# Applications ofq-Derivative Operator to the Subclass of Bi-Univalent Functions Involving q-Chebyshev Polynomials

Authors: Bilal Khan; Zhi-Guo Liu; Timilehin Gideon Shaba; Serkan Araci; Nazar Khan; Muhammad Ghaffar Khan
Journal: Journal of Mathematics
 (2022)

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## Abstract

In recent years, the usage of theq-derivative and symmetric q-derivative operators is significant. In this study, firstly, many known concepts of the q-derivative operator are highlighted and given. We then use the symmetric q-derivative operator and certain q-Chebyshev polynomials to define a new subclass of analytic and bi-univalent functions. For this newly defined functions’ classes, a number of coefficient bounds, along with the Fekete–Szegö inequalities, are also given. To validate our results, we give some known consequences in form of remarks.


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## Body

## 1. Introduction and Definitions

LetℋD denote the class of functions which are analytic in the open unit disk:(1)D=z:z∈ℂandz<1.LetA be the subclass of functions f∈ℋD, which satisfy the normalization condition given by(2)f0=f′0−1=0,that is, which are represented by the following Taylor–Maclaurin series expansion:(3)fz=z+∑k=2∞bkzk,z∈D.Also, letS be the class of functions in A, which are univalent in D.It is well known that every functionf∈S has an inverse f−1 defined by(4)f−1fz=z,z∈D,f−1fw=w,w<r0f;r0f≥14,where(5)f−1w=gw=w−b2w2+2b22−b3w3−5b23−5b2b3+b4w4+⋯.A function is said to be bi-univalent inD if both f and f−1 are univalent in D.LetΣ denote the class of bi-univalent function in D given by (3). Example of functions in the class Σ is(6)z1−z,log11−zandlog1+z1−z.However, the familiar Koebe function is an example of the classΣ. Other common examples of functions in S, such as(7)2z−z22andz1−z2,are also not members of Σ.Lewin [1] investigated a bi-univalent functions class Σ and showed that b2<1.51. Subsequently, Brannan and Clunie [2] conjectured that b2<2. Netanyahu [3], on the contrary, showed that(8)maxf∈Σb2=43.The coefficient for each of the Taylor–Maclaurin coefficientsa2n≥3,n∈ℕ is presumably still an open problem.Similar to the familiar subclassesS∗ζ and Kζ of star-like and convex functions of order ζ0≤ζ<1, respectively, Brannan and Taha [4] introduced certain subclasses of the bi-univalent function class Σ, namely, SΣ∗ζ and KΣζ of bi-star-like functions and bi-convex functions of order ζ0≤ζ<1, respectively. For each of the function classes SΣ∗ζ and KΣζ, they found nonsharp bounds on the first two Taylor–Maclaurin coefficients b2 and b3.Furthermore, lets1 and s2 be analytic functions in open unit disc D; then, the function s1 is subordinated to s2 and symbolically denoted as(9)s1z≺s2z,z∈D,if there occurs an analytic function w with properties that(10)w0=0andwz<1.Supposew holomorphic in D, such that(11)s1z=s2wz.If the functions2 is univalent in D, then the above condition is equivalent to(12)s1z≺s2z⟺s10=s20ands1D⊂s2D.Jackson [5] introduced and studied the q-derivative operator Dq of a function as follows:(13)Dqfz=fz−fqzz1−q=1zz+∑k=2∞1−qk1−qakzkand Dqf0=f′0. In case fz=zk, for k is a positive integer, the q-derivative of f is given by(14)Dqzk=zqk−zkzq−1=1−qk1−qzk−1,(15)limq⟶1−kq=limq⟶1−1−qk1−q=k,where z≠0,q≠0. For more details on the concepts of q-derivative, see [6, 7].The quantum (orq-) calculus is an essential tool for studying diverse families of analytic functions, and its applications in mathematics and related fields have inspired researchers. Srivastava [8] was the first person to apply it in the context of univalent functions. Numerous scholars conducted substantial work on q-calculus and examined its various applications due to the applicability of q-analysis in mathematics and other domains. For example, with the help of certain higher-order q -derivative operators, Khan et al. [7] defined and studied a number of subclasses of q-star-like functions. Also, Shi et al. [9] (see also [10]) used the q-differential operator and defined a new subclass of Janowski-type multivalent q-star-like functions. In [7, 9], a number of sufficient conditions and some other interesting properties have been examined. More importantly, the convolution theory enables us to investigate various properties of analytic functions. Due to the large range of applications of q-calculus and the importance of q-operators instead of regular operators, many researchers have explored q-calculus in depth. In addition, Srivastava [11] recently published survey-cum-expository review paper which is useful for researchers and scholars (see, for example, [12, 13]) working on these subjects. Also, Srivastava’s recent survey-cum-expository review article [11] further motivates the use of the q-analysis in geometric function theory, as well as commenting on the triviality of the so-called p,q-analysis involving an insignificant and redundant parameter p,q (see p. 340 of [11]).Utilizing the idea ofq-derivative operator, in 2013, Brahim et al. introduced and studied the symmetric q-derivative operator D˜qf for a function f as follows:(16)D˜qfz=fqz−fq−1zq−q−1z,z≠0,f′0,z=0.It is easy to see that(17)D˜qzk=kq˜zk−1,D˜qfz=1+∑k=2∞kq˜bkzk−1,where(18)kq˜=qk−q−kq−q−1.The relation betweenq-derivative operator and symmetric q-derivative operator is given by(19)D˜qfz=Dq2fq−1z.Supposef−1 is the inverse of f; then,(20)D˜qf−1w=1−2q˜b2w+3q˜2b22−b3w2−4q˜5b23−5b2b3+b4w3+⋯.Al Salam and Ismail [14] discovered a family of polynomials that can be understood as q-analogues of the second-order bivariate Chebyshev polynomials. In 2012, Johann Cigler introduced and studied the q-Chebyshev polynomials as follows.Definition 1 (see [15]).
The polynomials(21)Umt,y,q=Pm+1t,−1,y,q−q;qm=∑k=0n/2qk2m−kk1+qk+1⋯1+qm−kyktm−2kare called q-Chebyshev polynomial of the second kind.Theorem 1 (see [15]).
Theq-Chebyshev polynomials of the second kind satisfy(22)Umt,y,q=1+qmtUm−1t,y,q+qm−1yUm−2t,y,q,with initial values(23)U0t,y,q=1andU1t,y,q=1+qt.Remark 1.
It is clear that(24)Umt,−1,1=Umt,where Umt is the classical Chebyshev polynomial of the second kind.Now, making useq-Chebyshev polynomials, we define the following.Definition 2.
LetMz,t,y,q be defined as follows:(25)Mz,t,y,q=1+∑j=1∞Ujt,y,qzj.By using the principal of subordination and the symmetricq-derivative operator D˜q, we define the following subclasses of analytic and bi-univalent functions.Definition 3.
A functionf∈Σ given by (3) is said to be in the class M˜Σq,yt if the following conditions are satisfied:(26)D˜qfz≺Mz,t,y,q,12<t<1,0<q<1,z∈D,(27)D˜qf−1w≺Mw,t,y,q,12<t<1,0<q<1,w∈D.We note from (25) that(28)Mz,t,y,q=1+U1t,y,qz+U2t,y,qz2+U3t,y,qz3+⋯,where z∈D and t∈−1,1.Also, from (22), we have the following:(29)U1t,y,q=1+qtU2t,y,q=t21+q1+q2+qyU3t,y,q=1+q1+q21+q3t3+q1+q1+q2ytU4t,y,q=1+q1+q21+q31+q4t4+q1+q1+q21+q4+q2y2t+q4y.The goal of this research is to investigateq-Chebyshev polynomial expansions in order to derive initial coefficient estimates for some subclasses of analytic and bi-univalent functions defined by the symmetric q-derivative operator. In addition, Fekete–Szegö inequalities for the class M˜Σq,yt are established.Lemma 1 (see [16]).
Let the functionpbe given by(30)pz=1+p1z+p2z2+⋯,
be in the classD of functions with positive real part. Then,(31)pn≦2,n∈ℕ.This last inequality is sharp.


## 2. Coefficients Bounds forf∈M˜Σq,yt

Theorem 2.
Letf∈M˜Σq,yt. Then,(32)b2≤1+qt1+qt1+qt23q˜1+q−1+q22q2˜−qy2q2˜+1+q+2q2˜,(33)b3≤1+q2t22q2˜+1+qt3q˜,(34)b4≤51+q2t222q˜3q˜+1+qt4q˜+2t1+qt1+q2−1+2qy4q˜+1+qt1−2t1+q2+1+q21+q3t2+q1+q2y−2qy4q˜.Proof.
Letf∈σ given by (3) be in the class M˜Σq,yt. Then,(35)D˜qfz=Mωz,t,y,q,(36)D˜qf−1w=Mϖw,t,y,q.
Letp,y∈D be defined as(37)pz=1+ωz1−ωz=1+p1z+p2z2+p3z3+⋯⟹ωz=pz−1pz+1,z∈D.(38)yw=1+ϖw1−ϖw=1+y1w+y2w2+y3w3+⋯⟹ϖw=yw−1yw+1,w∈D.
It follows that, from (37) and (38),(39)ωz=12p1z+p2−p122z2+p3−p1p2+p134z3+⋯,(40)ϖw=12y1w+y2−y122w2+y3−y1y2+y134w3+⋯.
From (39) and (40), applying Mz,t,y,q as given in (25), we see that(41)Mωz,t,y,q=1+U1t,y,q2p1z+U1t,y,q2p2−p122+U2t,y,q4p12z2+U1t,y,q2·p3−p1p2+p134+U2t,y,q2p1p2−p122+U3t,y,q8p13z3+⋯,(42)Mϖw,t,y,q=1+U1t,y,q2y1w+U1t,y,q2y2−y122+U2t,y,q4y12w2+U1t,y,q2y3−y1y2+y134+U2t,y,q2y1y2−y122+U3t,y,q8y13w3+⋯.
From (35), (41) and (36), (42), we have(43)2q˜b2=U1t,y,q2p1,(44)3q˜b3=U1t,y,q2p2−p122+U2t,y,q4p12,(45)4q˜b4=U1t,y,q2p3−p1p2+p134+U2t,y,q2p1p2−p122+U3t,y,q8p13,(46)−2q˜b2=U1t,y,q2y1,(47)3q˜2b22−b3=U1t,y,q2y2−y122+U2t,y,q4y12,(48)−4q˜5b23−5b2b3+b4=U1t,y,q2y3−y1y2+y134+U2t,y,q2y1y2−y122+U3t,y,q8y13.
Adding (43) and (46), we have(49)p1=−y1,p12=y12andp13=−y13,(50)b22=U12t,y,qp12+y1282q2˜.
Also, adding (44) and (47) and applying (49) yields(51)23q˜b22=U1t,y,q2p2+y2−y12U1t,y,q−U2t,y,q.
Applying (49) in (50) gives(52)y12=42q2˜b22U12t,y,q.
Putting (52) into (51) with some calculations, we have(53)b22=U13t,y,qp2+y243q˜U12t,y,q−U2t,y,q−U1t,y,q2q2˜.
Applying triangular inequality and Lemma1, we have(54)b2≤1+qt1+qt1+qt23q˜1+q−1+q22q2˜−qy2q2˜+1+q+2q2˜.
Subtracting (47) from (44) with some calculations, we have(55)b3=b22+U1t,y,qp2−y243q˜,(56)b3=U12t,y,qp1242q2˜+U1t,y,qp2−y243q˜.
Applying triangular inequality and Lemma1, we have(57)b3≤1+q2t22q2˜+1+qt3q˜.
Subtracting (48) from (45), we have(58)24q˜b4=54q˜U12t,y,qp1p2−y282q˜3q˜+U1t,y,qp3−y32+U2t,y,q−U1t,y,qp1p2+y22+U1t,y,q−2U2t,y,q+U3t,y,qp134.
Applying triangular inequality and Lemma1, we have(59)b4≤51+q2t222q˜3q˜+1+qt4q˜+2t1+qt1+q2−1+2qy4q˜+1+qt1−2t1+q2+1+q21+q3t2+q1+q2y−2qy4q˜.


## 3. Fekete–Szegö Inequalities for the Function ClassM˜Σq,yt

Thenth coefficient of a function class S is well known to be restricted by n, and the coefficient limits give information about the functions geometric characteristics. The famous problem solved by Fekete–Szegö [17] is to determine the greatest value of the coefficient functional Ωσf/coloneqb3−σb22 over the class S for each σ∈0,1, which was demonstrated using the Loewner technique. In this section, we aim to determine the upper bounds of the coefficient functional b3−δb22 for the function class M˜Σq,yt.Theorem 3.
Letf∈M˜Σq,yt. Then, for someδ∈ℝ,(60)b3−δb22≤1+qt3q˜,δ−1≤Λqq−1,y,t3q˜1+q2t2,1+q3t3δ−1Λqq−1,y,t,δ−1≥Λqq−1,y,t3q˜1+q2t2,where(61)Λqq−1,y,t=1+qt23q2˜1+q−1+q22q2˜−qy2q2˜+1+qt2q2˜.Proof.
From (51) and (55), we have(62)b3−δb22=1−δU13t,y,qp2+y243q˜U12t,y,q−U2t,y,q−U1t,y,q2q2˜+U1t,y,qp2−y243q˜=U1t,y,qJδ+143q˜p2+Jδ−143q˜y2,where(63)Jδ=1−δU12t,y,q43q˜U12t,y,q−U2t,y,q−U1t,y,q2q2˜.
Applying Lemma1, we have(64)H22=b2b4−b32≤1+qt3q˜,0≤Jδ≤143q˜,41+qtJδ,Jδ≥143q˜.Remark 2.
Takingq=1 and y=−1 in Theorem 2 and Theorem 3, we have the results obtained by Altinkaya and Yalcin [18].


## 4. Conclusion

Recently, theq-derivative and symmetric q-derivative operators are particularly applicable in many diverse areas of mathematics and physics. In this study, firstly, many known concepts of the q-derivative operator have been highlighted and given. We have then used the symmetric q-derivative operator and certain q-Chebyshev Polynomials and have defined a new subclass of analytic and bi-univalent functions. For these newly defined functions’ classes, a number of coefficients bonds, along with the Fekete–Szeg ö inequalities, have also been given. To validate our results, we have given some known consequence in the form of Remarks.



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