# Applications of Fractionalq-Calculus to Certain Subclass of Analytic p-Valent Functions with Negative Coefficients

Authors: Ben Wongsaijai; Nattakorn Sukantamala
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

By making use of the concept of fractionalq-calculus, we firstly define q-extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class of q-analogue of p-valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, the q-extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.


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## Body

## 1. Introduction

The formulation of fractional calculus began shortly after the classical calculus was established. Since its definition is based on the concept of a noninteger order either integral or derivative, the fractional calculus had been considered as a subject in pure mathematics with no real applications for a long time. However, the role of fractional calculus has been changed in recent decades. Its applications take place in many fields of mathematical sciences. Extended from the fractional calculus, the fractionalq-calculus is the q-extension of the ordinary fractional calculus. Many results of the study on theory of q-calculus operators in recent decades have been applied in various areas such as problems in the ordinary fractional calculus, optimal control, solutions of q-difference equations, q-differential equations, q-integral equations, and q-transform analysis and also in the geometric function theory of complex analysis.In the field of geometric function theory, various subclasses of analytic functions have been studied from different viewpoints. The fractionalq-calculus is the important tools that are used to investigate subclasses of analytic functions. For example, the extension of the theory of univalent functions can be described by using the theory of q-calculus. In [1], Ismail et al. introduced the generalized class of starlike functions by using the q-difference operator and replaced the right-half plane by a suitable domain. In a similar way, Agrawal and Sahoo [2] introduced the generalized class of starlike functions of order α and Raghavendar and Swaminathan [3] also introduced the class of q-analogue to close-to-convex functions. Moreover, the q-calculus operators, such as fractional q-integral and fractional q-derivative operators, are used to construct several subclasses of analytic functions (see, e.g., [4–8]).In addition, the differential operators have been extensively investigated in the field of geometric function theory. The well-known differential operator defined on the class of analytic functions is introduced by Salagean [9]. This operator was successfully used by many authors and it led to the investigation of several properties of certain known and new classes of analytic functions (see, e.g., [10–14]). However, there are many generalized Salagean operators defined by several authors. In [15], Al-Oboudi defined the generalized Salagean operator by using the technique of convolution structure. In [16], Al-Oboudi and Al-Amoudi used the extension of fractional derivative and fractional integral to define linear multiplier fractional differential operator which yields the Al-Oboudi operator [15] and fractional differential operator. Moreover, Bulut [17] modified the Al-Oboudi and Al-Amoudi operator [16] by introducing nonnegative parameter l in that operator. Recently, Selvakumaran et al. [8] introduced the fractional q-differintegral operator by using the fractional q-calculus operators involving the generalized Al-Oboudi and Al-Amoudi operator [16]. For some recent investigations of these operators on the classes of analytic functions and related topics, such as coefficient estimate, distortion theorem, extreme points, and subordination, we refer to [18–23] and the references cited therein.This paper is organized as follows. In Section2, we propose the q-extension of the Bulut operator [17] which generalized Selvakumaran et al. operator [8]. We also define new class to Rq,λ,l,pδ,m(α) by using this new general differential operator together with q-analogue to p-valent closed-to-convex function. In Section 3, we give linear combination property and coefficient estimate for function belonging to Rq,λ,l,pδ,m(α). By making use of the coefficient estimate, the q-extension of geometric properties for function with negative coefficients TRq,λ,l,pδ,m(α) is given in Section 4. Then, we finish our paper by observations and concluding remarks.


## 2. Preliminaries and Definitions

Letp be a positive integer, and let Ap be the class of analytic functions and p-valent in the unit disk D={z∈C:z<1} that are of the form(1)f(z)=zp+∑k=p+1∞‍akzk.Let Tp be a subclass of Ap consisting of functions f(z) of the form (2)f(z)=zp-∑k=p+1∞‍akzk.In particular, we set A1≡A and T1≡T. For 0≤α<1, let R(α) be the subclass of A consisting of all functions which satisfy Ref′(z)>α in D. The functions in R(α) are called functions of bounded turning. All of those are univalent and close-to-convex in D (see [24]). Similarly, we denote by Rp(α), where 0≤α<1, the class of all functions in Ap which satisfy Ref′(z)/pzp-1>α (see more details in [25, 26]).For the convenience, we provide some basic definitions and concept details ofq-calculus which are used in this paper. In the theory of q-calculus, the q-shifted factorial is defined for α,q∈C, n∈N0≡N∪{0} as a product of n factors by (3)α;qn=1,n=0;1-α1-αq⋯1-αqn-1,n∈N,and in terms of the basic analogue of the gamma function (4)qα;qn=Γqα+n1-qnΓqα,n>0,where the q-gamma function [27, 28] is defined by(5)Γqx=q;q∞1-q1-xqx;q∞,0<q<1.We note that if q<1, the q-shifted factorial (3) remains meaningful for n=∞ as a convergent infinite product: (6)α;q∞=∏k=0∞1-αqk.Here, we recall the following q-analogue definitions given by Gasper and Rahman [27]. The recurrence relation for q-gamma function is given by (7)Γqx+1=xqΓqx,where [x]q=(1-qx)/(1-q), and is called q-analogue of x. It is well known that Γq(x)→Γ(x) as q→1-, where Γ(x) is the ordinary Euler gamma function.In view of the relation(8)limq→1-qα;qn1-qn=αn,we observe that the q-shifted factorial (3) reduces to the familiar Pochhammer symbol (α)n, where (α)n=α(α+1)(α+2)⋯(α+n-1).Letμ∈C be fixed. A set A⊂C is called a μ-geometric set if, for z∈A, μz∈A. Let f be a function defined on a q-geometric set. Jackson’s q-derivative and q-integral of a function on a subset of C are, respectively, given by (see Gasper and Rahman [27], pp. 19–22)(9)Dqfz=fz-fzqz1-q,z≠0,q≠0,∫0z‍f(t)dqt=z1-q∑k=0∞‍qkfzqk.In case f(z)=zn, the q-derivative of f(z), where n is a positive integer, is given by (10)Dqzn=zn-zqn1-qz=nqzn-1.As q→1- and n∈N, we have [n]q=1-qn/1-q=1+q+⋯+qn-1→n.We now recall the definition of the fractionalq-calculus operators of a complex-valued function f(z), which were recently studied by Purohit and Raina [29].Definition 1 (fractionalq-integral operator).
The fractionalq-integral operator Iq,zδ of a function f(z) of order δ(δ>0) is defined by (11)Iq,zδfz=Dq,z-δfz=1Γqδ∫0z‍z-tq1-δftdqt,where f(z) is analytic in a simply connected region in the z-plane containing the origin. Here, the term (z-tq)δ-1 is a q-binomial function defined by (12)z-tqδ-1=zδ-1∏k=0∞‍1-(tq/z)qk1-(tq/z)qδ+k-1=zδΦ10q-δ+1;-;q,tqδz.According to Gasper and Rahman [27], the series Φ10δ;-;q,z is single-valued when arg⁡z<π. Therefore, the function (z-tq)δ-1 in (12) is single-valued when arg⁡(-tqδ/z)<π, tqδ/z<1, and argz<π.Definition 2 (fractionalq-derivative operator).
The fractionalq-derivative operator Dq,zδ of a function f(z) of order δ(0≤δ<1) is defined by (13)Dq,zδfz=Dq,zIq,z1-δfz=1Γq1-δDq∫0z‍z-tq-δftdqt,where f(z) is suitably constrained and the multiplicity of (z-tq)-α is removed as in Definition 1 above.Definition 3 (extended fractionalq-derivative operator).
Under the hypotheses of Definition2, the fractional q-derivative for a function f(z) of order δ is defined by (14)Dq,zδf(z)=Dq,zmIq,zm-δf(z),where m-1≤δ<m,  m∈N0.In addition, the extension ofq-differintegral operator Ωqδ:Ap→Ap, for δ<p+1,  0<q<1, and n∈N, is defined by(15)Ωqδfz=Γqp+1-δΓqp+1zδDq,zδfz=zp+∑k=p+1∞Γqk+1Γqp-δ+1Γqp+1Γqk-δ+1akzk,where Dq,zδ in (15) represents, respectively, a fractional q-integral of f(z) of order δ when -∞<δ<0 and a fractional q-derivative of f(z) of order δ when 0≤δ<p+1. We note that when q→1-, the operator Ωqδ reduces to the operator Ωδ introduced by Owa and Srivastava [30].Now, we define theq-extension of Al-Oboudi type differential operator Dq,λ,l,pδ,m:Ap→Ap, for l,λ≥0, δ<p+1, and m∈N0, which is defined by(16)Dq,λ,l,pδ,0fz=fz,Dq,λ,l,pδ,1fz=pq-λpq+lpq+lΩqδfz+λpq+lzDqΩqδf(z),Dq,λ,l,pδ,2fz=Dq,λ,l,pδ,1Dq,λ,l,pδ,1fz,Dq,λ,l,pδ,2fz⋮Dq,λ,l,pδ,mf(z)=Dq,λ,l,pδ,1Dq,λ,l,pδ,m-1f(z).We note that if f∈A is given by (1), then by (16) we have(17)Dq,λ,l,pδ,mfz=zp+∑k=p+1∞‍Ψq,λ,l,pδ,mkakzk,where (18)Ψq,λ,l,pδ,mk=Γqk+1Γqp-δ+1Γqp+1Γqk-δ+1pq+kq-pqλ+lpq+lm.We note that, by setting appropriated values for the parameters in the operator Dq,λ,l,pδ,m, this operator reduces to many known differential operators. For example, in case l=0 the operator Dq,λ,0,pδ,m is exactly the Selvakumaran et al. operator Dq,λ,pδ,m in [8]. Also, when q→1- the operator Dλ,l,pδ,m reduces to the operator introduced by Bulut [17]. Moreover Bulut [17] noticed that, for suitable parameters l,λ,δ,p, and m, the operator Dλ,l,pδ,m generalizes many operators introduced by several authors, for instance, Salagean [9], Al-Oboudi [15], Al-Oboudi and Al-Amoudi [16], Acu and Owa [31], Acu et al. [32], Cătaş [33], Cho and Srivastava [34], Cho and Kim [35], Kumar et al. [36], Owa and Srivastava [30], and Uralegaddi and Somanatha [37].Next, we define theq-analogous to the function class Rp(α) by Rq,p(α). A function f∈Ap is said to be in the class Rq,p(α) of p-valently closed-to-convex with respect to q-differentiation if and only if(19)ReDqf(z)pqzp-1>α,z∈D,where 0<α<1. In particular, we set Rq,1(α)≡Rq(α). Note that the class Rq(α) generalizes the class Kq (with the function g(z)=z) which was introduced by Raghavendar and Swaminathan [3]. Moreover, we see that Dqf(z)→f′(z), as q→1-. This implies that an inequality ReDqf(z)/[p]qzp-1>α becomes Ref′(z)/pzp-1>α. Hence, the class Rq,p(α) clearly reduces to Rp(α) and satisfies(20)⋂0<q<1‍Rq,pα⊂Rpα⊂Rp.Furthermore, by using the operatorDq,λ,l,pδ,m defined by (16) and q-differentiation, we introduce a new class Rq,λ,l,pδ,m(α) as follows.Letδ<p+1, λ,l≥0, 0≤α<1, and m∈N0. Denote by Rq,λ,l,pδ,m(α) the class of all functions f∈Ap satisfying the condition (21)ReDqDq,λ,l,pδ,mfzpqzp-1>α,z∈D.Denote byTRq,λ,l,pδ,m(α) the class obtained by taking intersection of the class Rq,λ,l,pδ,m(α) with the class Tp. That is, (22)TRq,λ,l,pδ,m(α)≡Rq,λ,l,pδ,m(α)∩Tp.In particular, we set TRq,λ,l,pδ,0(α)≡TRq(α). The special cases of the class Rq,λ,l,pδ,m(α), as q→1-, have been studied by Bulut [38], Al-Oboudi in [15], and Tăut et al. [39] and the special cases of the class TRq,λ,l,pδ,m(α), as q→1-, have been proved by Altintas [40].


## 3. Main Results

### 3.1. General Properties

We begin to derive the linear combination property onRq,λ,l,pδ,m(α) in the following result.Theorem 4.
The classRq,λ,l,pδ,m(α) is convex.Proof.
Letf,g∈Rq,λ,l,pδ,m(α) of the form(23)f(z)=zp+∑k=p+1∞‍akzk,g(z)=zp+∑k=p+1∞‍bkzk.It is sufficient to show that the function h(z)=μf(z)+(1-μ)g(z), where 0≤μ≤1, is in the class Rq,λ,l,pδ,m(α). By (23), we see that (24)hz=zp+∑k=p+1∞‍μak+1-μbkzk.Hence(25)DqDq,λ,l,pδ,mh(z)pqzp-1=1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkμak+1-μbkzk-p=μRe1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p+(1-μ)Re1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p,where Ψq,λ,l,pδ,m(k) is defined by (18). Since f,g∈Rq,λ,l,pδ,m(α), we have (26)ReDqDq,λ,l,pδ,mfzpqzp-1=Re1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p>α,ReDqDq,λ,l,pδ,mgzpqzp-1=Re1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkbkzk-p>α.Applying (26) to (25), we obtain (27)ReDqDq,λ,l,pδ,mh(z)pqzp-1>μα+1-μα=α.Now, the proof is completed.Remark 5.
In casep=1, by letting q→1-, we obtain Theorem 4.1 in [38]. For q→1- with δ=0, we obtain Theorem 2.11 in [15]. Moreover, for δ=0, λ=1, and q→1-, we obtain Theorem 2.1 in [39].Next, we derive some sharp coefficient inequalities contained in the following theorem that are useful in the main results.Theorem 6.
Letf∈Ap be defined by (1) and satisfy the inequality (28)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkak≤1-α,where Ψq,λ,l,pδ,m(k) is defined in (18). Then, f∈Rq,λ,l,pδ,m(α). Moreover, the converse also holds if f∈Tp. The result is sharp.Proof.
Let the functionf∈Ap be defined by (1). To prove this, we consider(29)DqDq,λ,l,pδ,mfzpqzp-1-1=∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p≤∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p.By assumption (28), (29) can be rewritten as(30)DqDq,λ,l,pδ,mfzpqzp-1-1≤∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkak≤1-α.Therefore, we infer that f∈Rq,λ,l,pδ,m(α). To prove the converse, we let function f∈Tp be defined by (2) and belong to the class Rq,λ,l,pδ,m(α). Then, we have(31)RDqDq,λ,l,pδ,mfzpqzp-1=R1-∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p>α.Or, equivalently, (32)R∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p<1-α.In (32), by letting z→1- on the real axis, we obtain inequality (28) as desired. Finally, we note that assertion (28) is sharp, the extremal function being(33)fz=zp-pq1-αp+1qΨq,λ,l,pδ,mp+1zp+1.Now, the proof is completed.Corollary 7.
Iff∈TRq,λ,l,pδ,m(α), then for k=p+1,p+2,…, (34)ak≤1-αkqΨq,λ,l,pδ,mk,where Ψq,λ,l,pδ,m(k) is defined in (18).


## 3.1. General Properties

We begin to derive the linear combination property onRq,λ,l,pδ,m(α) in the following result.Theorem 4.
The classRq,λ,l,pδ,m(α) is convex.Proof.
Letf,g∈Rq,λ,l,pδ,m(α) of the form(23)f(z)=zp+∑k=p+1∞‍akzk,g(z)=zp+∑k=p+1∞‍bkzk.It is sufficient to show that the function h(z)=μf(z)+(1-μ)g(z), where 0≤μ≤1, is in the class Rq,λ,l,pδ,m(α). By (23), we see that (24)hz=zp+∑k=p+1∞‍μak+1-μbkzk.Hence(25)DqDq,λ,l,pδ,mh(z)pqzp-1=1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkμak+1-μbkzk-p=μRe1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p+(1-μ)Re1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p,where Ψq,λ,l,pδ,m(k) is defined by (18). Since f,g∈Rq,λ,l,pδ,m(α), we have (26)ReDqDq,λ,l,pδ,mfzpqzp-1=Re1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p>α,ReDqDq,λ,l,pδ,mgzpqzp-1=Re1+∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkbkzk-p>α.Applying (26) to (25), we obtain (27)ReDqDq,λ,l,pδ,mh(z)pqzp-1>μα+1-μα=α.Now, the proof is completed.Remark 5.
In casep=1, by letting q→1-, we obtain Theorem 4.1 in [38]. For q→1- with δ=0, we obtain Theorem 2.11 in [15]. Moreover, for δ=0, λ=1, and q→1-, we obtain Theorem 2.1 in [39].Next, we derive some sharp coefficient inequalities contained in the following theorem that are useful in the main results.Theorem 6.
Letf∈Ap be defined by (1) and satisfy the inequality (28)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkak≤1-α,where Ψq,λ,l,pδ,m(k) is defined in (18). Then, f∈Rq,λ,l,pδ,m(α). Moreover, the converse also holds if f∈Tp. The result is sharp.Proof.
Let the functionf∈Ap be defined by (1). To prove this, we consider(29)DqDq,λ,l,pδ,mfzpqzp-1-1=∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p≤∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p.By assumption (28), (29) can be rewritten as(30)DqDq,λ,l,pδ,mfzpqzp-1-1≤∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkak≤1-α.Therefore, we infer that f∈Rq,λ,l,pδ,m(α). To prove the converse, we let function f∈Tp be defined by (2) and belong to the class Rq,λ,l,pδ,m(α). Then, we have(31)RDqDq,λ,l,pδ,mfzpqzp-1=R1-∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p>α.Or, equivalently, (32)R∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkakzk-p<1-α.In (32), by letting z→1- on the real axis, we obtain inequality (28) as desired. Finally, we note that assertion (28) is sharp, the extremal function being(33)fz=zp-pq1-αp+1qΨq,λ,l,pδ,mp+1zp+1.Now, the proof is completed.Corollary 7.
Iff∈TRq,λ,l,pδ,m(α), then for k=p+1,p+2,…, (34)ak≤1-αkqΨq,λ,l,pδ,mk,where Ψq,λ,l,pδ,m(k) is defined in (18).


## 4. Geometric Properties for the ClassTRq,λ,l,pδ,m(α)

By observation, Theorem6 gives the necessary and sufficient conditions via coefficient bounded for functions to be in the multivalently analytic function class TRq,λ,l,pδ,m(α). Using this result, we will discuss standard properties for that class in sense of q-theory, such as distortion inequalities, inclusion relations, extreme points, radii of close-to-convexity, starlikeness and convexity, quasi-Hadamard property, and invariant properties. However, some of the mentioned properties can be obtained only in case 0≤δ<p+1 because the monotonicity of the sequence Ψq,λ,l,pδ,m(k) is required to prove those results. The following lemmas guarantee the monotone increasing property for the sequence Ψq,λ,l,pδ,m(k) in case 0≤δ≤p+1 and monotone decreasing in case δ<0.Lemma 8.
Let the sequenceakk=p+∞ be defined by (35)ak=Γqk+1Γqp+1-δΓqp+1Γqk+1-δ.(i)
If0≤δ<p+1, then akk=p+∞ is a nondecreasing sequence and 1≤ak for k≥p.(ii)
Ifδ<0, then akk=p+∞ is a decreasing sequence and ak≤1 for k≥p.Proof.
It is clear that the sequenceakk=p+∞ is nonnegative for δ<p+1. We have that (36)ak+1ak=Γqk+2Γqk+1-δΓqk+1Γqk+2-δ.So, by using (7), we get (37)ak+1ak=k+1qk+1-δq=1-qk+11-qk+1-δ.Since 0<q<1, we see that (38)1-qk+11-qk+1-δ<1,forδ<0,1-qk+11-qk+1-δ≥1,for0≤δ<p+1.Then, for 0≤δ<p+1, we conclude that akk=p+∞ is a nondecreasing sequence and satisfying 1=ap≤ak for all k≥p. Also, for δ<0, the sequence akk=p+∞ is a decreasing sequence and satisfying ak≤ap=1 for all k≥p.Lemma 9.
If0≤δ<p+1, then the sequence Ψq,λ,l,pδ,m(k) defined in (18) is an increasing sequence and satisfying 1≤Ψq,λ,l,pδ,m(k) for all k≥p.Proof.
The result is directly obtained by Lemma8 and the following inequality: (39)1≤pq+kq-pqλ+lpq+l,where l,λ≥0 and k≥p.


### 4.1. Distortion Inequalities

Next, we derive the distortion inequalities for functions in the multivalently analytic functions classTRq,λ,l,pδ,m(α) that will be given by the following results.Theorem 10.
For0≤δ<p+1, suppose that f∈Tp is defined by (2). If f∈TRq,λ,l,pδ,m(α), then (40)fz≤zp+pqp+1q1-αΨq,λ,l,pδ,mp+1zp+1,fz≥zp-pqp+1q1-αΨq,λ,l,pδ,mp+1zp+1.Generally,(41)Dqnfz≤pq!p-nq!zp-n+pqpq!p-n+1q!1-αΨq,λ,l,pδ,mp+1zp-n+1,(42)Dqnfz≥pq!p-nq!zp-n-pqpq!p-n+1q!1-αΨq,λ,l,pδ,mp+1zp-n+1,where [n]q!=[n]q[n-1]q⋯[1]q. The estimations in (40)–(42) are sharp.Proof.
Let the functionf∈Tp be defined by (2) and belong to the class Rq,λ,l,pδ,m(α). In virtue of Theorem 6 and Lemma 9, we have(43)p+1qpqΨq,λ,l,pδ,mp+1∑k=p+1∞‍ak≤∑k=p+1∞‍kqpqΨq,λδ,mkak≤1-α.From (43), the consequence is that(44)∑k=p+1∞‍ak≤1-αpqp+1q1Ψq,λ,l,pδ,mp+1.Since f∈Tp, it is easy to see that(45)zp-zp+1∑k=p+1∞‍ak≤fz≤zp+zp+1∑k=p+1∞‍ak.The conjunction of (44) and (45) yields assertions (40) of Theorem 10. Hence, (41) and (42) follow from(46)Dqnfz≤pq!p-nq!zp-n+p+1q!p-n+1q!zp-n+1∑k=p+1∞ak,Dqnfz≥pq!p-nq!zp-n-p+1q!p-n+1q!zp-n+1∑k=p+1∞‍ak.Finally, we note that assertions (40)–(42) are sharp, since equalities are attained by the function(47)fz=zp-pqp+1q1-αΨq,λ,l,pδ,mp+1zp+1.Now, the proof is completed.Remark 11.
By lettingz→1-, Theorem 10 demonstrates that the disk z<1 is mapped onto a domain that contains the disk(48)w<1-pqp+1q1-αΨq,λ,l,pδ,mp+1,under any multivalently analytic function f∈TRq,λ,l,pδ,m(α), and onto a domain that contains the disk(49)w<1-1-α1-qp1-qp+1,by any f∈TRq,p(α).


### 4.2. Inclusion Relation

In the following results, we obtain some inclusion relation for the parametersm, λ, and l of the class TRq,λ,l,pδ,m(α).Theorem 12.
If0≤l1≤l2 and 0≤λ1≤λ2, then(50)TRq,λ2,l2,pδ,mα⊂TRq,λ1,l1,pδ,mα,and TRq,λ2,l2,pδ,m(α)≠TRq,λ1,l1,pδ,m(α) if those parameters satisfy either l1≠l2 or λ1≠λ2.Proof.
The inclusion relation is directly obtained by Theorem6 and the inequality(51)∑k=p+1∞‍kqpqΨq,λ1,l1,pδ,mkak≤∑k=p+1∞‍kqpqΨq,λ2,l2,pδ,mkak.In case l1≠l2 or λ1≠λ2, we see that (52)f0z=zp-pqp+1q1-αΨq,λ1,l1,pδ,mp+1zp+1belongs to the class TRq,λ1,l1,pδ,m(α) but does not belong to the class TRq,λ2,l2,pδ,m(α), which implies that TRq,λ2,l2,pδ,m(α)≠TRq,λ1,l1,pδ,m(α). Now, the proof is completed.Applying Theorem6 and Lemma 9, we obtain another inclusion relation as follows.Theorem 13.
If0≤δ<p+1, then (53)TRq,λ,l,pδ,m+1(α)⊂TRq,λ,l,pδ,mα.


### 4.3. Extreme Points

Now, let us determine extreme points of the classTRq,λ,l,pδ,m(α).Theorem 14 (extreme points).
Letfp(z)=zp and(54)fkz=zp-pq1-αkqΨq,λ,l,pδ,mkzk,k≥p+1.Then f(z) is in the class TRq,λ,l,pδ,m(α) if and only if it can be expressed in the form(55)fz=μpzp+∑k=p+1∞‍μkfkz,where μk≥0 and ∑k=p∞‍μk=1.Proof.
Let the functionf(z)∈Tp be defined by (2). Since ∑k=p+1∞‍μk=1, we then have (56)fz=zp-∑k=p+1∞‍μkpq1-αkqΨq,λ,l,pδ,mkzk.Now, we obtain (57)1-α∑k=p+1∞kqΨq,λ,l,pδ,mkpq1-αμkpq1-αkqΨq,λ,l,pδ,mk=1-α∑k=p+1∞‍μk≤1-α.Thus, f∈TRq,λ,l,pδ,m(α) by Theorem 6. Conversely, suppose that f∈TRq,λ,l,pδ,m(α). We may set (58)μk=kqpqΨq,λ,l,pδ,mk1-αak,k≥p+1and μp=1-∑k=p+1∞‍μk. Then we have f(z)=μpzp-∑k=p+1∞‍μkfk(z). This completes the proof of Theorem 14.


### 4.4. Radii of Generalized Close-to-Convexity, Starlikeness, and Convexity

Now, the discussions on radii of generalized close-to-convexity, starlikeness, and convexity for the classTRq,λ,l,pδ,m(α) are given by the following results. In order to establish, we will also require the use of those classes of functions. First of all, a function f∈Ap is said to be in the class Sq,p∗(β) of p-valently starlike with respect to q-differentiation of order β(0≤β<p) if it satisfies the inequality(59)RezDqfzfz>β,z∈D.Furthermore, a function f∈Ap is said to be in the class Cq,p(β) of p-valently convex with respect to q-differentiation of order β(0≤β<p) if it satisfies the inequality(60)Re1+zDq2fzDqfz>β,z∈D.Both Sq,p∗(β) and Cq,p(β) were introduced by Selvakumaran et al. [8]. However, we consider the case 0≤β<[p]q instead of 0≤β<p. The definition of q-analogous of p-valently closed-to-convex was already recalled in (19).Theorem 15.
For0≤δ<p+1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently closed-to-convex with respect to q-differentiation of order α.Proof.
By Theorem13, we obtain(61)TRq,λ,l,pδ,mα⊂TRq,λ,l,pδ,m-1α⊂⋯⊂TRq,pα.This completes the proof.In general, for0≤α<β<1, the function f∈TRq,p(α) does not necessarily belong to the class TRq,p(β). We then derive the radii of generalized close-to-convexity order 0≤α<β<1 for the function f∈TRq,λ,l,pδ,m(α).Theorem 16.
For0≤δ<p+1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently closed-to-convex with respect to q-differentiation of order β in z<r1, where(62)r1=inf⁡k≥p+11-β1-αΨq,λ,l,pδ,mk1/k-p.Proof.
It is sufficient to show thatDqf(z)/[p]qzp-1-1<1-β. That is,(63)Dqfzpqzp-1-1≤∑k=p+1∞‍kqpqakzk-p≤1-β.Since f∈TRq,λ,l,pδ,m(α) and by application of Theorem 6, we obtain(64)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkak≤1-α.Hence, (63) is true if (65)z≤1-β1-αΨq,λ,l,pδ,mk1/k-p,k≥p+1.This completes the proof.Next, we obtain the radii of generalized starlikeness of orderβ in the following result.Theorem 17.
For0≤δ<p+1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently starlike with respect to q-differentiation of order β(0≤β<[p]q) in z<r2 where(66)
r
2
=
inf
⁡
k
≥
p
+
1
1
1
-
α
·
k
q
p
q
-
β
p
q
k
q
-
β
Ψ
q
,
λ
,
l
,
p
δ
,
m
k
1
/
k
-
p
.Proof.
We have to show thatzDqf(z)/f(z)-[p]q<[p]q-β. That is,(67)zDqfzfz-pq=zpqzp-1-∑k=p+1∞‍kqakzk-1zp-∑k=p+1∞‍akzk-pq≤∑k=p+1∞‍kq-pqakzk-p1-∑k=p+1∞‍akzk-p≤pq-β.Hence, (67) is true if(68)∑k=p+1∞kq-βakzk-p≤pq-β.By using (64), we can say (68) is true if(69)z<11-αkqpq-βpqkq-βΨq,λ,l,pδ,mk1/k-p,k≥p+1,which completes the proof.Corollary 18.
If0≤δ<p+1, then (70)TRq,λ,l,pδ,mα⊂Sq,p∗.Proof.
By Theorem17, we see that a function f∈TRq,λ,l,pδ,m(α) is p-valently starlike with respect to q-differentiation (β=0) in z<r2 where (71)r2=inf⁡k≥p+111-αΨq,λ,l,pδ,mk1/k-p≥inf⁡k≥p+111-α1/k-p·inf⁡k≥p+1Ψq,λ,l,pδ,mk1/k-p.It is easy to see that 1/1-α1/k-p is a decreasing sequence and limk→+∞1/1-α1/k-p=1. This implies(72)inf⁡k≥p+111-α1/k-p=limk→+∞11-α1/k-p=1.Moreover, by using Lemma 9, we obtain(73)inf⁡k≥p+1Ψq,λ,l,pδ,mk1/k-p≥1.Then, we have r2≥1. That is, f∈TRq,λ,l,pδ,m(α) is p-valently starlike with respect to q-differentiation in z∈D. The proof is completed.Next, we obtain the radii of generalized convexity of orderβ, where β≤2[p]q-[p-1]q-1, in the following result.Theorem 19.
For0≤δ<p+1 and β≤2[p]q-[p-1]q-1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently convex with respect to q-differentiation of order β(0≤β<[p]q) in z<r3, where (74)r3=inf⁡k≥p+111-α2pq-p-1q-β-11+k-1q-βhhhhhhhh·2pq-p-1q-β-11+k-1q-βΨq,λ,l,pδ,mk1/k-p.Proof.
We have to show that1+zDq2f(z)/Dqf(z)-[p]q<[p]q-β. That is,(75)1+zDq2fzDqfz-pq=1+zpqp-1qzp-2∑k=p+1∞‍hhhhhhhhhhhhh-∑k=p+1∞‍kqk+1qakzk-2hhhhhhhhhzpqp-1qzp-2-∑k=p+1∞‍kqk+1qakzk-2·pqzp-1-∑k=p+1∞‍kqakzk-1-1hhhhhhzpqp-1qzp-2-∑k=p+1∞‍kqk+1qakzk-2-pq≤pq1+p-1q-pq∑k=p+1∞‍hhhhhhh∑k=p+1∞‍+∑k=p+1∞‍kq1+k-1q-pqakzk-phhhhhh·pq-∑k=p+1∞‍kqakzk-p-1≤pq-β.Hence, (75) is true if(76)pq1+p-1q-pq+∑k=p+1∞‍kq1+k-1q-pqakzk-p≤pq-βpq-∑k=p+1∞‍kqakzk-p,or equivalently(77)∑k=p+1∞‍kq1+k-1q-βakzk-p≤pq2pq-p-1q-β-1.Since β≤2[p]q-[p-1]q-1, by using (64), we can say (77) is true if(78)z<11-α2pq-p-1q-β-11+k-1q-βΨq,λ,l,pδ,mk1/k-p,k≥p+1.The proof is completed.


### 4.5. Quasi-Hadamard Properties

In this section, we derive the quasi-Hadamard (convolution) properties. Before we derive the result, we recall the definition of the quasi-Hadamard properties. For any functionsfj∈Tp, j=1,2,3,…,n of the form(79)fjz=zp-∑k=p+1∞‍ak,jzk,the quasi-Hadamard product (f1⊗f2⊗⋯⊗fn)(z) is defined by (80)f1⊗f2⊗⋯⊗fnz=zp-∑k=p+1∞‍∏j=1n‍ak,jzk,z∈D.Next, we derive the quasi-Hadamard properties for the classTRq,λ,l,pδ,m(α). Using the techniques of Schild and Silverman [41] with Theorem 6, we prove the following results.Theorem 20.
For0≤δ<p+1, suppose that fj∈TRq,λ,l,pδ,m(αj), j=1,2,…,n; then (f1⊗f2⊗⋯⊗fn)∈TRq,λ,l,pδ,m(γ), where(81)γ=1-pqp+1qΨq,λ,l,pδ,mp+1n-1∏j=1n(1-αj).Proof.
To prove this theorem, we use the principle of mathematical induction onj. Let the functions fj∈Tp, for j=1,2 of the form(82)fjz=zp-∑k=p+1∞‍ak,jzk,for j=1,2 and k≥2. Since fj∈TRq,λ,l,pδ,m(αj) for j=1,2, by Theorem 6, we see that(83)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mk1-αjak,j≤1,forj=1,2.According to Theorem 6, it is sufficient to prove that (84)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mk1-γak,1ak,2≤1,where γ is defined in (81). Applying Cauchy-Schwarz’s inequality to (83) for j=1,2, we have the following inequality:(85)∑k=p+1∞‍kqpqΨq,λ,l,pδ,m(k)(1-α1)(1-α2)ak,1ak,2≤∑k=2∞‍kqpqΨq,λ,l,pδ,m(k)1-α1ak,1∑k=2∞‍kqpqΨq,λ,l,pδ,m(k)1-α2ak,2≤1.From (84) and (85), if the following inequality (86)1-α11-α21-γak,1ak,2≤1,for all k≥p+1, is satisfied, it can be concluded that (f1⊗f2)∈TRq,λ,l,pδ,m(γ). Now, applying Corollary 7, inequality (86) will be held if (87)pqkq1-α11-α2Ψq,λ,l,pδ,mk≤1-γ.By Lemma 9, we see that (88)pqkq1-α11-α2Ψq,λ,l,pδ,mk≤pqp+1q1-α11-α2Ψq,λ,l,pδ,mp+1:=1-γ.This yields our desired inequality (86). Now, we have (f1⊗f2)∈TRq,λ,l,pδ,m(γ). Next, we let the functions fj∈Tp, for j=1,2,…,N+1 and fj∈TRq,λδ,m(αj) for j=1,2,…,N+1. Suppose that (89)f1⊗f2⊗⋯⊗fN∈TRq,λ,l,pδ,mγ,where (90)γ:=1-pqp+1qΨq,λ,l,pδ,mp+1N-1∏j=1N(1-αj).Then, by means of the above technique, it can be shown that (91)f1⊗f2⊗⋯⊗fN⊗fN+1∈TRq,λδ,mγ′,where (92)γ′:=1-pqp+1q1-αN+11-γΨq,λ,l,pδ,mp+1=1-pqp+1qΨq,λ,l,pδ,mp+1N∏j=1N+1‍1-αj.By Lemma 9, we have 0<γ<1. This completes the proof of the theorem.


### 4.6. Invariant Properties

In the following results, we discuss invariant properties of the classTRq,λ,l,pδ,m(α) via Theorem 6. We consider the formerly studied operators in terms of the standard convolution formula; we choose g as a fixed function in Ap such that (f∗g)(z) exists for any f∈Ap. For various choices of g we get different linear operators that have been studied in the recent past.According to Theorem6, we easily obtain the following properties.Theorem 21.
Forδ<p+1, if the function g∈Ap is of the form (93)gz=zp+∑k=p+1∞‍μkzk,where μk≤1 for k≥p+1, then (f∗g)∈TRq,λ,l,pδ,m(α) for each f∈TRq,λ,l,pδ,m(α).Next, we recall the definition of Bernardi’s integral operator. For nonnegative real numberγ and f∈Ap, Bernardi-Libera’s integral operator Lγf:Ap→Ap is defined as follows:(94)Lγfz=γ+1zγ∫0z‍tγ-1ftdt,which was studied by Bernardi in [42]. Also, their properties for γ=1 are reported in [43, 44]. By using the concept of q-calculus, we introduce q-analogous to Bernardi’s integral operator defined by (95)Lq,γfz=γ+1qzγ∫0z‍tγ-1ftdqt.From (95), we have verified that (96)Lq,γfz=γ+pqzγz1-q∑j=0∞‍qjzqjγ-1fzqj=γ+pq1-q∑j=0∞‍qjγfzqj=γ+pq1-q∑j=0∞‍qjγ∑k=p∞‍qjkakzk,ap=1=γ+pq∑k=p∞‍∑j=0∞‍1-qqj(γ+k)akzk=zp+∑k=p+1∞‍γ+pqγ+kqakzk.That is, (97)Lq,γfz:=f∗gzp=z+∑k=p+1∞‍γ+pqγ+kqakzk,where g=zp+∑k=p+1∞‍[γ+p]q/[γ+k]qzk. It is clear that [γ+p]q/[γ+k]q≤1 for k≥p+1. Then, we obtain the invariant properties under integral operator Lq,γ as follows.Theorem 22.
Forδ<p+1, and γ>0, the class TRq,λ,l,pδ,m(α) is invariant under the integral operator Lq,γ defined in (95). That is,(98)Lq,γTRq,λ,l,pδ,mα⊂TRq,λ,l,pδ,mα.Moreover, in the view of the definition of fractionalq-integral and Lemma 8, we obtain the invariant properties under fractional q-integral.Theorem 23.
Forδ<p+1, the class TRq,λ,l,pδ,m(α) is invariant under the integral operator Iq,zδ defined in (11). That is, (99)Iq,zδTRq,λ,l,pδ,m(α)⊂TRq,λ,l,pδ,mα.


## 4.1. Distortion Inequalities

Next, we derive the distortion inequalities for functions in the multivalently analytic functions classTRq,λ,l,pδ,m(α) that will be given by the following results.Theorem 10.
For0≤δ<p+1, suppose that f∈Tp is defined by (2). If f∈TRq,λ,l,pδ,m(α), then (40)fz≤zp+pqp+1q1-αΨq,λ,l,pδ,mp+1zp+1,fz≥zp-pqp+1q1-αΨq,λ,l,pδ,mp+1zp+1.Generally,(41)Dqnfz≤pq!p-nq!zp-n+pqpq!p-n+1q!1-αΨq,λ,l,pδ,mp+1zp-n+1,(42)Dqnfz≥pq!p-nq!zp-n-pqpq!p-n+1q!1-αΨq,λ,l,pδ,mp+1zp-n+1,where [n]q!=[n]q[n-1]q⋯[1]q. The estimations in (40)–(42) are sharp.Proof.
Let the functionf∈Tp be defined by (2) and belong to the class Rq,λ,l,pδ,m(α). In virtue of Theorem 6 and Lemma 9, we have(43)p+1qpqΨq,λ,l,pδ,mp+1∑k=p+1∞‍ak≤∑k=p+1∞‍kqpqΨq,λδ,mkak≤1-α.From (43), the consequence is that(44)∑k=p+1∞‍ak≤1-αpqp+1q1Ψq,λ,l,pδ,mp+1.Since f∈Tp, it is easy to see that(45)zp-zp+1∑k=p+1∞‍ak≤fz≤zp+zp+1∑k=p+1∞‍ak.The conjunction of (44) and (45) yields assertions (40) of Theorem 10. Hence, (41) and (42) follow from(46)Dqnfz≤pq!p-nq!zp-n+p+1q!p-n+1q!zp-n+1∑k=p+1∞ak,Dqnfz≥pq!p-nq!zp-n-p+1q!p-n+1q!zp-n+1∑k=p+1∞‍ak.Finally, we note that assertions (40)–(42) are sharp, since equalities are attained by the function(47)fz=zp-pqp+1q1-αΨq,λ,l,pδ,mp+1zp+1.Now, the proof is completed.Remark 11.
By lettingz→1-, Theorem 10 demonstrates that the disk z<1 is mapped onto a domain that contains the disk(48)w<1-pqp+1q1-αΨq,λ,l,pδ,mp+1,under any multivalently analytic function f∈TRq,λ,l,pδ,m(α), and onto a domain that contains the disk(49)w<1-1-α1-qp1-qp+1,by any f∈TRq,p(α).


## 4.2. Inclusion Relation

In the following results, we obtain some inclusion relation for the parametersm, λ, and l of the class TRq,λ,l,pδ,m(α).Theorem 12.
If0≤l1≤l2 and 0≤λ1≤λ2, then(50)TRq,λ2,l2,pδ,mα⊂TRq,λ1,l1,pδ,mα,and TRq,λ2,l2,pδ,m(α)≠TRq,λ1,l1,pδ,m(α) if those parameters satisfy either l1≠l2 or λ1≠λ2.Proof.
The inclusion relation is directly obtained by Theorem6 and the inequality(51)∑k=p+1∞‍kqpqΨq,λ1,l1,pδ,mkak≤∑k=p+1∞‍kqpqΨq,λ2,l2,pδ,mkak.In case l1≠l2 or λ1≠λ2, we see that (52)f0z=zp-pqp+1q1-αΨq,λ1,l1,pδ,mp+1zp+1belongs to the class TRq,λ1,l1,pδ,m(α) but does not belong to the class TRq,λ2,l2,pδ,m(α), which implies that TRq,λ2,l2,pδ,m(α)≠TRq,λ1,l1,pδ,m(α). Now, the proof is completed.Applying Theorem6 and Lemma 9, we obtain another inclusion relation as follows.Theorem 13.
If0≤δ<p+1, then (53)TRq,λ,l,pδ,m+1(α)⊂TRq,λ,l,pδ,mα.


## 4.3. Extreme Points

Now, let us determine extreme points of the classTRq,λ,l,pδ,m(α).Theorem 14 (extreme points).
Letfp(z)=zp and(54)fkz=zp-pq1-αkqΨq,λ,l,pδ,mkzk,k≥p+1.Then f(z) is in the class TRq,λ,l,pδ,m(α) if and only if it can be expressed in the form(55)fz=μpzp+∑k=p+1∞‍μkfkz,where μk≥0 and ∑k=p∞‍μk=1.Proof.
Let the functionf(z)∈Tp be defined by (2). Since ∑k=p+1∞‍μk=1, we then have (56)fz=zp-∑k=p+1∞‍μkpq1-αkqΨq,λ,l,pδ,mkzk.Now, we obtain (57)1-α∑k=p+1∞kqΨq,λ,l,pδ,mkpq1-αμkpq1-αkqΨq,λ,l,pδ,mk=1-α∑k=p+1∞‍μk≤1-α.Thus, f∈TRq,λ,l,pδ,m(α) by Theorem 6. Conversely, suppose that f∈TRq,λ,l,pδ,m(α). We may set (58)μk=kqpqΨq,λ,l,pδ,mk1-αak,k≥p+1and μp=1-∑k=p+1∞‍μk. Then we have f(z)=μpzp-∑k=p+1∞‍μkfk(z). This completes the proof of Theorem 14.


## 4.4. Radii of Generalized Close-to-Convexity, Starlikeness, and Convexity

Now, the discussions on radii of generalized close-to-convexity, starlikeness, and convexity for the classTRq,λ,l,pδ,m(α) are given by the following results. In order to establish, we will also require the use of those classes of functions. First of all, a function f∈Ap is said to be in the class Sq,p∗(β) of p-valently starlike with respect to q-differentiation of order β(0≤β<p) if it satisfies the inequality(59)RezDqfzfz>β,z∈D.Furthermore, a function f∈Ap is said to be in the class Cq,p(β) of p-valently convex with respect to q-differentiation of order β(0≤β<p) if it satisfies the inequality(60)Re1+zDq2fzDqfz>β,z∈D.Both Sq,p∗(β) and Cq,p(β) were introduced by Selvakumaran et al. [8]. However, we consider the case 0≤β<[p]q instead of 0≤β<p. The definition of q-analogous of p-valently closed-to-convex was already recalled in (19).Theorem 15.
For0≤δ<p+1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently closed-to-convex with respect to q-differentiation of order α.Proof.
By Theorem13, we obtain(61)TRq,λ,l,pδ,mα⊂TRq,λ,l,pδ,m-1α⊂⋯⊂TRq,pα.This completes the proof.In general, for0≤α<β<1, the function f∈TRq,p(α) does not necessarily belong to the class TRq,p(β). We then derive the radii of generalized close-to-convexity order 0≤α<β<1 for the function f∈TRq,λ,l,pδ,m(α).Theorem 16.
For0≤δ<p+1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently closed-to-convex with respect to q-differentiation of order β in z<r1, where(62)r1=inf⁡k≥p+11-β1-αΨq,λ,l,pδ,mk1/k-p.Proof.
It is sufficient to show thatDqf(z)/[p]qzp-1-1<1-β. That is,(63)Dqfzpqzp-1-1≤∑k=p+1∞‍kqpqakzk-p≤1-β.Since f∈TRq,λ,l,pδ,m(α) and by application of Theorem 6, we obtain(64)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mkak≤1-α.Hence, (63) is true if (65)z≤1-β1-αΨq,λ,l,pδ,mk1/k-p,k≥p+1.This completes the proof.Next, we obtain the radii of generalized starlikeness of orderβ in the following result.Theorem 17.
For0≤δ<p+1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently starlike with respect to q-differentiation of order β(0≤β<[p]q) in z<r2 where(66)
r
2
=
inf
⁡
k
≥
p
+
1
1
1
-
α
·
k
q
p
q
-
β
p
q
k
q
-
β
Ψ
q
,
λ
,
l
,
p
δ
,
m
k
1
/
k
-
p
.Proof.
We have to show thatzDqf(z)/f(z)-[p]q<[p]q-β. That is,(67)zDqfzfz-pq=zpqzp-1-∑k=p+1∞‍kqakzk-1zp-∑k=p+1∞‍akzk-pq≤∑k=p+1∞‍kq-pqakzk-p1-∑k=p+1∞‍akzk-p≤pq-β.Hence, (67) is true if(68)∑k=p+1∞kq-βakzk-p≤pq-β.By using (64), we can say (68) is true if(69)z<11-αkqpq-βpqkq-βΨq,λ,l,pδ,mk1/k-p,k≥p+1,which completes the proof.Corollary 18.
If0≤δ<p+1, then (70)TRq,λ,l,pδ,mα⊂Sq,p∗.Proof.
By Theorem17, we see that a function f∈TRq,λ,l,pδ,m(α) is p-valently starlike with respect to q-differentiation (β=0) in z<r2 where (71)r2=inf⁡k≥p+111-αΨq,λ,l,pδ,mk1/k-p≥inf⁡k≥p+111-α1/k-p·inf⁡k≥p+1Ψq,λ,l,pδ,mk1/k-p.It is easy to see that 1/1-α1/k-p is a decreasing sequence and limk→+∞1/1-α1/k-p=1. This implies(72)inf⁡k≥p+111-α1/k-p=limk→+∞11-α1/k-p=1.Moreover, by using Lemma 9, we obtain(73)inf⁡k≥p+1Ψq,λ,l,pδ,mk1/k-p≥1.Then, we have r2≥1. That is, f∈TRq,λ,l,pδ,m(α) is p-valently starlike with respect to q-differentiation in z∈D. The proof is completed.Next, we obtain the radii of generalized convexity of orderβ, where β≤2[p]q-[p-1]q-1, in the following result.Theorem 19.
For0≤δ<p+1 and β≤2[p]q-[p-1]q-1, if f∈TRq,λ,l,pδ,m(α), then f is p-valently convex with respect to q-differentiation of order β(0≤β<[p]q) in z<r3, where (74)r3=inf⁡k≥p+111-α2pq-p-1q-β-11+k-1q-βhhhhhhhh·2pq-p-1q-β-11+k-1q-βΨq,λ,l,pδ,mk1/k-p.Proof.
We have to show that1+zDq2f(z)/Dqf(z)-[p]q<[p]q-β. That is,(75)1+zDq2fzDqfz-pq=1+zpqp-1qzp-2∑k=p+1∞‍hhhhhhhhhhhhh-∑k=p+1∞‍kqk+1qakzk-2hhhhhhhhhzpqp-1qzp-2-∑k=p+1∞‍kqk+1qakzk-2·pqzp-1-∑k=p+1∞‍kqakzk-1-1hhhhhhzpqp-1qzp-2-∑k=p+1∞‍kqk+1qakzk-2-pq≤pq1+p-1q-pq∑k=p+1∞‍hhhhhhh∑k=p+1∞‍+∑k=p+1∞‍kq1+k-1q-pqakzk-phhhhhh·pq-∑k=p+1∞‍kqakzk-p-1≤pq-β.Hence, (75) is true if(76)pq1+p-1q-pq+∑k=p+1∞‍kq1+k-1q-pqakzk-p≤pq-βpq-∑k=p+1∞‍kqakzk-p,or equivalently(77)∑k=p+1∞‍kq1+k-1q-βakzk-p≤pq2pq-p-1q-β-1.Since β≤2[p]q-[p-1]q-1, by using (64), we can say (77) is true if(78)z<11-α2pq-p-1q-β-11+k-1q-βΨq,λ,l,pδ,mk1/k-p,k≥p+1.The proof is completed.


## 4.5. Quasi-Hadamard Properties

In this section, we derive the quasi-Hadamard (convolution) properties. Before we derive the result, we recall the definition of the quasi-Hadamard properties. For any functionsfj∈Tp, j=1,2,3,…,n of the form(79)fjz=zp-∑k=p+1∞‍ak,jzk,the quasi-Hadamard product (f1⊗f2⊗⋯⊗fn)(z) is defined by (80)f1⊗f2⊗⋯⊗fnz=zp-∑k=p+1∞‍∏j=1n‍ak,jzk,z∈D.Next, we derive the quasi-Hadamard properties for the classTRq,λ,l,pδ,m(α). Using the techniques of Schild and Silverman [41] with Theorem 6, we prove the following results.Theorem 20.
For0≤δ<p+1, suppose that fj∈TRq,λ,l,pδ,m(αj), j=1,2,…,n; then (f1⊗f2⊗⋯⊗fn)∈TRq,λ,l,pδ,m(γ), where(81)γ=1-pqp+1qΨq,λ,l,pδ,mp+1n-1∏j=1n(1-αj).Proof.
To prove this theorem, we use the principle of mathematical induction onj. Let the functions fj∈Tp, for j=1,2 of the form(82)fjz=zp-∑k=p+1∞‍ak,jzk,for j=1,2 and k≥2. Since fj∈TRq,λ,l,pδ,m(αj) for j=1,2, by Theorem 6, we see that(83)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mk1-αjak,j≤1,forj=1,2.According to Theorem 6, it is sufficient to prove that (84)∑k=p+1∞‍kqpqΨq,λ,l,pδ,mk1-γak,1ak,2≤1,where γ is defined in (81). Applying Cauchy-Schwarz’s inequality to (83) for j=1,2, we have the following inequality:(85)∑k=p+1∞‍kqpqΨq,λ,l,pδ,m(k)(1-α1)(1-α2)ak,1ak,2≤∑k=2∞‍kqpqΨq,λ,l,pδ,m(k)1-α1ak,1∑k=2∞‍kqpqΨq,λ,l,pδ,m(k)1-α2ak,2≤1.From (84) and (85), if the following inequality (86)1-α11-α21-γak,1ak,2≤1,for all k≥p+1, is satisfied, it can be concluded that (f1⊗f2)∈TRq,λ,l,pδ,m(γ). Now, applying Corollary 7, inequality (86) will be held if (87)pqkq1-α11-α2Ψq,λ,l,pδ,mk≤1-γ.By Lemma 9, we see that (88)pqkq1-α11-α2Ψq,λ,l,pδ,mk≤pqp+1q1-α11-α2Ψq,λ,l,pδ,mp+1:=1-γ.This yields our desired inequality (86). Now, we have (f1⊗f2)∈TRq,λ,l,pδ,m(γ). Next, we let the functions fj∈Tp, for j=1,2,…,N+1 and fj∈TRq,λδ,m(αj) for j=1,2,…,N+1. Suppose that (89)f1⊗f2⊗⋯⊗fN∈TRq,λ,l,pδ,mγ,where (90)γ:=1-pqp+1qΨq,λ,l,pδ,mp+1N-1∏j=1N(1-αj).Then, by means of the above technique, it can be shown that (91)f1⊗f2⊗⋯⊗fN⊗fN+1∈TRq,λδ,mγ′,where (92)γ′:=1-pqp+1q1-αN+11-γΨq,λ,l,pδ,mp+1=1-pqp+1qΨq,λ,l,pδ,mp+1N∏j=1N+1‍1-αj.By Lemma 9, we have 0<γ<1. This completes the proof of the theorem.


## 4.6. Invariant Properties

In the following results, we discuss invariant properties of the classTRq,λ,l,pδ,m(α) via Theorem 6. We consider the formerly studied operators in terms of the standard convolution formula; we choose g as a fixed function in Ap such that (f∗g)(z) exists for any f∈Ap. For various choices of g we get different linear operators that have been studied in the recent past.According to Theorem6, we easily obtain the following properties.Theorem 21.
Forδ<p+1, if the function g∈Ap is of the form (93)gz=zp+∑k=p+1∞‍μkzk,where μk≤1 for k≥p+1, then (f∗g)∈TRq,λ,l,pδ,m(α) for each f∈TRq,λ,l,pδ,m(α).Next, we recall the definition of Bernardi’s integral operator. For nonnegative real numberγ and f∈Ap, Bernardi-Libera’s integral operator Lγf:Ap→Ap is defined as follows:(94)Lγfz=γ+1zγ∫0z‍tγ-1ftdt,which was studied by Bernardi in [42]. Also, their properties for γ=1 are reported in [43, 44]. By using the concept of q-calculus, we introduce q-analogous to Bernardi’s integral operator defined by (95)Lq,γfz=γ+1qzγ∫0z‍tγ-1ftdqt.From (95), we have verified that (96)Lq,γfz=γ+pqzγz1-q∑j=0∞‍qjzqjγ-1fzqj=γ+pq1-q∑j=0∞‍qjγfzqj=γ+pq1-q∑j=0∞‍qjγ∑k=p∞‍qjkakzk,ap=1=γ+pq∑k=p∞‍∑j=0∞‍1-qqj(γ+k)akzk=zp+∑k=p+1∞‍γ+pqγ+kqakzk.That is, (97)Lq,γfz:=f∗gzp=z+∑k=p+1∞‍γ+pqγ+kqakzk,where g=zp+∑k=p+1∞‍[γ+p]q/[γ+k]qzk. It is clear that [γ+p]q/[γ+k]q≤1 for k≥p+1. Then, we obtain the invariant properties under integral operator Lq,γ as follows.Theorem 22.
Forδ<p+1, and γ>0, the class TRq,λ,l,pδ,m(α) is invariant under the integral operator Lq,γ defined in (95). That is,(98)Lq,γTRq,λ,l,pδ,mα⊂TRq,λ,l,pδ,mα.Moreover, in the view of the definition of fractionalq-integral and Lemma 8, we obtain the invariant properties under fractional q-integral.Theorem 23.
Forδ<p+1, the class TRq,λ,l,pδ,m(α) is invariant under the integral operator Iq,zδ defined in (11). That is, (99)Iq,zδTRq,λ,l,pδ,m(α)⊂TRq,λ,l,pδ,mα.


## 5. Observation and Concluding Remark

In this section we briefly point out some consequences of the results derived in the proceeding sections. If we letq→1-, we observe that the function classes Rq,λ,l,pδ,m(α) and TRq,λ,l,pδ,m(α) provide the q-extensions of both known and newly obtained. By assigning appropriated values to the parameters m,l,γ,δ, and p, we can derive the corresponding results for several simpler subclasses of the class Tp from each of our theorems, especially as indicated in Altintas [40]. Therefore, it leads to the q-extension of the former results.Furthermore, we letf∈Tn,p, where Tn,p is a subclass of Tp consisting of functions of the form (100)fz=zp-∑k=n+p∞‍akzk.Theorem 6 can indeed be generalized further by considering the class of multivalent function Tn,p in place of Tp. That leads to the following corollary.Corollary 24.
Letf∈Tn,p be defined by (100); then f∈Rq,λ,l,pδ,m(α) if and only if f satisfies the inequality (101)∑k=n+p∞‍kqpqΨq,λ,l,pδ,mkak≤1-α,where Ψq,λ,l,pδ,m(k) is defined in (18). Moreover, the result is sharp.By previous argument in this paper, Corollary24 can fruitfully be used in investigating the geometric properties for several subclasses of Rq,λ,l,pδ,m(α)∩Tn,p. Then the q-extension of the former results, such as Sarangi and Uralegaddi [45], Aouf et al. [46], and Aouf [47], is obtained.We conclude this paper by remarking that the results presented in this paper give variousq-extension properties of different classes of analytic and multivalent function. Here, our results generalize several formerly known results. We introduce q-extension of the general differential operator, which is generalized from Bulut operator [17], in sense of q-theory. The new subclass of multivalently analytic function is proposed consequently by joining the class of q-analogue to close-to-convexity together with our q-extension of Bulut operator [17]. We discuss the linear combination property and coefficient estimate for the class Rq,λ,l,pδ,m(α). By making use of the coefficient estimate and the concept of q-theory, we obtain the q-extension of geometric properties for the class TRq,λ,l,pδ,m(α). We give the q-analogue to distortion properties and the radii of q-analogue starlikeness and convexity which were defined in [8]. Moreover, we consider the radii of q-analogue close-to-convexity that replace usual derivative by q-derivative operator. We also use the concept of q-theory to extend the Bernardi integral operator. As a consequence, the invariant property for TRq,λ,l,pδ,m(α) under such q-integral operator is obtained. Finally, the presented results can be extended to investigate the function on Tn,p which gives generalized formerly known and newly obtained results.



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*Source: 273236-2015-04-22.xml*