# On Janowski Analyticp,q-Starlike Functions in Symmetric Circular Domain

Authors: Muhammad Ghaffar Khan; Bakhtiar Ahmad; Basem Aref Frasin; Thabet Abdeljawad
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The main object of the present paper is to apply the concepts ofp,q-derivative by establishing a new subclass of analytic functions connected with symmetric circular domain. Further, we investigate necessary and sufficient conditions for functions belonging to this class. Convex combination, weighted mean, arithmetic mean, growth theorem, and convolution property are also determined.


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## Body

## 1. Introduction and Definitions

Quantum calculus orq-calculus is a generalization of classical calculus without the notation of limits. The theory of q-calculus is established by Jackson, for details see [1, 2]. Due to its numerous applications in various branches of applied sciences and mathematics, for example, physics, operator theory, numerical analysis, and differential equations, attracted researchers to this field. A detailed study on applications of q-calculus in operator theory may be found in [3]. The geometric interpretation of q-calculus has been recognized through studies on quantum groups. Starlikeness and convexity are two major properties of analytic functions. Ismail et al. [4] investigated the generalized starlike function S∗, and certain subclasses close-to-convex functions of q-Mittag-Leffler functions were studied by Srivastava and Bansal [5], also the reader is referred to [6–12] for more details.The foundation of quantum calculus is on one parameter, while the postquantum calculus or simplyp,q-calculus is the generalization of q-calculus based on two parameters. By setting p=1 in p,q-calculus, the q-calculus is obtained. The p,q-integer was considered by Chakrabarti and Jagannathan [13], also see the work [14–18]. The idea of q-starlike is extended to p,q-stalikeness by Raza et al. [19]. Before we define our new class in this field, we give some basics for a better understanding of the work to follow.LetA represent the family of function f that are analytic in the open unit disc D=z∈ℂ:z<1 having the series expansion
(1)fz=z+∑n=2∞anzn,z∈D.A functionfz of the form (1) is subordinate to function gz=z+∑n=2∞bnzn, symbolically represented fz≺gz, if there occur a Schwarz function wz with limitation that w0=0, and wz≤1, then fz=gwz. While the convolution of these functions can be defined by
(2)fz∗gz=z+∑n=2∞anbnzn,z∈D.For0<q<1, the q-derivative of a function f is defined by
(3)∂qfz=fz−fqzz1−q,z≠0,q≠1,where
(4)nq=1−qn1−q=1+∑l=1n−1ql,0,q=0,see [13] for details.Also for0<p<q<1, the p,q-derivative of a function f is defined in [2] as
(5)∂p,qfz=fpz−fqzzp−q,z≠0,p≠q.It can easily be seen that forn∈ℕ≔1,2,3,⋯ and z∈D,∂p,q∑n=1∞anzn=∑n=1∞np,qanzn−1,where(6)np,q=pn−qnp−q.We note that∂1,qfz=∂qfz (for more on this topic one should read [20–22]).Sakaguchi [23], in year 1956, established the class of starlike functions with respect to symmetrical points denoted by Ss∗ of holomorphic univalent functions in D if the below condition is satisfies
(7)Re2zf′zfz−f−z>0,z∈A.Motivated by the work of [19, 23, 24], we now define Sp,q∗l,m,C,D given below.Definition 1.
Let−D≤C<D≤1,0<p<q≤1 and −1≤m<l≤1, then the function f∈A is in the class Sp,q∗l,m,C,D if it satisfies
(8)l−mz∂p,qfzflz−fmz≺1+Cz1+Dz,z∈D,where the symbol “≺” indicates the well-known subordination.We note thatS1,q∗l,m,C,D=Sq∗l,m,C,D, where
(9)Sq∗l,m,C,D=f∈A:l−mz∂qfzflz−fmz≺1+Cz1+Dz,z∈D,and(10)limq→1−Sq∗1,−1,C,D=S∗C,D=f∈A:2zf′zfz−f−z≺1+Cz1+Dz,z∈D.Equivalently, a functionf∈A is in the Sp,q∗l,m,C,D if and only if
(11)l−mz∂p,qfz/flz−fmz−1Dl−mz∂p,qfz/flz−fmz−C<1,z∈D.In our main results, in the next section, we evaluate the criteria for functions belonging to this newly defined class. After that, the convex combination property for this class will be discussed. Then utilizing these results, the weighted mean and arithmetic mean properties will be investigated. Further, convolution type results will be discussed in the form of two theorems. At the end of this article, a conclusion and future work will be presented.


## 2. Main Results

Theorem 2.
Letf∈A be of the form (1). Then the function f∈Sp,q∗l,m,C,D, if and only if the following inequality holds
(12)∑n=2∞np,q1+D−C+1ln−mnl−man<D−C.Proof.
Let us suppose that the first inequality (12) holds. Then to show that f∈Sp,q∗l,m,C,D, we only need to prove the inequality (11). For this consider
(13)l−mz∂p,qfz/flz−fmz−1Dl−mz∂p,qfz/flz−fmz−C=∑n=2∞np,q−ln−mn/l−mαnznD−Cz−∑n=2∞Dnp,q−Cln−mn/l−mαnzn≤∑n=2∞np,q−ln−mn/l−mαnD−C−∑n=2∞Dnp,q−Cln−mn/l−mαn<1,where we used and this completes the direct part. Conversely, let f∈Sp,q∗l,m,C,D be of from (1). Then from (11), we have for z∈D,(14)l−mz∂p,qfz/flz−fmz−1Dl−mz∂p,qfz/flz−fmz−C=∑n=2∞np,q−ln−mn/l−mαnznD−Cz−∑n=2∞Dnp,q−Cln−mn/l−mαnzn<1
SinceRez<z<1, we have
(15)Re∑n=2∞np,q−ln−mn/l−mαnznD−Cz−∑n=2∞Dnp,q−Cln−mn/l−mαnzn<1
Now we choose values ofz on the real axis such that l−mz∂p,qfz/flz−fmz is real. Upon clearing the denominator in (15) and letting z⟶1− through real values, we obtain the required inequality (12).Theorem 3.
Letfi∈Sp,q∗l,m,C,D and having power series representations
(16)fiz=z+∑k=1∞ak,izk,fori=1,2,3,⋯,t.
ThenΦ∈Sp,q∗l,m,C,D, where
(17)Φz=∑i=1tωifizwith∑i=1tωi=1.Proof.
By Theorem2, one can write
(18)∑n=2∞np,q1+D−C+1ln−mn/l−mD−Can,i<1.
Therefore(19)Φz=∑i=1tωiz+∑n=2∞an,izn=z+∑i=1t∑n=2∞ωian,izn=z+∑n=2∞∑i=1tωian,izn;however,
(20)∑n=2∞np,q1+D−C+1ln−mn/l−mD−C∑i=1tωian,i=∑i=1tωi∑n=2∞np,q1+D−C+1ln−mn/l−mD−Can,i≤1,then Φ∈Sp,q∗l,m,C,D. Hence, the proof is completed.Theorem 4.
Iff1,f2∈Sp,q∗l,m,C,D, then their weighted mean ψk is also in Sp,q∗l,m,C,D, where ψk is defined by
(21)ψkz=1−kf1z+1+kf2z2.Proof.
From (21), one can easily write
(22)ψkz=z+∑n=2∞1−kan+1+kbn2zn.
To prove thatψk∈Sp,q∗l,m,C,D, it is enough to show that
(23)∑n=2∞np,q1+D−C+1ln−mn/l−mD−C.1−kan+1+kbn2<1.
For this, consider(24)∑n=2∞np,q1+D−C+1ln−mn/l−mD−C.1−kan+1+kbn2=1−j2.∑n=2∞np,q1+D−C+1ln−mn/l−mD−Can+1+j2.∑n=2∞np,q1+D−C+1ln−mn/l−mD−Cbn<1−k2+1+k2=1,where we have used inequality (12). Which completes the proof.Theorem 5.
Letfi∈Sp,q∗l,m,C,D, with i=1,2,⋯,j. Then, their arithmetic mean φ of fi(25)φz=1j∑i=1jfiz,is also in the class Sp,q∗l,m,C,D.Proof.
From (25), we can write
(26)φz=1j∑i=1jz+∑n=2∞an,izn=z+∑n=2∞1j∑i=1jan,izn.
Sincefi∈Sp,q∗l,m,C,D for every i=1,2,⋯,j, using (12), we have
(27)∑n=2∞np,q1+D−C+1ln−mn/l−mD−C.1j∑i=1jan,i=1j∑i=1j∑n=2∞np,q1+D−C+1ln−mn/l−mD−C.an,i≤1j∑i=1j1=1,which complete the proof.Theorem 6.
Letf∈Sp,q∗l,m,C,D. Then for z=r,0<r<1,(28)r−δp,ql,m,C,Dr2<fz<r+δp,ql,m,C,Dr2,where
(29)δp,ql,m,C,D=C−D2p,q1−D+C+1l+m.(30)r−γp,ql,m,C,Dr2<∂p,qfz<r+γp,ql,m,C,Dr2,where
(31)γp,ql,m,C,D=C−D1−D+C+1l+m.Proof.
To prove (28), consider
(32)fz≤r+∑n=2∞anrn,as 0<r<1 so rn<r2 hence
(33)fz<r+r2∑n=2∞an≤r+C−D2p,q1−D+C+1l+mr2.
Similarly,(34)fz≥r−∑n=2∞anrn>r−r2∑n=2∞an≥r−C−D2p,q1−D+C+1l+mr2.
Hence complete the proof of (28). Similarly, we can prove (30).Theorem 7.
Letfi∈Sp,q∗l,m,C,D, such that
(35)fiz=z+∑n=2∞an,izn,i=1,2,with condition an,2≤1, then f1∗f2∈Sp,q∗l,m,C,D.Proof.
Since form (35), we have
(36)fiz=z+∑n=2∞an,izn,i=1,2.
Then convolution is defined as(37)f1∗f2z=z+∑n=2∞an,1an,2zn.
Sincef2∈Sp,q∗l,m,C,D, with limitation that an,2≤1. Therefore
(38)∑n=2∞np,q1+D−C+1ln−mn/l−mD−Can,1an,2≤∑n=2∞np,q1+D−C+1ln−mn/l−mD−Can,1<1.
Hencef1∗f2∈Sp,q∗l,m,C,D.Theorem 8.
Letfz∈Sp,q∗l,m,C,D. Then
(39)1zfz∗1+Deiθz1−pz1−qz−1+Ceiθz1−lz1−mz≠0.Proof.
Letfz∈Sp,q∗l,m,C,D. Then by definition of subordination, there exists a Schwarz function wz, such that w0=0 and wz<1,(40)l−mz∂p,qfzflz−fmz=1+Cwz1+Dwz,equivalently,
(41)l−mz∂p,qfzflz−fmz≠1+Ceiθ1+Deiθ,(42)z∂p,qfz1+Deiθ−flz−fmzl−m1+Ceiθ≠0,using the relations
(43)z∂p,qfz=fz∗z1−pz1−qz,flz−fmzl−m=fz∗z1−lz1−mz,
now (42), becomes
(44)1zfz∗1+Deiθz1−pz1−qz−1+Ceiθz1−lz1−mz≠0,which completes the proof.


## 3. Conclusions

Utilizing the concepts of postquantum calculus, we defined a new subclass of analytic functions associated with symmetric circular domain. For this class, we investigated some useful results such as necessary and sufficient problem, convex combination, weight mean, arithmetic mean, distortion bounds, and convolution property. There are some problems open for researchers such as radii problems, extreme point theorem, analytic criteria, and integral mean of inequality. Moreover, this concept is new and can be extended to meromorphic functions and harmonic functions.



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