# Maclaurin Coefficient Estimates for New Subclasses of Bi-univalent Functions Connected with aq-Analogue of Bessel Function

Authors: Sheza M. El-Deeb
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this paper, we introduce new subclasses of the function classΣ of bi-univalent functions connected with a q-analogue of Bessel function and defined in the open unit disc. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients a2 and a3 for functions in these new subclasses.


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## Body

## 1. Introduction, Definitions, and Preliminaries

The theory ofq-calculus plays an important role in many areas of mathematical, physical, and engineering sciences. Jackson (see [1, 2]) was the first to have some applications of the q-calculus and introduced the q-analogue of the classical derivative and integral operators (see also [3]). Let A denote the class of analytic functions of the form
(1)fz=z+∑k=2∞akzk,z∈Δ≔z∈ℂ:z<1,and S be the subclass of A which are univalent functions in Δ.Ifg∈A is given by
(2)gz=z+∑k=2∞bkzk,z∈Δ,then, the Hadamard (or convolution) product of f and g is defined by
(3)f∗gz≔z+∑k=2∞akbkzk,z∈Δ.Iff and F are analytic functions in Δ, we say that fis subordinate toF, written f≺F, if there exists a Schwarz functionw, which is analytic in Δ, with w0=0, and wz<1 for all z∈Δ, such that fz=Fwz, z∈Δ. Furthermore, if the function F is univalent in Δ, then we have the following equivalence (see [4, 5]):
(4)fz≺Fz⇔f0=F0andfΔ⊂FΔ.TheBessel function of the first kind of orderν is defined by the infinite series (see [6])
(5)Jνz≔∑k≥0−1kz/22k+νk!Γk+ν+1z∈ℂ,ν∈Rwhere Γ stands for the Gamma function. Recently, Szász and Kupán [7] investigated the univalence of the normalized Bessel function of the first kind kν:Δ⟶ℂ defined by (see also [8–10])
(6)kνz≔2νΓν+1z1−ν2Jνz12=z+∑k=2∞−1k−1Γν+14k−1k−1!Γk+νzk,z∈Δ,ν∈R.For 0<q<1, the q-derivative operator for kν is defined by
(7)∂qkνz=∂qz+∑k=2∞−1k−1Γν+14k−1k−1!Γk+νzk≔kνqz−kνzzq−1=1+∑k=2∞−1k−1Γν+14k−1k−1!Γk+νk,qzk−1,z∈Δ,where
(8)k,q≔1−qk1−q=1+∑j=1k−1qj,0,q≔0.Using the definition formula (8), we will define the next two products:
(i)
For any nonnegative integerk, the q-shifted factorial is given by
(9)k,q!≔1,ifk=0,1,q2,q3,q⋯k,q,ifk∈ℕ.(ii)
For any positive numberr, the q-generalized Pochhammer symbol is defined by
(10)r,qk≔1,ifk=0,r,qr+1,q⋯r+k−1,q,ifk∈ℕ.Forν>0, λ>−1, and 0<q<1, El-Deeb and Bulboacă [11] define the function Iν,qλ:Δ⟶ℂ by
(11)Iν,qλz≔z+∑k=2∞−1k−1Γν+14k−1k−1!Γk+νk,q!λ+1,qk−1zk,z∈Δ.A simple computation shows that(12)Iν,qλz∗Mq,λ+1z=z∂qkνz,z∈Δ,where the function Mq,λ+1 is given by
(13)Mq,λ+1z≔z+∑k=2∞λ+1,qk−1k−1,q!zk,z∈Δ.Using the definition ofq-derivative along with the idea of convolutions, El-Deeb and Bulboacă [11] introduce the linear operator Nν,qλ:A⟶A defined by
(14)Nν,qλfz≔Iν,qλz∗fz=z+∑k=2∞ψkakzk,z∈Δ,ν>0,λ>−1,0<q<1,where
(15)ψk≔−1k−1Γν+14k−1k−1!Γk+ν·k,q!λ+1,qk−1.Remark 1.
From the definition relation (14), we can easily verify that the next relations hold for all f∈:
(16)iλ+1,qNν,qλfz=λ,qNν,qλ+1fz+qλz∂qNν,qλ+1fz,z∈Δ;iilimq→1−Nν,qλfz=Iν,1λ∗fz≕Iνλfz=z+∑k=2∞k!λ+1k−1−1k−1Γν+14k−1k−1!Γk+νakzk,z∈Δ.The Koebe one quarter theorem (see [12]) proves that the image of Δ under every univalent function f∈S contains a disk of radius 1/4. Therefore, every function f∈S has an inverse f−1 satisfying
(17)f−1fz=zz∈Δ,ff−1w=ww<r0f;r0f≥14,where
(18)f−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+⋯.A functionf∈A is said to be bi-univalent in Δ if both fz and f−1z are univalent in Δ. Let Σ denote the class of bi-univalent functions in Δ given by (1). For a brief history and interesting examples in the class Σ, see [13]. Brannan and Taha [14] (see also [15–17]) introduced certain subclasses of the bi-univalent functions class Σ similar to the familiar subclasses S∗α and Kα of starlike and convex functions of order α0≤α<1, respectively (see [13]). Thus, following Brannan and Taha [14], a function f∈A is said to be in the class SΣ∗α of strongly bi-starlike functions of order α0<α≤1 if each of the following conditions is satisfied:
(19)f∈Σ,argzf′zfz<απ20<α≤1,z∈Δargzg′wgw<απ20<α≤1,w∈Δ,where the function g is given by
(20)gw=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+…,and g is the extension of f−1 to Δ. The classes SΣ∗α and KΣα of bi-starlike functions of order α and biconvex functions of order α0<α≤1, corresponding to the function classes S∗α and Kα, were also introduced analogously. For each of the function classes SΣ∗α and KΣα, they found nonsharp estimates on the first two Taylor-Maclaurin coefficients a2 and a3 (for details, see [14, 17]).The objective of the present paper is to introduce new subclasses of the function classΣ and find estimates on the coefficients a2 and a3 for functions in these new subclasses of the function class Σ employing the techniques used earlier by Srivastava et al. [18].Now, we define the subclasses of functionsQΣqα,γ,λ,ν,PΣqβ,γ,λ,ν, and QΣh,p,γ,ν,q,λ as follows:Definition 2.
Letfz be given by (1), then fz is said to be in the class QΣqα,γ,λ,ν if the following conditions are satisfied:
(21)f∈ΣargzNν,qλfz′+γz2Nν,qλfz′′1−γNν,qλfz+γzNν,qλfz′<απ2ν>0;λ>−1;0<q<1;0<α≤1;0≤γ≤1,z∈Δ,(22)argwNν,qλgw′+γw2Nν,qλgw′′1−γNν,qλgw+γwNν,qλgw′<απ2ν>0;λ>−1;0<q<1;0<α≤1;0≤γ≤1,w∈Δ,where the function g is given by (2).
Puttingq⟶1−, we obtain that limq→1−QΣqα,γ,λ,ν≕GΣα,γ,λ,ν, where
(23)f∈Σ,argzIνλfz′+γz2Iνλfz′′1−γIνλfz+γzIνλfz′<απ2ν>0;λ>−1;0<α≤1;0≤γ≤1,z∈Δ,argwIνλgw′+γw2Iνλgw′′1−γIνλfgw+γwIνλgw′<απ2ν>0;λ>−1;0<α≤1;0≤γ≤1,w∈Δ.Definition 3.
Letfz be given by (1), then fz is said to be in the class PΣqβ,γ,λ,ν if the following conditions are satisfied:
(24)f∈ΣandRzNν,qλfz′+γz2Nν,qλfz′′1−γNν,qλfz+γzNν,qλfz′>βν>0;λ>−1;0<q<1;0≤β<1;0≤γ≤1,z∈Δ,(25)RwNν,qλgw′+γw2Nν,qλgw′′1−γNν,qλgw+γwNν,qλgw′>βν>0;λ>−1;0<q<1;0≤β<1;0≤γ≤1,w∈Δ,where the function g is given by (2).
Puttingq⟶1−, we obtain that limq→1−PΣqβ,γ,λ,ν≕HΣβ,γ,λ,ν, where
(26)f∈ΣandRzIνλfz′+γz2Iνλfz′′1−γIνλfz+γzIνλfz′>βν>0;λ>−1;0≤β<1;0≤γ≤1;z∈Δ,RwIνλgw′+γw2Iνλgw′′1−γIνλgw+γwIνλgw′>βν>0;λ>−1;0≤β<1;0≤γ≤1;w∈Δ.To prove our results, we need the following lemma.Lemma 4 (see [19], Lemma 3).
IfG∈P then ck≤2 for each k, where P is the family of all functions G analytic in Δ for which RGz>0,Gz=1+c1z+c2z2+⋯⋯ for z∈Δ.Definition 5.
Letfz be given by (1), hz and pw in P have the forms
(27)hz=1+h1z+h2z2+h3z3+.⋯,(28)pw=1+u1w+u2w2+u3w3+⋯,then fz is said to be in the class QΣh,p,γ,ν,q,λ if the following conditions are satisfied:
(29)f∈ΣandzNν,qλfz′+γz2Nν,qλfz′′1−γNν,qλfz+γzNν,qλfz′∈hΔν>0;λ>−1;0<q<1;0<α≤1;0≤γ≤1,z∈Δ,(30)wNν,qλgw′+γw2Nν,qλgw′′1−γNν,qλgw+γwNν,qλgw′∈pΔν>0;λ>−1;0<q<1;0<α≤1;0≤γ≤1,w∈Δ,where the function g is given by (2).


## 2. Coefficient Bounds for the Function ClassQΣqα,γ,λ,ν

Unless otherwise mentioned, we assume throughout this paper that(31)0<α≤1,0≤β<1,0≤γ≤1,ν>0,λ>−1,0<q<1,z∈Δ.Theorem 6.
Letfz be given by (1) which belongs to the class QΣqα,γ,λ,ν, then
(32)a2≤2α4α1+2γψ3+1−3α1+γ2ψ22,(33)a3≤α1+2γψ3+4α21+γ2ψ22.where ψk, k∈2,3, are given by (15).Proof.
It follows from (21) and (22) that
(34)zNν,qλfz′+γz2Nν,qλfz′′1−γNν,qλfz+γzNν,qλfz′=pzα,(35)wNν,qλgw′+γw2Nν,qλgw′′1−γNν,qλgw+γwNν,qλgw′=qwα,where pz and qw in P and have the forms
(36)pz=1+p1z+p2z2+p3z3+⋯,(37)qw=1+q1w+q2w2+q3w3+⋯.Now, equating the coefficients in (34) and (35), we get
(38)1+γψ2a2=p1α,(39)21+2γψ3a3−1+γ2ψ22a22=αp2+αα−12p12,(40)−1+γψ2a2=q1α,(41)21+2γ2a22−a3ψ3−1+γ2ψ22a22=αq2+αα−12q12.
From (38) and (40), we get
(42)p1=−q1,(43)21+γ2a22ψ22=α2p12+q12.
From (39), (41), and (43), we obtain
(44)a22=α2p2+q24α1+2γψ3+1−3α1+γ2ψ22.
Applying Lemma4 for the coefficients p2 and q2, we immediately have
(45)a2≤2α4α1+2γψ3+1−3α1+γ2ψ22.
This gives the bound ona2 as asserted in (32).
Next, in order to find the bound ona3, by subtracting (41) from (39), we get
(46)41+2γa3ψ3−41+2γψ3a22=αp2−q2+αα−12p12−q12.
It follows from (42), (43), and (46) that
(47)a3=α2p12+q1221+γ2ψ22+αp2−q241+2γψ3.
Applying Lemma4 once again for the coefficients p1,p2,q1, and q2, we immediately have
(48)a3≤4α21+γ2ψ22+α1+2γψ3.
This completes the proof of Theorem6.


## 3. Coefficient Bounds for the Function ClassPΣqβ,γ,λ,ν

Theorem 7.
Letfz be given by (1) which belongs to the class PΣqβ,γ,λ,ν, then
(49)a2≤21−β21+2γψ3−1+γ2ψ22.(50)a3≤1−β1+2γψ3+41−β21+γ2ψ22,where ψk, k∈2,3, are given by (15).Proof.
It follows from (24) and (25) that
(51)zNν,qλfz′+γz2Nν,qλfz′′1−γNν,qλfz+γzNν,qλfz′=β+1−βpz(52)wNν,qλgw′+γw2Nν,qλgw′′1−γNν,qλgw+γwNν,qλgw′=β+1−βqw,where pz and qw have the forms (36) and (37), respectively. Equating the coefficients in (51) and (52), we get
(53)1+γψ2a2=1−βp1,(54)21+2γψ3a3−1+γ2ψ22a22=1−βp2,(55)−1+γψ2a2=1−βq1,(56)21+2γ2a22−a3ψ3−1+γ2ψ22a22=1−βq2.
From (53) and (55), we get
(57)p1=−q1,(58)21+γ2ψ22a22=1−β2p12+q12.
From (54) and (56), we obtain
(59)a22=1−βp2+q241+2γψ3−21+γ2ψ22.
Applying Lemma4 for the coefficients p2 and q2, we immediately have
(60)a2≤21−β21+2γψ3−1+γ2ψ22.
This gives the bound ona2 as asserted in (49).
Next, in order to find the bound ona3, by subtracting (56) from (54), we get
(61)41+2γψ3a3−41+2γψ22a22=1−βp2−q2.
It follows from (58) and (61) that
(62)a3=1−βp2−q241+2γψ3+1−β2p12+q1221+γ2ψ3.
Applying Lemma4 once again for the coefficients p1,p2,q1, and q2, we immediately have
(63)a3≤1−β1+2γψ3+41−β21+γ2ψ22.This completes the proof of Theorem6.


## 4. General Coefficient Bounds for the Function ClassQΣqh,p,γ,λ,ν

This section begins by finding the estimates on the coefficientsa2 and a3 for functions in the class QΣqh,p,γ,λ,ν.Theorem 8.
Letfz be given by (1) which belongs to the class QΣqh,p,γ,λ,ν, then
(64)a2≤minh′02+p′0221+γ2ψ22,h′′0+u′′041+2γψ3−21+γ2ψ22,(65)a3≤minh′02+u′0221+γ2ψ22+h′′0+u′′041+2γψ3,41+2γψ3−1+γ2ψ22h′′0+1+γ2ψ22u′′041+2γψ321+2γψ3−1+γ2ψ22,where ψk, k∈2,3, are given by (15).Proof.
It follows from (29) and (30) that
(66)zNν,qλfz′+γz2Nν,qλfz′′1−γNν,qλfz+γzNν,qλfz′=hΔ,(67)wNν,qλgw′+γw2Nν,qλgw′′1−γNν,qλgw+γwNν,qλgw′=pΔ,where hz and pw are given by (27) and (28). Now, equating the coefficients of z2 and z3 in (66) and (67), we get
(68)1+γψ2a2=h1,(69)21+2γψ3a3−1+γ2ψ22a22=h2,(70)−1+γψ2a2=u1,(71)21+2γ2a22−a3ψ3−1+γ2ψ22a22=u2.From (68) and (70), we get
(72)h1=−u1,(73)a22=h12+u1221+γ2ψ22.
From (69) and (71), we obtain
(74)a22=h2+u241+2γψ3−21+γ2ψ22.
Therefore, we obtain from the equations (73) and (74) that
(75)a2≤h′02+u′0221+γ2ψ22,a2≤h′′0+u′′041+2γψ3−21+γ2ψ22,respectively, this gives the bound on a2 as asserted in (64).
Next, in order to find the bound ona3, by subtracting (71) from (69), we get
(76)41+2γa3ψ3−41+2γψ3a22=h2−u2.
It follows from (73) into (76) that
(77)a3=h12+u1221+γ2ψ22+h2−u241+2γψ3.We immediately have
(78)a3≤h′02+u′0221+γ2ψ22+h′′0+u′′041+2γψ3.
On the other hand, upon substituting the value ofa22 from (74) into (76), we get
(79)a3=41+2γψ3−1+γ2ψ22h2+1+γ2ψ22u241+2γψ321+2γψ3−1+γ2ψ22.
Consequently, we get(80)a3≤41+2γψ3−1+γ2ψ22h′′0+1+γ2ψ22u′′041+2γψ321+2γψ3−1+γ2ψ22.This completes the proof of Theorem8.



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