# A Subfamily of Univalent Functions Associated withq-Analogue of Noor Integral Operator

Authors: Muhammad Arif; Miraj Ul Haq; Jin-Lin Liu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

The main objective of the present paper is to define a new subfamily of analytic functions using subordinations along with the newly definedq-Noor integral operator. We investigate a number of useful properties such as coefficient estimates, integral representation, linear combination, weighted and arithmetic means, and radius of starlikeness for this class.


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## Body

## 1. Introduction and Definitions

In recent years,q-analysis (q-calculus) has motivated the researchers a lot due to its numerous applications in mathematics and physics. Jackson [1, 2] was the first to give some application of q-calculus and also introduced the q-analogue of derivative and integral operator. Later on, Aral and Gupta [3, 4] defined the q-Baskakov-Durrmeyer operator by using q-beta function while in papers [5, 6] the authors discussed the q-generalization of complex operators known as q-Picard and q-Gauss-Weierstrass singular integral operators. Using convolution of normalized analytic functions, Kanas and Raducanu [7] defined q-analogue of Ruscheweyh differential operator and studied some of its properties. The application of this differential operator was further studied by Aldweby and Darus [8] and Mahmood and Sokół [9]. The aim of the current paper is to define a q-analogue of the Noor integral operator involving convolution concepts and then give some interesting applications of this operator.Let us denote the open unit disk byD=z∈C:z<1 and the symbol A denotes the family of those analytic functions f which has the following Taylor series representation:(1)fz=z+∑k=2∞akzk,z∈D.For two functions f and g that are analytic in D and have the form (1), we define the convolution of these functions by(2)fz∗gz=z+∑k=2∞akbkzk,z∈D.For0<q<1, the q-derivative of a function f∈A is defined by(3)∂qfz=fqz-fzzq-1,z≠0.It can easily be seen that for n∈N≔1,2,3,… and z∈D(4)∂q∑n=1∞anzn=∑n=1∞n,qanzn-1,where(5)n,q=1-qn1-q=1+∑l=1n-1ql,0,q=0.For any nonnegative integer n, the q-number shift factorial is defined by(6)n,q!=1,n=0,1,q2,q3,q⋯n,q,n∈N.Also the q-generalized Pochhammer symbol for x>0 is given by(7)x,qn=1,n=0,x,qx+1,q⋯x+n-1,q,n∈N.Forμ>-1, we define the function Fq,μ+1-1z by(8)Fq,μ+1-1z∗Fq,μ+1z=z∂qfz,where the functionFq,μ+1z is given by(9)Fq,μ+1z=z+∑n=2∞μ+1,qn-1n-1,q!zn,z∈D.It is quite clear that the series defined in (9) is convergent absolutely in D. Using the definition of q-derivative along with the idea of convolutions, we now define the integral operator Iqμ:A→A by(10)Iqμfz=Fq,μ+1-1z∗fz=z+∑n=2∞ψn-1anzn,z∈D,with(11)ψn-1=n,q!μ+1,qn-1.From (10), we can easily get the identity(12)μ+1,qIqμfz=μ,qIqμ+1fz+qμz∂qIqμ+1fz.We note that Iq0fz=z∂qfz, Iq1fz=fz, and(13)limq→1-⁡Iqμfz=z+∑n=2∞n!μ+1n-1anzn.This shows that, by taking q→1-, the operator defined in (10) reduces to the familiar Noor integral operator introduced in [10, 11]. Also for more details on the q-analogue of differential and integral operators, see the work [12–14].Motivated from the work studied in [7, 15–17], we now define subfamilies of the set A by using the operator Iqμ as follows.Definition 1.
Let-1≤B<A≤1 and 0<q<1. Then the function f∈A is in the class Qqμ,A,B if it satisfies(14)z∂qIqμfzIqμfz≺1+Az1+Bz,z∈D,where the notion “≺” denotes the familiar subordinations.
Equivalently, a functionf∈A is in the class Qqμ,A,B, if and only if(15)z∂qIqμfz/Iqμfz-1A-Bz∂qIqμfz/Iqμfz<1,z∈D.We will assume throughout our discussion, unless otherwise stated, that(16)μ>-1,-1≤B<A≤1,0<q<1and all coefficients ak are positive.We need the following result in the proof of a result.Lemma 2 (see [18]).
Let-1≤B2≤B1<A1≤A2≤1. Then(17)1+A1z1+B1z≺1+A2z1+B2z.


## 2. Main Results

Theorem 3.
Letf∈A be given by (1). Then the function f is in the family Qqμ,A,B, if and only if(18)∑n=2∞n,q1-B-1+Aanψn-1<A-B.Proof.
Let us assume first that inequality (18) holds. To show f∈Qqμ,A,B, we only need to prove the inequality (15). For this, consider(19)z∂qIqμfz/Iqμfz-1A-Bz∂qIqμfz/Iqμfz=∑n=2∞ψn-1n,q-1anznA-Bz+∑n=2∞ψn-1A-Bn,qanzn≤∑n=2∞ψn-1n,q-1anA-B-∑n=2∞ψn-1A-Bn,qan<1,where we have used (4), (10), and (18) and this completes the direct part. Conversely, let f∈Qqμ,A,B be of the form (1). Then from (15) along with (10), we have, for z∈D,(20)z∂qIqμfz/Iqμfz-1A-Bz∂qIqμfz/Iqμfz=∑n=2∞ψn-1n,q-1anznA-Bz+∑n=2∞ψn-1A-Bn,qanzn<1.Since Rez<z, we have(21)Re∑n=2∞ψn-1n,q-1anznA-B+∑n=2∞ψn-1A-Bn,qanzn<1.Now we choose values of z on the real axis such that z∂qIqμfz/Iqμfz is real. Upon clearing the denominator in (21) and letting z→1- through real values, we obtain the required inequality (18).Theorem 4.
Letf∈Qqμ,A,B. Then(22)Iqμfz=exp⁡∫0z1t1-Aϕt1-Bϕtdqt,with ϕz<1 and z∈D.Proof.
Letf∈Qqμ,A,B and setting(23)z∂qIqμfzIqμfz=vz,with(24)vz≺1+Az1+Bz,equivalently, we can write(25)vz-1A-Bvz<1,or in other way, we have(26)vz-1A-Bvz=ϕz,ϕz<1,z∈D.Thus we can rewrite(27)z∂qIqμfzIqμfz=1-Aϕt1-Bϕt,and further by simple computation of integration, the proof is completed.Theorem 5.
Letfi∈Qqμ,A,B and have the form(28)fiz=z+∑k=1∞ak,izk,fori=1,2,…,l.Then F∈Qqμ,A,B, where(29)Fz=∑i=1lcifizwith∑i=1lci=1.Proof.
By the virtue of Theorem3, one can write(30)∑n=2∞n,q1-B-1+Aψn-1A-Ban,i<1.Therefore(31)Fz=∑i=2lciz+∑n=2∞an,izn=z+∑i=2l∑n=2∞cian,izn=z+∑n=2∞∑i=2lcian,izn;however(32)∑n=2∞n,q1-B-1+Aψn-1A-B∑i=2lan,ici=∑i=2l∑n=2∞n,q1-B-1+Aψn-1A-Ban,ici≤1;then F∈Qqμ,A,B. Hence the proof is complete.Theorem 6.
Iff and g belong to Qqμ,A,B, then their weighted mean hj is also in Qqμ,A,B, where hj is defined by(33)hjz=1-jfz+1+jgz2.Proof.
From (33), we can easily write(34)hjz=z+∑n=2∞1-jan+1+jbn2zn.To prove that hj∈Qqμ,A,B, we need to show that(35)∑n=2∞n,q1-B-1+AA-B1-jan+1+jbn2ψn-1<1.For this, consider(36)∑n=2∞n,q1-B-1+AA-B1-jan+1+jbn2ψn-1=1-j2∑n=2∞n,q1-B-1+AA-Bψn-1an+1+j2∑n=2∞n,q1-B-1+AA-Bψn-1bn<1-j2+1+j2=1,where we have used inequality (18). Hence the result follows.Theorem 7.
Letfi with i=1,2,…,λ belong to the class Qqμ,A,B. Then the arithmetic mean h of fi is given by(37)hz=1λ∑i=1λfizand is also in the class Qqμ,A,B.Proof.
From (37), we can write(38)hz=1λ∑i=1λz+∑n=2∞an,izn=z+∑n=2∞1λ∑i=1λan,izn.Since fi∈Qqμ,A,B for every i=1,2,…,λ, using (38) and (18), we have(39)∑n=2∞ψn-1n,q1-B-1+A1λ∑i=1λan,i=1λ∑i=1λ∑n=2∞ψn-1n,q1-B-1+Aan,i≤1λ∑i=1λA-B=A-B,and this completes the proof.Theorem 8.
Letf∈Qqμ,A,B. Then f is in the family S∗β of starlike functions of order β0≤β<1 for z<r1, where(40)r1=1-βn,q1-B-1+An,q!n-βA-Bμ+1,qn-11/n-1.Proof.
Letf∈Qqμ,A,B. To prove f∈S∗β, we only need to show(41)zf′z/fz-1zf′z/fz+1-2β<1.Using (1) along with some simple computation yields(42)∑n=2∞n-β1-βanzn-1<1.Since f∈Qqμ,A,B, from (18), we can easily obtain(43)∑n=2∞n,q!μ+1,qn-1n,q1-B-1+AA-Ban<1.Now inequality (42) will be true, if the following holds:(44)∑n=2∞n-β1-βanzn-1<∑n=2∞n,q!μ+1,qn-1n,q1-B-1-AA-Ban,which implies that(45)zn-1<1-βn,q1-B-1+An,q!n-βA-Bμ+1,qn-1,and thus we get the needed result.Theorem 9.
Let-1≤B2≤B1<A1≤A2≤1 and Iqμ+1fz≠0 in D, and this satisfies(46)μ+1,qqμIqμfzIqμ+1fz-μ,qμ+1,q≺1+A1z1+B1z.Then f∈Qqμ+1,A2,B2.Proof.
SinceIqμ+1fz≠0 in D, therefore let us define the function pz by(47)z∂qIqμ+1fzIqμ+1fz=pzz∈D.By the virtue of identity (12), we obtain(48)1qμμ+1,qIqμfzIqμ+1fz-μ,q=pz.Therefore, using (46), we have(49)z∂qIqμ+1fzIqμ+1fz=pz≺1+A1z1+B1z,and now, using Lemma 2, we have f∈Qqμ+1,A2,B2.



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*Source: 3818915-2018-02-01.xml*