𝑧−𝑡)^−𝜇 is removed, as in the definition of 𝐷_𝑧^−𝜇𝑓(𝑧); 𝐷_𝑧^𝑛+𝜇𝑑𝑓(𝑧)∶=^𝑛𝑑𝑧^𝑛𝐷^𝜇_𝑧𝑓(𝑧)(0≤𝜇<1;𝑛∈ℕ₀=ℕ∪{0}). (2.23) By virtue of (2.21), (2.22), (2.23) and in terms of Gamma function, it is wellknown (see for details [11]) that

𝐷_𝑧^−𝜇𝑧^𝑘=Γ(𝑘+1)𝑧Γ(𝑘+𝜇+1)^𝑘+𝜇𝐷(𝑘∈ℕ,𝜇>0),_𝜇𝑧𝑧^𝑘=Γ(𝑘+1)𝑧Γ(𝑘−𝜇+1)^𝑘−𝜇𝐷(𝑘∈ℕ,0≤𝜇<1),_{𝑧𝑞+𝜇}𝑧^𝑘=𝑑^𝑞𝑑𝑧^𝑞𝐷^𝜇_𝑧𝑧^𝑘=Γ(𝑘+1)𝑧Γ(𝑘−𝑞−𝜇+1)^{𝑘−(𝑞+𝜇)}, (2.24) where 𝑞∈ℕ₀, 𝑘∈ℕ, 0≤𝜇<1, and 𝑞≤𝑘 for 𝜇=0.

In Section 4, we will investigate the integral operator 𝐽_𝛿,𝑝 defined by (𝐽_𝛿,𝑝𝑓)(𝑧)=𝛿+𝑝𝑧^𝑝𝑧₀𝑡^𝛿−1𝑓(𝑡)𝑑𝑡, (2.25) where 𝑓∈𝒜(𝑝), 𝛿>−𝑝, and 𝑝∈ℕ. For 𝑝=1 and 𝛿=0, the operator was first defined by Bernardi [12]. Later on several authors studied the operator 𝐽_𝛿,𝑝; for example, see [1, 5].

3. Distortion Inequalities of Convolution Operators

Theorem 3.1. Let the function 𝑓 defined by (1.2) be in the class 𝒯_{𝛿,𝑝,𝛾}(𝛽). Then |𝑞_{𝛿,𝑝,𝛾}1(𝑧)|≥1−2𝐸_𝜂,𝛽|𝑧|,|𝑞_{𝛿,𝑝,𝛾}1(𝑧)|≤1+2𝐸_𝜂,𝛽|𝑧| (3.1) for some 𝜂∈ℝ. Here, 𝑞_{𝛿,𝑝,𝛾}(𝑧) and 𝐸_𝜂,𝛽 are defined, respectively, by (2.1) and (2.14).

Proof. From (2.2), we have |𝑞_{𝛿,𝑝,𝛾}(𝑧)|≥1−|^∞_𝑘=1(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾)𝑝(1)_𝑘𝑎_𝑝+𝑘𝑧^𝑘|≥1−^∞_𝑘=1(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾)𝑝(1)_𝑘𝑎_𝑝+𝑘|𝑧|^𝑘≥1−|𝑧|^∞_𝑘=1(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾)𝑝(1)_𝑘𝑎_𝑝+𝑘. (3.2) Making use of Lemma 2.1, we get |𝑞_{𝛿,𝑝,𝛾}1(𝑧)|≥1−2𝐸_𝜂,𝛽|𝑧| (3.3) for some 𝜂∈ℝ. Similarly, |𝑞_{𝛿,𝑝,𝛾}(𝑧)|≤1+|𝑧|^∞_𝑘=1(𝛿+𝑝)_𝑘(1+(𝑘𝛾/𝑝))(1)_𝑘𝑎_𝑝+𝑘1≤1+2𝐸_𝜂,𝛽|𝑧| (3.4) for some 𝜂∈ℝ. This completes the proof.

We next obtain distortion inequalities for the fractional operaters 𝐷^𝜇_𝑧 and 𝐷_𝑧^−𝜇.

Theorem 3.2. Suppose 𝜇≤(1+𝑘+𝑝)/(𝑘+𝑏+2) and 𝑝<𝑏+1. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then for some 𝜂∈ℝ, one has |𝐷^𝜇_𝑧𝐺(𝑧)|≥Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇(1−𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1−𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽),|𝐷^𝜇_𝑧𝐺(𝑧)|≤Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇(1+𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1−𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽), (3.5) where 0≤𝜇<1, 𝑧∈𝒰, 𝑝∈ℕ, and the operator 𝐺(𝑧)∶=𝐺^𝑝_𝑓(𝑎,𝑏;𝑧) was defined by (2.19).

Proof. By using (2.19), we deduce that 𝐷^𝜇_𝑧𝐺(𝑧)=Γ(𝑝+1)𝑧Γ(𝑝−𝜇+1)^𝑝−𝜇−^∞_𝑘=1(1+𝑎)(1+𝑏)Γ(𝑘+𝑝+1)𝑎(𝑘+𝑎+1)(𝑘+𝑏+1)Γ(𝑘+𝑝−𝜇+1)^𝑝+𝑘𝑧^{𝑝+𝑘−𝜇}. (3.6) Then Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧𝐺(𝑧)=1−^∞_𝑘=1𝜃(𝑘)𝑎_𝑝+𝑘𝑧^𝑘, (3.7) where 𝜃(𝑘)=(1+𝑎)(1+𝑏)Γ(𝑘+𝑝+1)Γ(𝑝−𝜇+1)(𝑘+𝑎+1)(𝑘+𝑏+1)Γ(𝑘+𝑝−𝜇+1)Γ(𝑝+1)(𝑘,𝑝∈ℕ,0≤𝜇<1). (3.8) Since 𝜃(𝑘) is a decreasing function of 𝑘, when 𝜇≤(1+𝑘+𝑝)/(𝑘+𝑏+2), then 0<𝜃(𝑘)≤𝜃(1)=(1+𝑎)(1+𝑏)(1+𝑝).(2+𝑎)(2+𝑏)(1−𝜇+𝑝) (3.9) Also, according to Lemma 2.1 and 𝛿+𝑝≥1, we have 1𝑝(𝛿+𝑝)(𝑝+𝛾)^∞_𝑘=1𝑎_𝑝+𝑘=Γ(𝛿+𝑝+1)(1+(𝛾/𝑝))Γ(𝛿+𝑝)^∞_𝑘=1𝑎_𝑝+𝑘≤^∞_𝑘=1(𝛿+𝑝)_𝑘(1+(𝑘𝛾/𝑝))(1)_𝑘𝑎_𝑝+𝑘≤12𝐸_𝜂,𝛽 (3.10) for some 𝜂∈ℝ. Then ^∞_𝑘=1𝑎_𝑝+𝑘≤𝑝𝐸2(𝛿+𝑝)(𝑝+𝛾)^𝜂,𝛽 (3.11) for some 𝜂∈ℝ. From (3.7) and (3.9), we obtain Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧𝐺(𝑧)≥1−𝜃(1)^∞_𝑘=1𝑎_𝑝+𝑘𝑧^𝑘. (3.12) In view of (3.11), we conclude that |Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧𝐺(𝑧)|≥1−𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1−𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽 (3.13) for some 𝜂∈ℝ, and |Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧𝐺(𝑧)|≤1+𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1−𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽 (3.14) for some 𝜂∈ℝ, which yield (3.5).

By letting 𝑝=1 and 𝑏=𝑎−1>0 in Theorem 3.2, we deduce the following consequence.

Corollary 3.3. If 𝑓∈𝒯_𝛿,1,𝑏(𝛽), then for 0≤𝜇<1, 𝑧∈𝒰, 𝛿≥0 and some 𝜂∈ℝ|𝐷^𝜇_𝑧1𝐺(𝑧)|≥Γ(2−𝜇)|𝑧|^1−𝜇1(1−(3+𝑏)(2−𝜇)(1+𝛿)|𝑧|𝐸_𝜂,𝛽),|𝐷^𝜇_𝑧1𝐺(𝑧)|≤Γ(2−𝜇)|𝑧|^1−𝜇1(1+(3+𝑏)(2−𝜇)(1+𝛿)|𝑧|𝐸_𝜂,𝛽). (3.15)

Theorem 3.4. Let 𝜇>0, 𝑧∈𝒰 and 𝑝∈ℕ. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷_𝑧^−𝜇𝐺(𝑧)|≥Γ(𝑝+1)Γ(𝑝+𝜇+1)|𝑧|^𝑝+𝜇(1−𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1−𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽),|𝐷_𝑧^−𝜇𝐺(𝑧)|≤Γ(𝑝+1)Γ(𝑝+𝜇+1)|𝑧|^𝑝+𝜇(1+𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1−𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽) (3.16)(3.17) for some 𝜂∈ℝ. The operator 𝐺(𝑧)∶=𝐺^𝑝_𝑓(𝑎,𝑏;𝑧) was defined by (2.19).

Proof. In view of (2.19) and (2.22), we have Γ(𝑝+𝜇+1)𝑧Γ(𝑝+1)^{−(𝜇+𝑝)}𝐷_𝑧^−𝜇𝐺(𝑧)=1−^∞_𝑘=1𝜏(𝑘)𝑎_𝑝+𝑘𝑧^𝑘, (3.18) where 𝜏(𝑘)=(1+𝑎)(1+𝑏)Γ(𝑘+𝑝+1)Γ(𝑝+𝜇+1).(𝑘+𝑎+1)(𝑘+𝑏+1)Γ(𝑘+𝑝+𝜇+1)Γ(𝑝+1) (3.19) Since 𝜏 is a decreasing function of 𝑘, it follows that 0<𝜏(𝑘)≤𝜏(1)=(1+𝑎)(1+𝑏)(1+𝑝).(2+𝑎)(2+𝑏)(1+𝜇+𝑝) (3.20) By using (3.11), (3.18), and (3.20), we get |Γ(𝑝+𝜇+1)𝑧Γ(𝑝+1)^{−(𝜇+𝑝)}𝐷_𝑧^−𝜇𝐺(𝑧)|≥1−𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1+𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽,|Γ(𝑝+𝜇+1)𝑧Γ(𝑝+1)^{−(𝜇+𝑝)}𝐷_𝑧^−𝜇𝐺(𝑧)|≤1+𝑝(1+𝑎)(1+𝑏)(1+𝑝)2(2+𝑎)(2+𝑏)(1+𝜇+𝑝)(𝛿+𝑝)(𝛾+𝑝)|𝑧|𝐸_𝜂,𝛽 (3.21) for some 𝜂∈ℝ. The last two inequalities yield (3.16) and (3.17), respectively.

Letting 𝛿=0, 𝑝=1, and 𝑎=𝑏+1=𝜇+2 in Theorem 3.4, we get the following result.

Corollary 3.5. Let 𝐺¹_𝑓(𝜇+2,𝜇+1;𝑧) be defined by (2.19). If 𝑓∈𝒯_𝛾(𝛽), then |𝐷_𝑧^−𝜇𝐺¹_𝑓1(𝜇+2,𝜇+1;𝑧)|≥Γ(𝜇+2)|𝑧|^𝜇+11(1−(𝜇+4)(𝛾+1)|𝑧|𝐸_𝜂,𝛽),|𝐷_𝑧^−𝜇𝐺¹_𝑓1(𝜇+2,𝜇+1;𝑧)|≤Γ(𝜇+2)|𝑧|^𝜇+11(1+(𝜇+4)(𝛾+1)|𝑧|𝐸_𝜂,𝛽) (3.22) for some 𝜂∈ℝ, 𝜇>0, 𝑧∈𝒰, and 𝑝∈ℕ.

We next prove the distortion theorems involving fractional calculus and generalized convolution operator defined by (2.20).

Theorem 3.6. Suppose 0≤𝜇<1, 𝑧∈𝒰, 𝑝∈ℕ, and 𝜂∈ℝ. Also, let 𝑎≤1, 𝑏≤𝑝−𝜇+1, and 𝑐≥𝑝+1. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)|≥Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇(1−𝑎𝑏(𝑝+1)2𝑐(1−𝜇+𝑝)(𝛿+𝑝)|𝑧|𝐸_𝜂,𝛽),|𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)|≤Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇(1+𝑎𝑏(𝑝+1)2𝑐(1−𝜇+𝑝)(𝛿+𝑝)|𝑧|𝐸_𝜂,𝛽) (3.23) for some 𝜂∈ℝ. Here, the operator 𝐻(𝑓)(𝑧) is defined by (2.20).

Proof. By making use of (2.20), we have 𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)=𝐷^𝜇_𝑧{𝑧^𝑝−^∞_𝑘=1(𝑎)_𝑘(𝑏)_𝑘(1+(𝑝/𝛾))_𝑘(𝑝/𝛾)_𝑘(𝑐)_𝑘(1)_𝑘𝑎_𝑝+𝑘𝑧^𝑝+𝑘}=Γ(𝑝+1)𝑧Γ(𝑝−𝜇+1)^𝑝−𝜇−^∞_𝑘=1(𝑎)_𝑘(𝑏)_𝑘(1+(𝑝/𝛾))_𝑘Γ(𝑝+𝑘+1)(𝑝/𝛾)_𝑘(𝑐)_𝑘(1)_𝑘𝑎Γ(𝑝+𝑘−𝜇+1)^𝑝+𝑘𝑧^{𝑝+𝑘−𝜇}. (3.24) It is easy to verify that 1+𝑘𝛾𝑝=(1+(𝑝/𝛾))_𝑘(𝑝/𝛾)_𝑘. (3.25) This implies that Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)=1−^∞_𝑘=1𝜆(𝑘)(1+𝑘𝛾𝑝)𝑎_𝑝+𝑘𝑧^𝑘, (3.26) where 𝜆(𝑘)=(𝑎)_𝑘(𝑏)_𝑘Γ(𝑝+𝑘+1)Γ(𝑝−𝜇+1)(𝑐)_𝑘(1)_𝑘.Γ(𝑝+𝑘−𝜇+1)Γ(𝑝+1) (3.27) Since 𝜆 is a decreasing function of 𝑘, when 𝑎≤1, 𝑐≥𝑝+1 and 𝑏≤𝑝−𝜇+1, we get 0<𝜆(𝑘)≤𝜆(1)=𝑎𝑏(𝑝+1).𝑐(𝑝−𝜇+1) (3.28) From Lemma 2.1 and 𝛿+𝑝≥1, we obtain (𝛿+𝑝)^∞_𝑘=1(1+𝑘𝛾𝑝)𝑎_𝑝+𝑘≤^∞_𝑘=1(𝛿+𝑝)_𝑘(1+(𝑘𝛾/𝑝))(1)_𝑘𝑎_𝑝+𝑘≤12𝐸_𝜂,𝛽 (3.29) for some 𝜂∈ℝ. It follows from (3.26) and (3.28) that |Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)|≥1−𝑎𝑏(𝑝+1)2𝑐(𝑝−𝜇+1)(𝛿+𝑝)|𝑧|𝐸_𝜂,𝛽,|Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)|≤1+𝑎𝑏(𝑝+1)2𝑐(𝑝−𝜇+1)(𝛿+𝑝)|𝑧|𝐸_𝜂,𝛽 (3.30) for some 𝜂∈ℝ, which yield (3.23).

We state an obvious variant of Theorem 3.6 as follows.

Corollary 3.7. Let the function 𝑓 defined by (1.2) be in the class 𝒯_{𝛿,𝑝,𝛾}(𝛽). Also let 𝑎≤1, 𝑏≤𝑝−𝜇+1, 𝑐≥𝑝+1, and 𝛿+𝑝>1. Then |𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)|≥Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇1(1−2|𝑧|𝐸_𝜂,𝛽),|𝐷^𝜇_𝑧𝐻(𝑓)(𝑧)|≤Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇1(1+2|𝑧|𝐸_𝜂,𝛽) (3.31) for some 𝜂∈ℝ, 0≤𝜇<1, 𝑧∈𝒰, and 𝑝∈ℕ.

The proof of Theorem 3.8 is much akin to that of Theorem 3.6, and so it is omitted here.

Theorem 3.8. Let 𝑎≤1, 𝑏≤𝑝+𝜇+1 and 𝑐≥𝑝+1. Also, let 𝜇>0, 𝑝∈ℕ, and 𝑧∈𝒰. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷_𝑧^−𝜇𝐻(𝑓)(𝑧)|≥Γ(𝑝+1)Γ(𝑝+𝜇+1)|𝑧|^𝑝+𝜇(1−𝑎𝑏(𝑝+1)2𝑐(1+𝜇+𝑝)(𝛿+𝑝)|𝑧|𝐸_𝜂,𝛽),|𝐷_𝑧^−𝜇𝐻(𝑓)(𝑧)|≤Γ(𝑝+1)Γ(𝑝+𝜇+1)|𝑧|^𝑝+𝜇(1+𝑎𝑏(𝑝+1)2𝑐(1+𝜇+𝑝)(𝛿+𝑝)|𝑧|𝐸_𝜂,𝛽) (3.32) for some 𝜂∈ℝ.

Next we prove the following.

Theorem 3.9. Let 𝑎≤1, 𝑏≤𝑝−𝜇+1, and 𝑐≥𝑝+1. Also, let 0≤𝜇<1, 𝑝∈ℕ, and 𝑧∈𝒰. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷^𝜇_𝑧(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))|≥Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇(1−𝑎𝑏(𝑝+1)2𝑐(1−𝜇+𝑝)|𝑧|𝐸_𝜂,𝛽),|𝐷^𝜇_𝑧(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))|≤Γ(𝑝+1)Γ(𝑝−𝜇+1)|𝑧|^𝑝−𝜇(1+𝑎𝑏(𝑝+1)2𝑐(1−𝜇+𝑝)|𝑧|𝐸_𝜂,𝛽) (3.33) for some 𝜂∈ℝ.

Proof. We have 𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧)=𝑧^𝑝−^∞_𝑘=1𝐴_𝑘𝑧^𝑝+𝑘, (3.34) where 𝐴_𝑘=(𝛿+𝑝)_𝑘(𝑎)_𝑘(𝑏)_𝑘(1+(𝑝/𝛾))_𝑘(1)_𝑘(𝑝/𝛾)_𝑘(𝑐)_𝑘(1)_𝑘𝑎_𝑝+𝑘. (3.35) Therefore 𝐷^𝜇_𝑧(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))=Γ(𝑝+1)𝑧Γ(𝑝−𝜇+1)^𝑝−𝜇−^∞_𝑘=1𝐴_𝑘Γ(𝑝+𝑘+1)𝑧Γ(𝑝+𝑘−𝜇+1)^{𝑝+𝑘−𝜇}. (3.36) So, from (3.25), we have Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))=1−^∞_𝑘=1(𝛿+𝑝)_𝑘(𝑝+𝑘𝛾)𝑝(1)_𝑘𝜆(𝑘)𝑎_𝑝+𝑘𝑧^𝑘, (3.37) where 𝜆(𝑘) is defined by (3.27). Since 𝜆 is a decreasing function of 𝑘, when 𝑎≤1,𝑏≤𝑝−𝜇+1 and 𝑐≥𝑝+1, then 0<𝜆(𝑘)≤𝜆(1)=𝑎𝑏(𝑝+1).𝑐(𝑝−𝜇+1) (3.38) From (3.37), (3.38), and Lemma 2.1, we find that |Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))|≥1−𝑎𝑏(𝑝+1)2𝑐(1−𝜇+𝑝)|𝑧|𝐸_𝜂,𝛽,|Γ(𝑝−𝜇+1)𝑧Γ(𝑝+1)^𝜇−𝑝𝐷^𝜇_𝑧(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))|≤1+𝑎𝑏(𝑝+1)2𝑐(1−𝜇+𝑝)|𝑧|𝐸_𝜂,𝛽 (3.39) for some 𝜂∈ℝ. The above inequalities lead us to the desired inequalities (3.33).

The proof of the following theorem is similar to Theorem 3.9, and so it is omitted here.

Theorem 3.10. Let 𝑎≤1, 𝑏≤𝑝+𝜇+1, and 𝑐≥𝑝+1. Also, let 𝜇>0, 𝑧∈𝒰, and 𝑝∈ℕ. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷_𝑧^−𝜇(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))|≥Γ(𝑝+1)Γ(𝑝+𝜇+1)|𝑧|^𝑝+𝜇(1−𝑎𝑏(𝑝+1)2𝑐(1+𝜇+𝑝)|𝑧|𝐸_𝜂,𝛽),|𝐷_𝑧^−𝜇(𝐷^{𝛿+𝑝−1}𝐻(𝑓)(𝑧))|≤Γ(𝑝+1)Γ(𝑝+𝜇+1)|𝑧|^𝑝+𝜇(1+𝑎𝑏(𝑝+1)2𝑐(1+𝜇+𝑝)|𝑧|𝐸_𝜂,𝛽) (3.40) for some 𝜂∈ℝ.

Upon setting 𝛿=0 and 𝑝=1 in Theorems 3.6, 3.8, 3.9, and 3.10, we arrive at the following result.

Corollary 3.11. Let 𝑎≤1, 𝑐≥2, 𝜇>0, 𝑧∈𝒰, and 𝑝∈ℕ. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷^𝜇_𝑧1𝐻(𝑓)(𝑧)|≥Γ(2−𝜇)|𝑧|^1−𝜇(1−𝑎𝑏𝑐(2−𝜇)|𝑧|𝐸_𝜂,𝛽),|𝐷^𝜇_𝑧1𝐻(𝑓)(𝑧)|≤Γ(2−𝜇)|𝑧|^1−𝜇(1+𝑎𝑏𝑐(2−𝜇)|𝑧|𝐸_𝜂,𝛽) (3.41) for some 𝜂∈ℝ and 𝑏≤2−𝜇. Furthermore |𝐷_𝑧^−𝜇1𝐻(𝑓)(𝑧)|≥Γ(2+𝜇)|𝑧|^1+𝜇(1−𝑎𝑏𝑐(2+𝜇)|𝑧|𝐸_𝜂,𝛽),|𝐷_𝑧^−𝜇1𝐻(𝑓)(𝑧)|≤Γ(2+𝜇)|𝑧|^1+𝜇(1+𝑎𝑏𝑐(2+𝜇)|𝑧|𝐸_𝜂,𝛽) (3.42) for some 𝜂∈ℝ and 𝑏≤2+𝜇.

Remark 3.12. Under the hypothesis of Corollary 3.11, 𝐷^𝜇_𝑧𝐻(𝑓)(𝑧) and 𝐷_𝑧^−𝜇𝐻(𝑓)(𝑧) are included in disks with its center at origin and radii 𝑟 and 𝑅, respectively, given by 1𝑟=Γ(2−𝜇)(1+𝑎𝑏𝐸𝑐(2−𝜇)^𝜂,𝛽1)(0≤𝜇<1,𝑏≤2−𝜇,forsome𝜂∈ℛ),𝑅=Γ(2+𝜇)(1+𝑎𝑏𝐸𝑐(2+𝜇)^𝜂,𝛽)(𝜇>0,𝑏≤2+𝜇,forsome𝜂∈ℛ). (3.43)

4. Distortion Inequalities of Integral Operator

In this section, we obtain the distortion theorems involving the integral operator 𝐽_𝛿,𝑝 of functions in the class 𝒯_{𝛿,𝑝,𝛾}(𝛽) and fractional calculus operator.

Theorem 4.1. Let 0≤𝜇<1/(𝛿+𝑝+1), 𝑧∈𝒰, and 𝑝∈ℕ. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷^𝜇_𝑧(𝑧^−𝛿(𝐽_𝛿,𝑝1𝑓)(𝑧))|≥Γ(2−𝜇)|𝑧|^−𝜇𝑝(1−2(𝛿+𝑝+1)(𝑝+𝛾)|𝑧|𝐸_𝜂,𝛽),|𝐷^𝜇_𝑧(𝑧^−𝛿(𝐽_𝛿,𝑝1𝑓)(𝑧))|≤Γ(2−𝜇)|𝑧|^−𝜇𝑝(1+2(𝛿+𝑝+1)(𝑝+𝛾)|𝑧|𝐸_𝜂,𝛽) (4.1) for some 𝜂∈ℝ. The operator (𝐽_𝛿,𝑝𝑓)(𝑧) is defined by (2.21).

Proof. Using the definition (2.25), for function 𝑓∈𝒦(𝑝) of the form (1.2), we have (𝐽_𝛿,𝑝𝑓)(𝑧)=𝑧^𝛿−^∞_𝑘=1𝛿+𝑝𝑎𝑝+𝑘+𝛿^𝑝+𝑘. (4.2) So 𝐷^𝜇_𝑧(𝑧^−𝛿(𝐽_𝛿,𝑝1𝑓)(𝑧))=𝑧Γ(2−𝜇)^−𝜇−^∞_𝑘=1(𝛿+𝑝)Γ(𝑘+1)𝑎(𝛿+𝑝+𝑘)Γ(𝑘−𝜇+1)^𝑝+𝑘𝑧^𝑘−𝜇. (4.3) Therefore, we obtain Γ(2−𝜇)𝑧^𝜇𝐷^𝜇_𝑧(𝑧^−𝛿(𝐽_𝛿,𝑝𝑓)(𝑧))=1−^∞_𝑘=1𝜙(𝑘)𝑎_𝑝+𝑘𝑧^𝑘, (4.4) where 𝜙(𝑘)=(𝛿+𝑝)Γ(𝑘+1)Γ(2−𝜇).(𝛿+𝑝+𝑘)Γ(𝑘−𝜇+1) (4.5) Since 𝜙(𝑘) is a decreasing function of 𝑘, when 𝜇<1/(𝛿+𝑝+1), then 0<𝜙(𝑘)≤𝜙(1)=𝛿+𝑝𝛿+𝑝+1. (4.6) By using (3.11), (4.4), and (4.6), we get |Γ(2−𝜇)𝑧^𝜇𝐷^𝜇_𝑧(𝑧^−𝛿(𝐽_𝛿,𝑝𝑝𝑓)(𝑧))|≥1−2(𝛿+𝑝+1)(𝑝+𝛾)|𝑧|𝐸_𝜂,𝛽,|Γ(2−𝜇)𝑧^𝜇𝐷^𝜇_𝑧(𝑧^−𝛿(𝐽_𝛿,𝑝𝑝𝑓)(𝑧))|≤1+2(𝛿+𝑝+1)(𝑝+𝛾)|𝑧|𝐸_𝜂,𝛽 (4.7) for some 𝜂∈ℝ, which prove the inequalities (4.1).

The proof of the following theorem is similar to Theorem 4.1, and so it is omitted here.

Theorem 4.2. Let 𝜇>0, 𝑝∈ℕ, and 𝑧∈𝒰. If 𝑓∈𝒯_{𝛿,𝑝,𝛾}(𝛽), then |𝐷_𝑧^−𝜇(𝑧^−𝛿(𝐽_𝛿,𝑝1𝑓)(𝑧))|≥Γ(𝜇+1)|𝑧|^𝜇𝑝(1−2(𝛿+𝑝+1)(𝜇+1)(𝑝+𝛾)|𝑧|𝐸_𝜂,𝛽),|𝐷_𝑧^−𝜇(𝑧^−𝛿(𝐽_𝛿,𝑝1𝑓)(𝑧))|≤Γ(𝜇+1)|𝑧|^𝜇𝑝(1+2(𝛿+𝑝+1)(𝜇+1)(𝑝+𝛾)|𝑧|𝐸_𝜂,𝛽) (4.8) for some 𝜂∈ℝ.

Acknowledgment

The authors thank the referee for some useful suggestions for improvement of the article.