# Applications of Some Generalized Janowski Meromorphic Multivalent Functions

Authors: Bakhtiar Ahmad; Muhammad Ghaffar Khan; Maslina Darus; Wali Khan Mashwani; Muhammad Arif
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, the ideas of post-quantum calculus and meromorphic multivalent functions are combined and some applications of these functions are discussed. We introduce a new subclass of meromorphic multivalent functions in association with Janowski domain. We investigate and study some useful geometric properties of this class of functions such as sufficiency criteria, distortion problem, growth theorem, radii of starlikeness and convexity, convex combination, and coefficient estimates for this class.


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## Body

## 1. Introduction and Definitions

In this article, we introduce a new subclass of meromorphic multivalent functions in parameterα, in post-quantum analogue. The quantum calculus (q-calculus) is the generalization of classical calculus by replacing the notion of limits with a parameter q. In the field of geometric function theory (GFT), the q-generalization of different classes of analytic and meromorphic functions is the current focus of various prominent researchers. Many generalizations have been made and various properties are discussed for these generalizations. This motivation in the recent past is due to its numerous mathematical and physical applications. Jackson started this area of mathematics with the generalization of derivative and integral in q-analogue, which are known as q-derivative and q-integral [1, 2]. The latest advancements in this field can be traced to Srivastava, who along with Bansal [3], investigated a certain family of q-Mittag-Leffer functions. Aldawish and Darus [4] gave results on starlikeness of q-generalized functions. Some results such as coefficient estimates for q-starlike and q-convex functions were evaluated by Seody and Aouf [5]. Similar results for some other subclasses of q-starlike functions were evaluated by Fan et al. in [6]. Mahmood and Sokół in [7] introduced a class in conical domain associated with Ruscheweyh q-differential operator and investigated its various properties. Shi et al. [8] used a generalized operator in the introduction and study of a class of analytic functions. Wang et al. [9] gave a useful generalization of Choi–Saigo–Srivastava operator in q-analogue. For other interesting results, the reader is referred to the work published in [10, 11]. Similarly, the trend was carried to meromorphic functions by various researchers. Ahmad et al. [12] gave the investigation of q-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain and obtained some interesting results. Arif and Ahmad [13] introduced and studied q-analogue of a meromorphic multivalent operator and presented some interested results. The convex generalization of meromorphic multivalent functions in q-analogue can be seen in [14]. Further work by Ahmad and Arif [15] generalized a subclass of meromorphic multivalent close to convex functions via a q-operator.We extend the idea fromq-calculus to post-quantum calculus (P,Q-calculus) for the meromorphic multivalent functions. In P,Q-calculus, there is P,Q-number with two independent variables P and Q. This was first considered by Jagannathan and Chakrabarti [16]. They defined the P,Q-number and generalized several forms of q-oscillator algebras, which are known in the field of physics. In the P,Q-analogue of mathematical concepts, the work of White and Wachs [17] is of most importance. They obtain the generating function of the joint distribution of pairs of statistics known as P,Q-Stirling number. In the field of GFT, the recent research work on generalizing various classes in P,Q-analogue can be seen in [18, 19]. Hence, various applied and theoretical problems have led the way to the creation of P,Q-calculus. Since 1991, many researchers in the fields of mathematics and physics have contributed to the development of P,Q-calculus.Motivated from the above discussed work, in which many ideas have been generalized to post-quantum analogue, we introduce a new class of meromorphic multivalent functions associated with Janowski domain. To better understand this class, we characterize with some results relating to bounds of its coefficients in its power series. Furthermore, we give well-known results of distortion and growth for these functions. In the next section, the region in the open unit disc is evaluated where these functions would behave like starlike and convex functions. After that, an important result of convex combinations is presented. In Section7, we discuss conclusion and potential future work of this research.We give few basic concepts ofP,Q-calculus in the following which are closely related to our work.AnyP,Q-number can be defined mathematically as(1)nP,Q≔Pn−QnP−Q=Pn−1+Pn−1Q+Pn−1Q+⋯+Qn−1,where 0<Q<P≤1 and P≠Q.To make things a bit simpler, we will use the following notation throughout this article:(2)Fn=nP,Q.Some elementary properties of theP,Q-numbers are discussed in the following lines and can be verified easily:(i)
Fn1+n2=Qn2Fn1+Pn1Fn2=Pn2Fn1+Qn1Fn2.(ii)
Fn=P+QFn−1+PQFn−2.(iii)
TheP,Q-factorial is defined by(3)Fn!=FnFn−1Fn−2,…,F1,forn≥1 while F0!=1.(iv)
Similarly,P,Q-derivative is defined mathematically as(4)DP,Qfz=fPz−fQzP−Qz,withz≠0 and P≠Q.(v)
With simple calculations, one can easily deduce the following identity:(5)DP,Qzn=Fnzn−1.For the special value, whenP=1, this derivative becomes the well-known q-derivative in q-calculus.In this article, we introduce a new classℳP,Qp,α,L,M. The class ℳP,Qp,α,L,M is a subclass of meromorphic multivalent functions in P,Q-analogue in Janowski domain which generalizes some important classes both in classical and in q-analogue of this field. Then, we investigate some important properties of this class ℳP,Qp,α,L,M.LetAp denote the class of all meromorphic multivalent functions f that are analytic in the punctured disc D=z∈ℂ:0<z<1, and it is represented by(6)fz=1zp+∑k=1∞ak+pzk+p,z∈D.Next, we recall the definition of subordination, and for two functionsh1,h2∈A, we say h1 is subordinated to h2 and is symbolically written as(7)h1≺h2,if there exist an analytic function wz with the property(8)wz≤z,w0=0,such that(9)h1z=h2wz.Furthermore, ifh2∈S, then the above condition becomes(10)h1≺h2⇔h10=h20,h1D=h2D.Definition 1.
Let−1≤M<L≤1,α≥0,0<P,Q≤1, and P≠Q. Then, the function f∈Ap will be in the class ℳP,Qp,α,L,M if it satisfies the condition(11)−11−αPpQpzDP,QfzFpfz+αP2p+1Q2p+1z2DP,Q2fzFpFp+1fz≺1+Lz1+Mz.Equivalently, a functionfz∈Ap will be in the class ℳP,Qp,α,L,M if and only if the following inequality holds true(12)PpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz+1−α1−αL+MPpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz<1.We note that, for various specific values ofL,M,Q,P, and α, we get the following well-known subclasses.(1)
Ifα=0,P=1, and p=1, we get the class of meromorphic q-starlike functions associated with Janowski functions studied by Mahmood et al. [20](2)
ForP=1,Q⟶1−, and p=1, our class generalizes the class of meromorphic α-convex functions introduced by Ali and Ravichandran [21](3)
PuttingL=1,α=0, P=1,Q⟶1−, and M=−1, we get ℳSp∗, the class of meromorphic p-valent starlike functions investigated by Royster [22](4)
Forα=0,P=1,Q⟶1−,L=1,M=−1, and p=1, we get the class of meromorphic starlike functions studied by Pommerenke [23]For our main results, we will need the following lemmaLemma 1 (see [24]).
For a functionhz, if analytic in D and is represented by the following,(13)hz=1+∑n=1∞cnzn,and kz is analytic and convex in D with(14)kz=1+∑n=1∞knzn,and hz≺kz, then(15)cn≤k1,where n∈N.Now, we give some properties for classℳP,Qp,α,L,M.


## 2. Sufficiency Criteria

In this section, we derive the condition under which functions will belong to this classℳP,Qp,α,L,M.Theorem 1.
Iff belongs to Ap, represented by (6), then the function f will be in ℳP,Qp,α,L,M if and only if the following inequality holds:(16)∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1an+p≤L−M1−αFpFp+1.Proof.
Let us suppose that the first inequality (16) holds. Then, to show that f is in the class ℳP,Qp,α,L,M, we will prove inequality (12). For this, consider(17)PpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz+1−α1−αL+MPpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz=Fp+1PpQpzDP,Qfz+αP2p+1Q2p+1z2DP,Q2fz+1−αFpFp+1fzL1−αFpFp+1fz+MFp+1PpQpzDP,Qfz+αP2p+1Q2p+1z2DP,Q2fz.
Now, with the help of (4)–(6), we get, after some simplification and using the notations,(18)Ψ1n=1−αFpFp+1+PpQpFpFp+n+αP2p+1Q2p+1Fp+nFp+n−1,(19)Ψ2n=L1−αFpFp+1+MPpQpFpFp+n+MαP2p+1Q2p+1Fp+nFp+n−1,for simplicity; then, (17) is equal to(20)∑n=1∞Ψ1nap+nzp+n1−αL−MFpFp+1/zp+∑n=1∞Ψ2nap+nzp+n=∑n=1∞Ψ1nan+pzn+2pL−M1−αFpFp+1+∑n=1∞Ψ2nan+pzn+2p≤∑n=1∞Ψ1nan+pzn+2pL−M1−αFpFp+1−∑n=1∞Ψ2nan+pzn+2p=∑n=1∞Ψ1nan+pzn+2pL−M1−αFpFp+1−∑n=1∞Ψ2nan+pzn+2p.
Asz<1, hence, the above is(21)≤∑n=1∞Ψ1nan+pL−M1−αFpFp+1−∑n=1∞Ψ2nan+p<1,where we use inequality (16), and this completes the direct part.
Conversely, letf∈ℳP,Qp,α,L,M be given by (6). Then, from (12), for z∈D, consider(22)PpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz+1−α1−αL+MPpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz=∑n=1∞Ψ1nan+pzn+2pL−M1−αFpFp+1+∑n=1∞Ψ2nan+pzn+2p.
Sinceℜez≤z, we have(23)ℜe∑n=1∞Ψ1nan+pzn+2pL−M1−αFpFp+1+∑n=1∞Ψ2nan+pzn+2p≤PpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz+1−α1−αL+MPpQpzDP,Qfz/Fpfz+αP2p+1Q2p+1z2DP,Q2fz/FpFp+1fz<1.
Now, choose values ofz on the real axis so that(24)11−αPpQpzDP,QfzFpfz+αP2p+1Q2p+1z2DP,Q2fzFpFp+1fzis real. Upon clearing the denominator in (23) and letting z⟶1− through real values, we obtain (16).


## 3. Coefficient Estimates

In this section, we estimate the upper bounds of the coefficients of the functions in their series representation for this class.Theorem 2.
Letf belong to ℳP,Qp,α,L,M and be of form (6). Then,(25)ap+1≤FpFp+11−αL−MΨ11,ap+2≤FpFp+11−αL−MΨ12+FpFp+11−αL−M2Ψ12Ψ11,ap+3≤FpFp+11−αL−MΨ13+FpFp+11−αL−M2Ψ13Ψ11+FpFp+11−αL−M2Ψ13Ψ12+FpFp+11−αL−M3Ψ13Ψ12Ψ11,where Ψ1n is defined in (18).Proof.
Forf∈Ap, it will belong to ℳP,Qp,α,L,M if the following holds true:(26)−11−αPpQpzDp,QfzFpfz+αP2p+1Q2p+1z2DP,Q2fzFpFp+1fz≺1+Lz1+Mz.
Now, if(27)hz=−11−αPpQpzDP,QfzFpfz+αP2p+1Q2p+1z2DP,Q2fzFpFp+1fz,then(28)hz≺1+Lz1+Mz,and it will be of the form(29)hz=1+∑n=1∞dnzn.
However,(30)1+Lz1+Mz=1+L−Mz+⋯.
Then, by Lemma1, we obtain(31)dn≤L−M.
From (27), we can easily obtain(32)−11−αFp+1PpQpzDP,Qfz+αP2p+1Q2p+1z2DP,Q2fz=FpFp+1hzfz.
Now, putting the series expansions ofhz and fz in this and simplifying and comparing the coefficients of zp+n on both sides of the above equation, then(33)−Fp+nFp+1PpQp+αP2p+1Q2p+1Fp+n−11−αap+n=FpFp+1ap+n+FpFp+1ap+n−1d1+ap+n−2d2+⋯+ap+1dn−1,which implies that(34)−Fp+nFp+1PpQp+αP2p+1Q2p+1Fp+n−1+1−αFpFp+11−αap+n=FpFp+1ap+n−1d1+ap+n−2d2+⋯+ap+1dn−1.
Taking absolute on both sides,(35)−Fp+nFp+1PpQp+αP2p+1Q2p+1Fp+n−1+1−αFpFp+11−αap+n=FpFp+1ap+n−1d1+ap+n−2d2+⋯+ap+1dn−1.
Using the triangle inequality,(36)Fp+nFp+1PpQp+αP2p+1Q2p+1Fp+n−1+1−αFpFp+11−αap+n≤FpFp+1ap+n−1d1+ap+n−2d2+⋯+ap+1dn−1.
Now, using (31), we obtain(37)Fp+nFp+1PpQp+αP2p+1Q2p+1Fp+n−1+1−αFpFp+11−αap+n≤FpFp+1L−M1+ap+1+ap+2+⋯+ap+n−1,which implies that(38)ap+n≤1−αFpFp+1L−MFp+nFp+1PpQp+aP2p+1Q2p+1Fp+n−1+1−αFpFp+1×1+ap+1+ap+2+⋯+ap+n−1.
Now, by puttingn=1,2, and 3 and using notation (18) for simplicity, we get the required result.


## 4. Distortion and Growth-Type Inequalities

Here, we discuss distortion-type inequality first. As inP,Q-analogue, we use P,Q-derivative in the distortion result instead of simple derivative. Furthermore, we prove the growth theorem for this class.Theorem 3.
Iff∈ℳP,Qp,α,L,M and is of form (6), then, for z=r,(39)FpQpPprp+1−J1rp≤DP,Qfz≤FpQpPprp+1+J1rp,where(40)J1=1−αL−MFpFp+11−α1+LFp+1+MPpQpFp+1+αP2p+1Q2p+1Fp.Proof.
From (6) with the help of (4) and (5), we can write(41)DP,Qfz=−FpQpPpzp+1+∑n=1∞Fp+nan+pzn+p−1,⟹DP,Qfz≤FpQpPpzp+1+∑n=1∞Fp+nan+pzn+p−1.
Sincez=r<1, so rn+p−1≤rp, and hence,(42)DP,Qfz≤FpQpPprp+1+rp∑n=1∞Fp+nan+p.
Similarly,(43)DP,Qfz≥FpQpPprp+1−rp∑n=1∞Fp+nan+p.
Since we know from (16) that(44)∑n=1∞1+L1−αFpFp+1Fp+n+1+MQpPpFp+1+αP2p+1Q2p+1Fp+n−1Fp+nan+p≤1−αL−MFpFp+1and(45)1+L1−αFp+1+MQpPpFp+1+αP2p+1Q2p+1Fp∑n=1∞Fp+nan+p≤∑n=1∞1+L1−αFpFp+1Fp+n+1+MQpPpFp+1+αP2p+1Q2p+1Fp+n−1Fp+nan+p,hence, by transitivity, we obtain(46)1+L1−αFp+QpPpFp+1+αP2p+1Q2p+1Fp1+M∑n=1∞Fp+nan+p≤L−M1−αFpFp+1,which implies that(47)∑n=1∞Fp+nan+p≤1−αL−MFpFp+11−α1+LFp+1+MQpPpFp+1+αP2p+1Q2p+1Fp.
Now, we put this inequality in (42) and (43),(48)DP,Qfz≤FpQpPprp+1+rp1−αL−MFpFp+11−α1+LFp+1+MQpPpFp+1+αP2p+1Q2p+1Fp.
Similarly,(49)DP,Qfz≥FpQpPprp+1−rp1−αL−MFpFp+11−α1+LFp+1+MQpPpFp+1+αP2p+1Q2p+1Fp,from which we obtain the required result.In the next result, we discuss the growth theorem for this class.Theorem 4.
Letf∈ℳP,Qp,α,L,M and is of form (6). Then, for z=r,(50)1rp−J2rp≤fz≤1rp+J2rp,where(51)J2=1−αL−MFp1−α1+LFp+1+MPpQpFp+1+αP2p+1Q2p+1Fp.Proof.
Consider(52)fz=1zp+∑n=1∞an+pzn+p≤1zp+∑n=1∞an+pzn+p=1rp+∑n=1∞an+prn+p.
Asz=r<1, so rn+p−1≤rp and(53)fz≤1rp+rp∑n=1∞an+p.
Similarly,(54)fz≥1rp−rp∑n=1∞an+p.
Since we know from (16) that(55)∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+n−1Fp+nan+p≤1−αL−MFpFp+1and(56)1+L1−αFp+PpQpFp+1+αP2p+1Q2p+1Fp1+MFp+1∑n=1∞an+p≤∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+n−1Fp+nan+p,hence(57)1+L1−αFp+PpQpFp+1+αP2p+1Q2p+1Fp1+MFp+1∑n=1∞an+p≤1−αL−MFpFp+1,which gives(58)∑n=1∞an+p≤1−αL−MFp1−α1+LFp+1+MPpQpFp+1+αP2p+1Q2p+1Fp.
Now, by putting this value in (53) and (54),(59)fz≤1rp+rp1−αL−MFp1−α1+LFp+1+MPpQpFp+1+αP2p+1Q2p+1Fp.
Similarly,(60)fz≥1rp−rp1−αL−MFp1−α1+LFp+1+MPpQpFp+1+αP2p+1Q2p+1Fp,and from this, we get the required result.


## 5. Radii of Convexity and Starlikeness

In this section, we find the region where these functions will behave like convex and starlike functions of some orderσ.Theorem 5.
Letf∈ℳP,Qp,α,L,M; then, f will be in the class ℳCpσ for z<r1, where(61)r1=pp−σ1−α1+LFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1p+nn+p+σ1−αL−MFpFp+11/n+2p.Proof.
Letf∈ℳP,Qp,α,L,M. To prove f∈ℳCpσ, we only need to show(62)zf″z/f′z+p+1zf″z/f′z+1+2σ−p<1.
Using (6) along with some simple computation yields(63)∑n=1∞p+nn+p+σpp−σan+pzn+2p<1.
From (16), we can easily obtain that(64)∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1an+p≤1−αL−MFpFp+1⟹∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−11−αL−MFpFp+1an+p<1.
Now, inequality (63) will be true if the following holds(65)∑n=1∞p+nn+p+σpp−σan+pzn+2p<∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−11−αL−MFpFp+1an+p,which implies that(66)p+nn+p+σpp−σzn+2p≤1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−11−αL−MFpFp+1,so(67)zn+2p<pp−σ1−α1+LFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1p+nn+p+σ1−αL−MFpFp+1,from which we obtain(68)z<pp−σ1−α1+LFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1p+nn+p+σ1−αL−MFpFp+11/n+2p=r1,and thus the required condition.Theorem 6.
Letf∈ℳP,Qp,α,L,M; then, f will be in the class ℳSp∗σ for z<r2, where(69)r2=p−σ1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1n+p+σ1−αL−MFpFp+11/n+2p.Proof.
We know thatf belongs to ℳSp∗σ if and only if(70)zf′z/fz+pzf′z/fz−p−2σ≤1.
Using (6) and upon simplification yield(71)∑n=1∞n+p+σp−σan+pzn+2p<1.
Now, from (16), we can easily obtain(72)⟹∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−11−αL−MFpFp+1an+p<1.
For inequality (71) to be true, it will be enough if(73)∑n=1∞n+p+σp−σan+pzn+2p<∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−11−αL−MFpFp+1an+p.
However, this implies that(74)n+p+σp−σzn+2p<1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−11−αL−MFpFp+1,which gives(75)zn+2p<p−σ1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1n+p+σ1−αL−MFpFp+1,and hence,(76)z<p−σ1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1n+p+σ1−αL−MFpFp+11/n+2p=r2.
Thus, the required result is proved.


## 6. Convex Combination

Theorem 7.
The classℳP,Qp,α,L,M is closed under convex combination.Proof.
Let us consider that we have two functions,(77)fiz=1zp+∑k=1∞ak+p,izk+p,i=1,2,z∈D,to prove our result, and we will show that(78)hz=γf1z+1−γf2z,0≤γ≤1,is in the class ℳP,Qp,α,L,M.
As(79)hz=1zp+∑k=1∞γak+p,1+1−γak+p,2zn,if we consider(80)∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1γak+p,1+1−γak+p,2≤γ∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1ak+p,1+1−γ∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1ak+p,2,by using Theorem 1 for both f1z and f1z, we have(81)∑n=1∞1+L1−αFpFp+1+1+MPpQpFp+1Fp+n+αP2p+1Q2p+1Fp+nFp+n−1γak+p,1+1−γak+p,2≤γ1−αL−MFpFp+1+1−γ1−αL−MFpFp+1=1−αL−MFpFp+1,and hence, by Theorem 1, hz∈ℳP,Qp,α,L,M.


## 7. Conclusion and Future Study

In this article, the concepts of post-quantum calculus are used to define a subfamily of meromorphic multivalent functions in association with Janowski functions. Some interesting results relating to geometry of these functions are proved for this class. This class not only generalizes some well-known classes in classical GFT but also covers many classes inq-analogue. This combination of post-quantum Calculus and GFT is an innovative concept and will of course lead to many new ideas and applications in the fields of mathematics. Various classes in the field of GFT such as analytic functions and harmonic functions can be generalized using the ideas and methods presented in this article.



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