# Convexity of Certainq-Integral Operators of p-Valent Functions

Authors: K. A. Selvakumaran; S. D. Purohit; Aydin Secer; Mustafa Bayram
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

By applying the concept (and theory) of fractionalq-calculus, we first define and introduce two new q-integral operators for certain analytic functions defined in the unit disc 𝒰. Convexity properties of these q-integral operators on some classes of analytic functions defined by a linear multiplier fractional q-differintegral operator are studied. Special cases of the main results are also mentioned.


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## Body

## 1. Introduction and Preliminaries

The subject of fractional calculus has gained noticeable importance and popularity due to its established applications in many fields of science and engineering during the past three decades or so. Much of the theory of fractional calculus is based upon the familiar Riemann-Liouville fractional derivative (or integral). The fractionalq-calculus is the extension of the ordinary fractional calculus in the q-theory. Recently, there was a significant increase of activity in the area of the q-calculus due to applications of the q-calculus in mathematics, statistics, and physics. For more details, one may refer to the books [1–4] on the subject. Recently, Purohit and Raina [5–7] have added one more dimension to this study by introducing certain subclasses of functions which are analytic in the open disk 𝒰, by using fractional q-calculus. Purohit [8] also studied similar work and considered new classes of multivalently analytic functions in the open unit disk.The aim of this paper is to consider a linear multiplier fractionalq-differintegral operator and to define certain new subclasses of functions which are p-valent and analytic in the open unit disk. The results derived include convexity properties of these q-integral operators on some classes of analytic functions. Special cases of the main results are also mentioned.Let𝒜p denote the class of functions f(z) of the form
(1)f(z)=zp+∑n=p+1∞‍anzn,(p∈ℕ={1,2,3,…}),
which are analytic and p-valent in the open unit disk 𝒰={z∈ℂ:|z|<1}. A function f∈𝒜p is said to be p-valently starlike of order α(0≤α<p) if and only if
(2)ℜ{zf′(z)f(z)}>α,(z∈𝒰).
We denote by 𝒮p*(α) the class of all such functions. On the other hand, a function f∈𝒜p is said to be in the class 𝒞p(α) of p-valently convex of order α(0≤α<p) if and only if
(3)ℜ{1+zf′′(z)f′(z)}>α,(z∈𝒰).
Note that 𝒮p*(0)=𝒮p* and 𝒞p(0)=Cp are, respectively, the classes of p-valently starlike and p-valently convex functions in 𝒰. Also, we note that 𝒮1*(0)=𝒮* and 𝒞1(0)=𝒞 are, respectively, the usual classes of starlike and convex functions in 𝒰. A function f∈𝒜p is said to be in the class 𝒰𝒮p(α,k) of k-uniformly p-valent starlike of order α(-1≤α<p) if it satisfies
(4)ℜ{zf′(z)f(z)-α}≥k|zf′(z)f(z)-p|,(k≥0,z∈𝒰).
Furthermore, a function f∈𝒜p is said to be in the class 𝒰𝒞p(α,k) of k-uniformly p-valent convex of order α(-1≤α<p) if it satisfies
(5)ℜ{1+zf′′(z)f′(z)-α}≥k|1+zf′′(z)f′(z)-p|,1111.1(k≥0,z∈𝒰).
For uniformly starlike and uniformly convex functions we refer to the papers [9–11]. Note that 𝒰𝒮1(α,k)=𝒰𝒮𝒯(α,k) and 𝒰𝒞1(α,k)=𝒰𝒞𝒱(α,k), where the classes 𝒰𝒮𝒯(α,k) and 𝒰𝒞𝒱(α,k) are, respectively, the classes of k-uniformly starlike of order α(0≤α<1) and k-uniformly convex of order α(0≤α<1) studied in [12].For the convenience of the reader, we now give some basic definitions and related details ofq-calculus which are used in the sequel.For any complex numberα the q-shifted factorials are defined as
(6)(α;q)0=1,(α;q)n=∏k=0n-1‍(1-αqk),n∈ℕ,
and in terms of the basic analogue of the gamma function
(7)(qα;q)n=Γq(α+n)(1-q)nΓq(α),(n>0),
where the q-gamma function is defined by
(8)Γq(x)=(q,q)∞(1-q)1-x(qx;q)∞,(0<q<1).
If |q|<1, the definition (6) remains meaningful for n=∞ as a convergent infinite product:
(9)(α;q)∞=∏j=0∞‍(1-αqj).
In view of the relation
(10)lim⁡q→1-(qα;q)n(1-q)n=(α)n,
we observe that the q-shifted factorial (6) reduces to the familiar Pochhammer symbol (α)n, where (α)n=α(α+1)⋯(α+n-1). Also, the q-derivative and q-integral of a function on a subset of ℂ are, respectively, given by (see [2] pp. 19–22)
(11)Dqf(z)=f(z)-f(zq)(1-q)z,(z≠0,q≠0),∫0z‍f(t)dqt=z(1-q)∑k=0∞‍qkf(zqk).
Therefore, the q-derivative of f(z)=zn, where n is a positive integer, is given by
(12)Dqzn=zn-(zq)n(1-q)z=[n]qzn-1,
where
(13)[n]q=1-qn1-q=qn-1+⋯+1
and is called the q-analogue of n. As q→1, we have [n]q=qn-1+⋯+1→1+⋯+1=n.Theq-analogues to the function classes 𝒮p*(α), 𝒞p(α), 𝒰𝒮p(α,k), and 𝒰𝒞p(α,k) are given as follows.A functionf∈𝒜p is said to be in the class 𝒮q,p*(α) of p-valently starlike with respect to q-differentiation of order α(0≤α<p) if it satisfies
(14)ℜ{zDq(f(z))f(z)}>α,(z∈𝒰).
Also, a function f∈𝒜p is said to be in the class 𝒞q,p(α) of p-valently convex with respect to q-differentiation of order α(0≤α<p) if it satisfies
(15)ℜ{1+zDq2(f(z))Dq(f(z))}>α,(z∈𝒰).
On the other hand, a function f∈𝒜p is said to be in the class 𝒰𝒮q,p(α,k) of k-uniformly p-valent starlike with respect to q-differentiation of order α(-1≤α<p) if it satisfies
(16)ℜ{zDq(f(z))f(z)-α}≥k|zDq(f(z))f(z)-[p]q|,111111111(k≥0,z∈𝒰).
Furthermore, a function f∈𝒜p is said to be in the class 𝒰𝒞q,p(α,k) of k-uniformly p-valent convex with respect to q-differentiation of order α(-1≤α<p) if it satisfies
(17)ℜ{1+zDq2(f(z))Dq(f(z))-α}≥k|1+zDq2(f(z))Dq(f(z))-[p]q|,111111111111.(k≥0,z∈𝒰).In the following, we define the fractionalq-calculus operators of a complex-valued function f(z), which were recently studied by Purohit and Raina [5].Definition 1 (fractionalq-integral operator).
The fractionalq-integral operator Iq,zδ of a function f(z) of order δ is defined by
(18)Iq,zδf(z)≡Dq,z-δf(z)=1Γq(δ)∫0z‍(z-tq)δ-1f(t)dqt,(δ>0),
where f(z) is analytic in a simply connected region of the z-plane containing the origin and the q-binomial function (z-tq)δ-1 is given by
(19)(z-tq)δ-1=zδ-1Φ01[q-δ+1;-;q,tqδz].
The series Φ01[δ;-;q,z] is single valued when |arg(z)|<π and |z|<1 (see for details [2], pp. 104–106); therefore, the function (z-tq)δ-1 in (18) is single valued when |arg(-tqδ/z)|<π, |tqδ/z|<1, and |arg(z)|<π.Definition 2 (fractionalq-derivative operator).
The fractionalq-derivative operator Dq,zδ of a function f(z) of order δ is defined by
(20)Dq,zδf(z)≡Dq,zIq,z1-δf(z)=1Γq(1-δ)×Dq,z∫0z‍(z-tq)-δf(t)dqt,(0≤δ<1),
where f(z) is suitably constrained and the multiplicity of (z-tq)-δ is removed as in Definition 1.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
(21)Dq,zδf(z)=Dq,zmIq,zm-δf(z),
where m-1≤δ<1, m∈ℕ0=ℕ∪{0}, and ℕ denotes the set of natural numbers.Remark 4.
It follows from Definition2 that
(22)Dq,zδzn=Γq(n+1)Γq(n+1-δ)zn-δ(δ≥0,n>-1).


## 2. The Operator𝒟q,p,λδ,m

UsingDq,zδ, we define a q-differintegral operator Ωq,pδ:𝒜p→𝒜p as follows:
(23)Ωq,pδf(z)=Γq(p+1-δ)Γq(p+1)zδDq,zδf(z)=zp+∑n=p+1∞‍Γq(p+1-δ)Γq(n+1)Γq(p+1)Γq(n+1-δ)anzn,11111.(δ<p+1;n∈ℕ;0<q<1;z∈𝒰),
where Dq,zδf(z) in (23) 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. Here we note that Ωq,p0f(z)=f(z).We now define a linear multiplier fractionalq-differintegral operator 𝒟q,p,λδ,m as follows:
(24)𝒟q,p,λδ,0f(z)=f(z),𝒟q,p,λδ,1f(z)=(1-λ)Ωq,pδf(z)11111111111+λz[p]qDq(Ωq,pδf(z)),(λ≥0),𝒟q,p,λδ,2f(z)=𝒟q,p,λδ,1(𝒟q,p,λδ,1f(z)),⋮𝒟q,p,λδ,mf(z)=𝒟q,p,λδ,1(𝒟q,p,λδ,m-1f(z)),m∈ℕ.
If f(z)∈𝒜p is given by (1), then by (24) we have
(25)𝒟q,p,λδ,mf(z)=zp+∑n=p+1∞‍(Γq(p+1-δ)Γq(n+1)Γq(p+1)Γq(n+1-δ)[1-λ+[n]q[p]qλ])m×anzn.
It can be seen that, by specializing the parameters, the operator 𝒟q,p,λδ,m reduces to many known and new integral and differential operators. In particular, when δ=0, p=1, and q→1 the operator 𝒟q,p,λδ,m reduces to the operator introduced by AL-Oboudi [13] and if δ=0, p=1, λ=1, and q→1 it reduces to the operator introduced by Sǎlǎgean [14].By using the operator𝒟q,p,λδ,mf(z) defined by (24) and q-differentiation, we introduce two new subclasses of analytic functions 𝒰𝒮q,p,λδ,m(α,k) and 𝒰𝒞q,p,λδ,m(α,k) as follows.A functionf∈𝒜p is said to be in the class 𝒰𝒮q,p,λδ,m(α,k) if and only if
(26)ℜ{zDq(𝒟q,p,λδ,mf(z))𝒟q,p,λδ,mf(z)-α}≥k|zDq(𝒟q,p,λδ,mf(z))𝒟q,p,λδ,mf(z)-[p]q|,(-1≤α<p,k≥0).Furthermore, a functionf∈𝒜p is said to be in the class 𝒰𝒞q,p,λδ,m(α,k) if and only if
(27)ℜ{1+zDq2(𝒟q,p,λδ,mf(z))Dq(𝒟q,p,λδ,mf(z))-α}≥k|1+zDq2(𝒟q,p,λδ,mf(z))Dq(𝒟q,p,λδ,mf(z))-[p]q|,(-1≤α<p,k≥0).It is interesting to note that the classes𝒰𝒮q,p,λδ,m(α,k) and 𝒰𝒞q,p,λδ,m(α,k) generalize several well-known subclasses of analytic functions. For instance, if q→1, then(1)
𝒰𝒮1,p,λδ,0(α,k)=𝒰𝒮p(α,k),(2)
𝒰𝒮1,p,10,1(α,k)=𝒰𝒞1,p,λδ,0(α,k)=𝒰𝒞p(α,k),(3)
𝒰𝒮1,p,λδ,0(α,0)=𝒮p*(α),(4)
𝒰𝒮1,p,10,1(α,0)=𝒰𝒞1,p,λδ,0(α,0)=𝒞p(α).


## 3. Thep-Valent q-Integral Operators Fq and Gq

We now introduce two newp-valent q-integral operators as follows.Definition 5.
Letm=(m1,m2,…,mn)∈ℕ0n, γ=(γ1,γ2,…,γn)∈ℝ+n and fi∈𝒜p for all i={1,2,…,n}, n∈ℕ. Then Fq(z):𝒜pn→𝒜p is defined as
(28)Fq(z)=ℱq,p,λδ,γ,m(f1,f2,…,fn)=∫0z‍[p]qtp-1∏i=1n‍(𝒟q,p,λδ,mifi(t)tp)γidqt,
and Gq(z):𝒜pn→𝒜p is defined as
(29)Gq(z)=𝒢q,p,λδ,γ,m(f1,f2,…,fn)=∫0z‍[p]qtp-1∏i=1n‍(Dq(𝒟q,p,λδ,mifi(t))[p]qtp-1)γidqt,
where 𝒟q,p,λδ,mifi(t) is given by (24).It is interesting to observe that several well-known and new integral operators are special cases of the operatorsFq(z) and Gq(z). We list a few of them in the following remarks.Remark 6.
Lettingmi=0 for all i={1,2,…,n} and q→1, the q-integral operator Fq(z) reduces to the operator Fp(z) studied by Frasin in [15]. Upon setting p=1, δ=0, λ=1, and q→1, we obtain the integral operator DkF(z) studied by Breaz et al. in [16]. For p=1, m1=m2=⋯=mn=0, and q→1, the operator Fq(z) reduces to the operator Fn(z) which was studied by D. Breaz and N. Breaz in [17]. Observe that when p=n=1, m1=0, γ1=γ, and q→1, we obtain the integral operator Iγ(f)(z) studied by Pescar and Owa in [18]. Also, for p=n=1, m1=0, γ1=1, and q→1, the q-integral operator Fq(z) reduces to the Alexander integral operator I(f)(z) studied in [19].Remark 7.
Lettingmi=0 for all i={1,2,…,n} and q→1, the q-integral operator Gq(z) reduces to the operator Gp(z) studied by Frasin in [15]. For p=1, m1=m2=⋯=mn=0 and q→1, the operator Gq(z) reduces to the operator Gγ1,γ2,…,γn(z) which was studied by Breaz et al. (see [20]). Also, for p=n=1, m1=0, γ1=1, and q→1, the q-integral operator Gq(z) reduces to the integral operator G(z) introduced and studied by Pfaltzgra (see [21]).In this paper, we obtain the order of convexity with respect toq-differentiation of the q-integral operators Fq(z) and Gq(z) on the classes 𝒰𝒮q,p,λδ,m(α,k) and 𝒰𝒞q,p,λδ,m(α,k). As special cases, the order of convexity of the operators ∫0z‍(f(t)/t)γdt and ∫0z‍(f′(t))γdt is also given.


## 4. Convexity of the OperatorFq

First, we prove the following convexity result with respect toq-differentiation of the operator Fq.Theorem 8.
Letm=(m1,m2,…,mn)∈ℕ0n, γ=(γ1,γ2,…,γn)∈ℝ+n, -1≤αi<p, ki>0, and fi∈𝒰𝒮q,p,λδ,mi(αi,ki) for all i={1,2,…,n}, n∈ℕ. If
(30)0≤p+∑i=1n‍γi(αi-p)<p,
then the q-integral operator Fq(z) defined by (28) is p-valently convex with respect to q-differentiation of order p+∑i=1n‍γi(αi-p).Proof.
From (28), we observe that Fq(z)∈𝒜p. On the other hand, it is easy to verify that
(31)Dq(Fq(z))=[p]qzp-1∏i=1n‍(𝒟q,p,λδ,mifi(z)zp)γi.
Now by logarithmic q-differentiation we have
(32)ln⁡qq-1Dq2(Fq(z))Dq(Fq(z))=ln⁡qq-1[p-1z+∑i=1n‍γi(Dq(𝒟q,p,λδ,mifi(z))𝒟q,p,λδ,mifi(z)-pz)].
Therefore,
(33)1+zDq2(Fq(z))Dq(Fq(z))=p+∑i=1n‍γi(zDq(𝒟q,p,λδ,mifi(z))𝒟q,p,λδ,mifi(z)-αi)+∑i=1n‍γi(αi-p).
Taking the real parts on both sides of the above equation, we have
(34)ℜ{1+zDq2(Fq(z))Dq(Fq(z))}=p+∑i=1n‍γi(αi-p)111111+∑i=1n‍γiℜ(zDq(𝒟q,p,λδ,mifi(z))𝒟q,p,λδ,mifi(z)-αi).
Since fi∈𝒰𝒮q,p,λδ,mi(αi,ki) for all i={1,2,…,n}, from (26) we get
(35)ℜ{1+zDq2(Fq(z))Dq(Fq(z))}≥p+∑i=1n‍γi(αi-p)+∑i=1n‍γiki|zDq(𝒟q,p,λδ,mifi(z))𝒟q,p,λδ,mifi(z)-[p]q|.
As ∑i=1n‍γiki|(zDq(𝒟q,p,λδ,mifi(z))/𝒟q,p,λδ,mifi(z))-[p]q|>0, for all i={1,2,…,n}, we obtain from the above
(36)ℜ{1+zDq2(Fq(z))Dq(Fq(z))}>p+∑i=1n‍γi(αi-p).
This completes the proof.Corollary 9.
Letm=(m1,m2,…,mn)∈ℕ0n, γ=(γ1,γ2,…,γn)∈ℝ+n, -1≤αi<p, ki>0 and fi∈𝒰𝒮q,p,λδ,mi(αi,ki) for all i={1,2,…,n}, n∈ℕ. If
(37)|zDq(𝒟q,p,λδ,mifi(z))𝒟q,p,λδ,mifi(z)-[p]q|>-p+∑i=1nγi(αi-p)∑i=1nγiki,
for all i={1,2,…,n}, then the q-integral operator Fq(z) defined by (28) is p-valently convex with respect to q-differentiation in 𝒰.Proof.
From (35) and (37) we easily find that Fq∈𝒞q,p.Lettingq→1, p=n=1, m1=0, γ1=γ, α1=α, k1=k, and f1=f in Theorem 8, we have the following.Corollary 10.
Letγ>0, -1≤α<1, k>0, and f∈𝒰𝒮𝒯(α,k). If 0≤1+γ(α-1)<1, then the integral operator ∫0z‍(f(t)/t)γdt is convex of order 1+γ(α-1) in 𝒰.


## 5. Convexity of the OperatorGq

Now, we prove the following convexity result with respect toq-differentiation of the operator Gq.Theorem 11.
Letm=(m1,m2,…,mn)∈ℕ0n, γ=(γ1,γ2,…,γn)∈ℝ+n, -1≤αi<p, ki>0, and fi∈𝒰𝒞q,p,λδ,mi(αi,ki) for all i={1,2,…,n}, n∈ℕ. If
(38)0≤p+∑i=1n‍γi(αi-p)<p,
then the q-integral operator Gq(z) defined by (29) is p-valently convex with respect to q-differentiation of order p+∑i=1n‍γi(αi-p).Proof.
From (29), we observe that Gq(z)∈𝒜p. On the other hand, it is easy to verify that
(39)Dq(Gq(z))=[p]qzp-1∏i=1n‍(Dq(𝒟q,p,λδ,mifi(z))[p]qzp-1)γi.
Now by logarithmic q-differentiation we have
(40)ln⁡qq-1Dq2(Gq(z))Dq(Gq(z))=ln⁡qq-1[p-1z+∑i=1n‍γi(Dq2(𝒟q,p,λδ,mifi(z))Dq(𝒟q,p,λδ,mifi(z))-p-1z)].
Therefore,
(41)1+zDq2(Gq(z))Dq(Gq(z))=p+∑i=1n‍γi(1+Dq2(𝒟q,p,λδ,mifi(z))Dq(𝒟q,p,λδ,mifi(z))-αi)+∑i=1n‍γi(αi-p).
Taking the real parts on both sides of the above equation, we have
(42)ℜ{1+zDq2(Gq(z))Dq(Gq(z))}=p+∑i=1n‍γi(αi-p)+∑i=1n‍γiℜ(1+Dq2(𝒟q,p,λδ,mifi(z))Dq(𝒟q,p,λδ,mifi(z))-αi).
Since fi∈𝒰𝒞q,p,λδ,mi(αi,ki) for all i={1,2,…,n}, from (27) we get
(43)ℜ{1+zDq2(Gq(z))Dq(Gq(z))}≥p+∑i=1n‍γi(αi-p)+∑i=1n‍γiki|1+Dq2(𝒟q,p,λδ,mifi(z))Dq(𝒟q,p,λδ,mifi(z))-[p]q|.
As ∑i=1n‍γiki|1+(Dq2(𝒟q,p,λδ,mifi(z))/Dq(𝒟q,p,λδ,mifi(z)))-[p]q|>0, for all i={1,2,…,n}, we obtain from the above
(44)ℜ{1+zDq2(Gq(z))Dq(Gq(z))}>p+∑i=1n‍γi(αi-p).
This completes the proof.Corollary 12.
Letm=(m1,m2,…,mn)∈ℕ0n, γ=(γ1,γ2,…,γn)∈ℝ+n, -1≤αi<p, ki>0, and fi∈𝒰𝒞q,p,λδ,mi(αi,ki) for all i={1,2,…,n}, n∈ℕ. If
(45)|1+Dq2(𝒟q,p,λδ,mifi(z))Dq(𝒟q,p,λδ,mifi(z))-[p]q|>-p+∑i=1nγi(αi-p)∑i=1nγiki,
for all i={1,2,…,n}, then the q-integral operator Gq(z) defined by (29) is p-valently convex with respect to q-differentiation in 𝒰.Proof.
From (43) and (45) we easily find that Gq∈𝒞q,p.Lettingq→1, p=n=1, m1=0, γ1=γ, α1=α, k1=k, and f1=f in Theorem 11, we have the following.Corollary 13.
Letγ>0, -1≤α<1, k>0, and f∈𝒰𝒞𝒱(α,k). If 0≤1+γ(α-1)<1, then the integral operator ∫0z‍(f′(t))γdt is convex of order 1+γ(α-1) in 𝒰.We remark in conclusion that, by suitably specializing the parameters in Theorems8 and 11, we can deduce the results obtained in [15, 22, 23].



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