# Applications of a Newq-Difference Operator in Janowski-Type Meromorphic Convex Functions

Authors: Bakhtiar Ahmad; Muhammad Ghaffar Khan; Mohamed Kamal Aouf; Wali Khan Mashwani; Zabidin Salleh; Huo Tang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The main aim of the present article is the introduction of a new differential operator inq-analogue for meromorphic multivalent functions which are analytic in punctured open unit disc. A subclass of meromorphic multivalent convex functions is defined using this new differential operator in q-analogue. Furthermore, we discuss a number of useful geometric properties for the functions belonging to this class such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity. Also, algebraic property of closure is discussed of functions belonging to this class. Integral representation problem is also proved for these functions.


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## Body

## 1. Introduction and Definitions

LetAp denote the family of all meromorphic p-valent functions f that are analytic in the punctured disc D=z∈ℂ:0<z<1 and obeying the normalization
(1)fz=1zp+∑n=p+1∞anzn,z∈D.Also, letMCpα denote the well-known family of meromorphic p-valent convex functions of order α0≤α<p and defined as
(2)fz∈MCpα⇔Rezf′z′f′z<−α.For0<q<1, the q-difference operator or q-derivative of a function f is defined by
(3)∂qfz=fqz−fzzq−1,z≠0,q≠1.It can easily be seen that forn∈ℕ, where ℕ stands for the set of natural numbers and z∈D,
(4)∂q∑n=1∞anzn=∑n=1∞n,qanzn−1,where
(5)n,q=1−qn1−q=1+∑l=1nql,0,q=0.For any nonnegative integern, the q-number shift factorial is defined by
(6)n,q!=1,n=0,1,q2,q3,q⋯n,q,n∈ℕ.Also, theq-generalized Pochhammer symbol for x∈ℝ is given by
(7)x,qn=1,n=0,x,qx+1,q⋯x+n−1,q,n∈ℕ.In (3), if q→1−, then this operator becomes the conventional derivative in the classical calculus, so the limits can be generalized by introducing the parameter q, with condition 0<q<1, and all such concepts, which have been developed thus, are known as quantum calculus (q-calculus). Many physical phenomena are better explained using this generalized operator, and as a result, this field attracted a lot of the researchers due to its various applications in the branches of mathematics and physics (see details in [1, 2]). Jackson [3, 4] was the pioneer of this field, who gave some applications of q-calculus and introduced the q-analogues of derivative and integral. Aral and Gupta [1, 2, 5] defined an operator, which is known as q-Baskakov Durrmeyer operator by using q-beta functions. The generalization of complex operators known as q-Picard and q-Gauss-Weierstrass singular integral operators was discussed by Aral and Anastassiu in [6–8]. Later, Kanas and Ra˘ducanu [9] introduced the q-analogue of a Ruscheweyh differential operator and studied its various properties. More applications of this operator can be seen in the paper [10]. Huda and Darus [11] utilized the q-analogue of a Liu-Srivastava operator and defined an integral operator. In somewhat similar way, Mohammed and Darus [12] introduced a generalized operator along with investigating a class of functions relating to q-hypergeometric functions. Later, Seoudy [13] estimated coefficient bounds for some q-starlike and q-convex functions of complex order. Recently, Arif and Ahmad defined a new q-differential operator for meromorphic multivalent functions and investigated classes related to q-meromorphic starlike and convex functions in their articles [14, 15].In this article, we introduce a newq-differential operator for meromorphic functions and use this operator to define and study some properties of a new family of meromorphic multivalent functions associated with circular domain.We now define the differential operatorDμ,q:Ap→Ap by
(8)Dμ,qfz=1+p,qμfz+μqpz∂qfz,where μ≥0.Using (1), we can easily obtain
(9)Dμ,qfz=1zp+∑n=p+1∞1+p,qμ+μqpn,qanzn.We take(10)Dμ,q0fz=fz,Dμ,q2fz=Dμ,qDμ,qfz=1zp+∑n=p+1∞1+p,qμ+μqpn,q2anzn.In a similar way, form∈ℕ∪0, we get
(11)Dμ,qmfz=1zp+∑n=p+1∞1+p,qμ+μqpn,qmanzn.From (8) and (11), we get the following identity:
(12)Dμ,qm+1fz=μqpz∂qDμ,qmfz+1+p,qμDμ,qmfz.We now define a subfamilyMCμ,qp,m,A,B of Ap by using the operator Dμ,qm as follows.Definition 1.
For−1≤B<A≤1and0<q<1,we definef∈Apto be in the classMCμ,qp,m,A,B,if it satisfies(13)−qp∂qz∂qDμ,qmfzp,q∂qDμ,qmfz≺1+Az1+Bz.
Here, the relation symbol “≺” is used for the subordinations.We see that for particular values ofp,m,A,B,μ, and q, we get some of the well-known classes few of which are listed below:
(1)
Form=0 and q→1−, we get the class of meromorphic multivalent convex functions associated with Janowski functions denoted by MCp∗A,B(2)
ForA=1,B=−1, and m=0, we get MCp,q∗, the class of meromorphic multivalent convex functions in q-analogue(3)
ForA=1,B=−1,m=0, and q→1−, we get the class of meromorphic multivalent convex functions denoted by MCp∗(4)
ForA=1,B=−1,m=0,p=1, and q→1−, we get MC∗, the class of meromorphic convex functionsIt can easily be verified that a functionf∈Ap will be in the class MCμ,qp,m,A,B, if and only if
(14)qp∂qz∂qDμ,qmfz/p,q∂qDμ,qmfz+1A+Bqp∂qz∂qDμ,qmfzqp∂qz∂qDμ,qmfz/p,q∂qDμ,qmfzp,q∂qDμ,qmfz<1.The following lemma is used in our main results.Lemma 2 (see [16]).
Lethz be analytic in D and have the form
(15)hz=1+∑n=1∞dnzn,and kz is analytic and convex in D with series representation
(16)kz=1+∑n=1∞knzn.So ifhz≺kz, then dn≤k1, for n∈ℕ=1,2,⋯.


## 2. Main Results and Their Consequences

In this section, we start with sufficiency criteria for this newly defined class and then, we give the coefficient estimates for the functions belonging to this class. The following lemma is proved which will be used in this section.Lemma 3.
Suppose that the sequenceAp+nn=1∞ is defined by
(17)Ap+n=p,qA−Bψ0ϕ1ψ1n=1,Ap+n=p,qA−Bϕnψn∑k=0n−1ψkAp+kn≥2.Then,(18)Ap+n=ψ0ψn·p,qA−Bϕ1∏k=1n−1p,qA−B+ϕkϕk+1n≥2.Proof.
From (17), we have
(19)ϕnψnAp+n=p,qA−B∑k=0n−1ψkAk.Thus, we obtain that(20)ϕn+1ψn+1Ap+n+1=p,qA−B∑k=0nψkAk=p,qA−BψnAp+n+p,qA−B∑k=0n−1ψkAk=p,qA−BψnAp+n+ϕnψnAp+n=p,qA−B+ϕnψnAp+n.From (20), we find that
(21)Ap+n+1Ap+n=p,qA−B+ϕnψnϕn+1ψn+1n≥1.Thus,(22)Ap+n=Ap+nAp+n−1·Ap+n−1Ap+n−2⋯Ap+2Ap+1·Ap+1=p,qA−B+ϕn−1ψn−1ϕnψn⋯p,qA−B+ϕ1ψ1ϕ2ψ2·p,qA−Bψ0ϕ1ψ1=ψ0ψn·p,qA−Bϕ1∏k=1n−1p,qA−B+ϕkϕk+1n≥2.In conjunction with (17), we complete the proof of Lemma 3.Theorem 4.
Iff∈Apis of the form (1), then it will be in the classMCμ,qp,m,A,Bif and only if the inequality(23)∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman≤p,q2A−Bqpis satisfied.Proof.
Forf∈MCμ,qp,m,A,B, we need to prove the inequality (14). For this, consider
(24)H≔qp∂qz∂qDμ,qmfz/p,q∂qDμ,qmfz+1A+Bqp∂qz∂qDμ,qmfzqp∂qz∂qDμ,qmfz/p,q∂qDμ,qmfzp,q∂qDμ,qmfz=qp∂qz∂qDμ,qmfz+p,q∂qDμ,qmfzAp,q∂qDμ,qmfz+Bqp∂qz∂qDμ,qmfz.
By using (8) and with the help of (3) and (11),
(25)H≔∑n=p+1∞1+p,qμ+μqpn,qmqpn,q2+p,qn,qanzn−1−A−Bp,q2A−Bp,q2/qpzp+1+∑n=p+1∞1+p,qμ+μqpn,qmAp,qn,q+Bqpn,q2anzn−1=∑n=p+1∞1+p,qμ+μqpn,qmqpn,q2+p,qn,qanzn+p−A−Bp,q2A−Bp,q2/qp+∑n=p+1∞1+p,qμ+μqpn,qmAp,qn,q+Bqpn,q2anzn+p≤∑n=p+1∞1+p,qμ+μqpn,qmqpn,q2+p,qn,qanA−Bp,q2A−Bp,q2/qp−∑n=p+1∞1+p,qμ+μqpn,qmAp,qn,q+Bqpn,q2an.
Now, if we use the inequality (23), then
(26)H<1,and this completes the direct part of the proof.Conversely, letf∈MCμ,qp,m,A,B and be of the form (1); then, from (14), we have for z∈D,
(27)qp∂qz∂qDμ,qmfz/p,q∂qDμ,qmfz+1A+Bqp∂qz∂qDμ,qmfzqp∂qz∂qDμ,qmfz/p,q∂qDμ,qmfzp,q∂qDμ,qmfz=∑n=p+1∞1+p,qμ+μqpn,qmqpn,q2+p,qn,qanzn+p−A−Bp,q2A−Bp,q2/qp+∑n=p+1∞1+p,qμ+μqpn,qmAp,qn,q+Bqpn,q2anzn+p.SinceRez≤z, we have
(28)Re∑n=p+1∞1+p,qμ+μqpn,qmqpn,q2+p,qn,qanzn+pA−Bp,q2A−Bp,q2/qp+∑n=p+1∞1+p,qμ+μqpn,qmAp,qn,q+Bqpn,q2anzn+p<1.Now if the values ofz are chosen on the real axis, then qp∂qz∂qDμ,qmfz/p,q∂qDμ,qmfz is real. Using some calculations in the inequality (28) and letting z→1− through real values, we finally get (23).Theorem 5.
Iff∈MCμ,qp,m,A,B and is of the form (1), then
(29)ap+1≤p,qA−Bψ0ϕ1ψ1,(30)ap+n≤ψ0ψn·p,qA−Bϕ1∏k=1n−1p,qA−B+ϕkϕk+1n≥2,where
(31)ϕn≔qpp+n,q2+p,qp+n,q,(32)ψn≔1+p,qμ+μqpp,qp+n,qm.Proof.
Iff∈Ap is in the class MCμ,qp,m,A,B, then it satisfies
(33)−qp∂qz∂qDμ,qmfzp,q∂qDμ,qmfz≺1+Az1+Bz.Now, let(34)hz=−qp∂qz∂qDμ,qmfzp,q∂qDμ,qmfz.Since(35)Rehz>0,so hz is in the class P with its representation which is given by
(36)hz=1+∑n=1∞dnzn.Now,(37)hz≺1+Az1+Bz.But(38)1+Az1+Bz=1+A−Bz+⋯.Now, using Lemma2, we get
(39)dn≤A−B,now putting the series expansions of hz and fz in (34), simplifying and comparing the coefficients of zn+p on both sides
(40)−qp1+p,qμ+μqpn+p,qmp+n,q2ap+n=p,qp+n,q1+p,qμ+μqpn+p,qmap+n+p,q∑i=0n−11+p,qμ+μqpp+i,qmp+i,qap+idn−i,which implies that
(41)−1+p,qμ+μqpn+p,qmqpp+n,q2+p,qp+n,qap+n=p,q∑i=0n−11+p,qμ+μqpp+i,qmp+n,qap+idn−i.Now, by taking absolute on both sides with using the triangle inequality and using (39), we obtain
(42)1+p,qμ+μqpp+n,qmqpp+n,q2+p,qp+n,qap+n≤p,qA−B∑i=0n−11+p,qμ+μqpp+i,qmap+i.Using the notation (31) and (32) implies that
(43)ap+1≤p,qA−Bψ0ϕ1ψ1,(44)ap+n≤p,qA−Bϕnψn∑k=0n−1ψkap+kn≥2.Now, we define the sequenceAp+nn=1∞ as follows:
(45)Ap+1=p,qA−Bψ0ϕ1ψ1n=1,Ap+n=p,qA−Bϕnψn∑k=0n−1ψkAp+kn≥2.In order to prove that(46)ap+n≤Ap+nn≥2,we use the principle of mathematical induction. It is easy to verify that
(47)ap+1≤Ap+1=p,qA−Bψ0ϕ1ψ1.Thus, assuming that(48)ap+j≤Ap+jj=2,3,⋯,n,we find from (44) and (48) that
(49)ap+n+1≤p,qA−Bϕnψn∑k=0nψkap+k≤p,qA−Bϕnψn∑k=0nψkAp+k=Ap+n+1.Therefore, by the principle of mathematical induction, we have(50)ap+n≤Ap+nn≥2.By means of Lemma3 and (45), we know that
(51)Ap+n=ψ0ψn·p,qA−Bϕ1∏k=1n−1p,qA−B+ϕkϕk+1n≥2.Combining (50) and (51), we readily get the coefficient estimates (30).


## 3. Closure Theorems

Let the functionsfkz,k=1,2,3,⋯,l be defined by
(52)fkz=1zp+∑n=p+1∞an,kznz∈D,an,k≥0.Theorem 6.
Let the functionsfkzk=1,2,3,⋯,l defined by (52) be in the class MCμ,qp,m,A,B. Then, the function F∈MCμ,qp,m,A,B,where(53)Fz=∑k=1lλkfkzλk≥0,∑k=1lλk=1.Proof.
From (53), we have
(54)Fz=1zp+∑n=p+1∞∑k=1lλkan,kzn.By Theorem4, we have
(55)∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qm∑k=1lλkan,k=∑k=1lλkn=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman,k≤∑k=1lλkp,q2A−Bqp=p,q2A−Bqp.Hence, by Theorem4, F∈MCμ,qp,m,A,B.Theorem 7.
The classMCμ,qp,m,A,B is closed under convex combination.Proof.
Let the functionfkzk=1,2 given by (52) be in the class MCμ,qp,m,A,B. It is enough to show that
(56)hz=αf1z+1−αf2z,0≤α≤1,is in the class MCμ,qp,m,A,B. Since for 0≤α≤1,(57)hz=1zp+∑n=p+1∞αan,1+1−αan,2zn,by Theorem 4, we have
(58)∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμαan,1+1−αan,2=αn=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman,1+1−αn=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman,2≤αp,q2A−Bqp+1−αp,q2A−Bqp=p,q2A−Bqp.
Hence, by Theorem4, hz∈MCμ,qp,m,A,B.Theorem 8.
Let the functionfkzk=1,2 given by (52) belong to MCμ,qp,m,A,B; then, their weighted mean hjz is also in the class MCμ,qp,m,A,B, where hjz is defined by
(59)hjz=1−jf1z+1+jf2z2.Proof.
From (59), one can easily write
(60)hjz=1zp+∑n=p+1∞1−jan,1+1+jan,22zn.To provehjz∈MCμ,qp,m,A,B, we consider
(61)∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qm1−jan,1+1+jan,22=1−j2∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman,1+1+j2∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman,2≤1−j2p,q2A−Bqp+1+j2p,q2A−Bqp=p,q2A−Bqp.Hence, by Theorem4, hjz∈MCμ,qp,m,A,B.


## 4. Distortion Theorem

In the next two results, we shall discuss the growth and distortion theorems for our newly defined class of functions.Theorem 9.
Iffis in the classMCμ,qp,m,A,Band has the form (1), then forz=r, we have(62)1rp−ϑ1rp≤fz≤1rp+ϑ1rp,where
(63)ϑ1=p,q2A−Bqpqpp+1,q21+B+1+Ap+1,qp,q1+p,qμ+μqpp+1,qm.Proof.
As(64)fz=1zp+∑n=p+1∞anzn≤1zp+∑n=p+1∞anzn,for z=r<1, we have rn<rp for n≥p+1 and
(65)fz≤1rp+rp∑n=p+1∞an.
Similarly,(66)fz≥1rp−rp∑n=p+1∞an.
Now, iff∈MCμ,qp,m,A,B, then by (23),
(67)∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman≤p,q2A−Bqp.
But we know that(68)qpp+1,q21+B+1+Ap+1,qp,q1+p,qμ+μqpp+1,qm∑n=p+1∞an≤∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman.
Hence,(69)qpp+1,q21+B+1+Ap+1,qp,q1+p,qμ+μqpp+1,qm∑n=p+1∞an≤p,q2A−Bqp,which implies that
(70)∑n=p+1∞an+p≤p,q2A−Bqpqpp+1,q21+B+1+Ap+1,qp,q1+p,qμ+μqpp+1,qm.
Now, by putting this value in (65) and (66), we get the required proof.Theorem 10.
Letf∈MCμ,qp,m,A,B and have the form (1). Then, for z=r,
(71)p,qmqmp+ζrm+p−ϑ2rp≤∂qmfz≤p,qmqmp+ζrm+p+ϑ2rp,where
(72)ϑ2=p,q2A−Bqpqpp+1,q1+B+1+Ap,q1+p,qμ+μqpp+1,qm,ζ=n=1mn.Proof.
From the help of (3) and (4), we can write
(73)∂qmfz=−1mp,qmqmp+ζzp+m+∑n=p+1∞n−m−1,qm+1anzn−m.
Sincez=r<1 implies that rn−m≤rp for m≤n and n≥p+1, hence
(74)∂qmfz≤p,qmqmp+ζrm+p+rp∑n=p+1∞n−m−1,qm+1an.
Similarly,(75)∂qmfz≥p,qmqmp+ζrm+p−rp∑n=p+1∞n−m−1,qm+1an.
Sincef is in the class MCμ,qp,m,A,B, so by (23), we have the inequality
(76)qpp+1,q1+B+1+Ap,q1+p,qμ+μqpp+1,qm∑n=p+1∞n,qan≤p,q2A−Bqp,from which it can be deduced that
(77)∑n=p+1∞n,qan≤p,q2A−Bqpqpp+1,q1+B+1+Ap,q1+p,qμ+μqpp+1,qm,but it can easily be seen that
(78)∑n=p+1∞n−m−1,qap+n≤∑n=p+1∞n,qan,which implies
(79)∑n=p+1∞n−m−1,qap+n≤p,q2A−Bqpqpp+1,q1+B+1+Ap,q1+p,qμ+μqpp+1,qm.
Now, using this inequality in (74) and (75), we obtain the required proof.


## 5. Integral Representation

Theorem 11.
Let the functionfgiven by (1) be in the classMCμ,qp,m,A,B.Then, the functionGzrepresented by(80)Gz=1+γ1zp+γp∫0zfttdtγ≥0,z∈D,is in the class MCμ,qp,m,A,B.Proof.
From (1),
(81)fz=z−p+∑n=p+1∞anzn.Then,(82)Gz=1+γ1zp+γp∫0zt−p+∑n=p+1∞antntdt=1+γ1zp+γp−1p1zp+∑n=p+1∞annzn=z−p+∑n=p+1∞anγpnzn.Consider(83)∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qmanγpn≤∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qmγpan≤p,q2A−Bqp,since γp≤1.Therefore, by Theorem4, Gz∈MCμ,qp,m,A,B.


## 6. Radius Problems

The following results are about the radii of convexity and starlikeness for the functions of the classMCμ,qp,m,A,B.Theorem 12.
Iff∈MCμ,qp,m,A,B, thenf∈MCpαforz<r1, where(84)r1=qppp−αqpn+p,q21+B+1+An+p,qp,q1+p,qμ+μqpn+p,qmp+nn+p+αp,q2A−B1/n+2p.Proof.
Letf∈MCμ,qp,m,A,B. To prove f∈MCpα, we only need to show
(85)zf′′z+p+1f′zzf′′z+1+2α−pf′z<1.Using (1) along with some simple computation yields
(86)∑n=1∞p+np+n+αpp−αan+pzn+2p<1.Asf is in the class MCμ,qp,m,A,B, so we have from (23)
(87)∑n=p+1∞qpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman≤p,q2A−Bqp⇒∑n=p+1∞qpqpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qman<1.Equivalently,(88)∑n=1∞qpqpn+p,q21+B+1+An+p,qp,q1+p,qμ+μqpn+p,qman+p<1.Now, inequality (86) will be true, if the following holds:
(89)∑n=1∞p+1n+p+αpp−αan+pzn+2p<∑n=1∞qpqpn+p,q21+B+1+An+p,qp,q1+p,qμ+μqpn+p,qmp,q21−Aan+p,which implies that
(90)zn+2p<qppp−αqpn+p,q21+B+1+An+p,qp,q1+p,qμ+μqpn+p,qmp+nn+p+αp,q2A−B,and so
(91)z<qppp−αqpn+p,q21+B+1+An+p,qp,q1+p,qμ+μqpn+p,qmp+nn+p+αp,q2A−B1/n+2p=r1.We get the required condition.Theorem 13.
Letf∈MCμ,qp,m,A,B. Then,f∈MSp∗αforz<r2, where(92)r2=qpp−αqpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qmn+p+αp,q2A−B1/n+2p.Proof.
We know thatf∈MSp∗α, if and only if
(93)zf′z+pfzzf′z−p−2αfz<1.Using (1) and with some simplification, we get
(94)∑n=1∞n+p+αp−αan+pzn+2p<1.Now, from (23), we can easily obtain that
(95)∑n=p+1∞qpqpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qmp,q2A−Ban<1.For inequality (94) to hold, it will be enough if
(96)∑n=1∞n+p+αp−αan+pzn+2p<∑n=1∞qpqpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qmp,q2A−B,which implies that
(97)zn+2p<qpp−αqpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qmn+p+αp,q2A−B,and hence,
(98)z<qpp−αqpn,q21+B+1+An,qp,q1+p,qμ+μqpn,qmn+p+αp,q2A−B1/n+2p=r2.Thus, we obtain the required result.


## 7. Conclusion

The applications ofq-calculus have been the focus point in the recent times in various branches of mathematics. This article introduces a new operator in q-analogue for meromorphic multivalent functions. Then, a new subclass of multivalent convex functions is defined and studied for some of its geometric properties like sufficient conditions, coefficient estimates, and distortion. Also, problems of closure and integral representation are discussed in detail. Many other classes can be defined using this operator which will open a lot of new opportunities for research in this and related fields.



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*Source: 5534357-2021-04-15.xml*