# On the Partial Sums of the q-Generalized Dini Function

Authors: Alaa H. El-Qadeem; Mohamed A. Mamon; Ibrahim S. Elshazly
Journal: Journal of Mathematics
 (2022)

---

## Abstract

Our objective in this paper is to introduce a q-analog of the generalized Dini function. Also, we investigate the lower bound for the ratio of the q-generalized Dini function to its sequences of partial sums.


---

## Body

## 1. Introduction and Basic Concepts

LetA denote the class of functions f that have the following Maclaurin’s form,(1)fz=z+∑n=2∞anzn,which is analytic and univalent in the open unit disc U=z:z<1 and satisfies the normalization conditions f0=f′0−1=0.Special functions play an inspired role in applied mathematics and physics. The widespread use of these functions has attracted many researchers to work in many directions. Lately, many authors studied the geometric properties of some special functions such as starlikness, univalence, and convexity, see [1–6]. There are several results related to partial sums of analytic univalent functions that were developed by the authors in [7–9]. Specifically, the authors in [10] investigated the partial sums of the generalized Bessel function, and then, a lot of authors followed them in studying the same problem for different special functions such as Bessel [11, 12], Struve [13], Lommel [14], Wright [15], and Mittag-Leffler [16], see also [17].Our aim in this study is to develop a q-analog of the generalized Dini function, which is inspired by early studies on analytic and special functions. We also provide lower bounds for the ratio ofq-generalized Dini function to its sequences of partial sums, for m∈ℕ=1,2,3,…, ψν,b,caz;qm=z+∑n=1mζnzn+1. We will investigate the following:(2)ℜψν,b,caz;qψν,b,caz;qm,ℜψν,b,caz;qmψν,b,caz;q,ℜψν,b,caz;q′ψν,b,caz;qm′,andℜψν,b,caz;qm′ψν,b,caz;q′.To introduce the main results, we would like to recall some fundamentals and concepts related to geometric function theory and the definition of q-generalized Bessel function. At first, let us consider the following second-order linear homogenous differential equation (for more details, see [18–20]):(3)z2w″z+bzw′z+cz2−ν2+1−bνwz=0,b,c,ν∈ℂ.The functionwν,b,cz is known as the generalized Bessel function of the first kind of order ν, which is a particular solution of equation (3). The function wν,b,cz has the following series representation:(4)wν,b,cz=∑n=0∞−cnn!Γν+n+b+1/2z22n+ν,where Γ stands for the Gamma function.Now, letq∈0,1 and a∈ℂ; the q-shifted factorial is defined by(5)a;qn=1,n=0,∏k=0n−11−aqk,n∈ℕ.The limit ofa;qn as n tends to infinity exists and is denoted by a;q∞:(6)a;q∞=∏k=0∞1−aqk.The multipleq-shifted factorial for complex numbers a1,a2,…,ar is defined by(7)a1,a2,…,ar;qn=∏j=1raj;qn.Ifaqα≠q−n, for all n∈ℕ0≔ℕ∪0, we define a;qα to be(8)a;qα=a;q∞aqα;q∞.Definition 1.
Letb,ν∈ℝ and c∈ℂ; the q-generalized Bessel function is defined by(9)ωv,b,cz;q=qk;q∞q;q∞∑n=0∞−cn1−qb−1/2q;qnqk;qnz22n+ν=1q;qk−1∑n=0∞−cn1−qb−1/2q,qk;qnz22n+ν,where k=ν+b+1/2≠0,−1,−2,…, z∈U and q∈0,1.
Now, we introduce theq-generalized Dini function ϕν,b,caz;q in terms of ωv,b,cz;q.Definition 2.
Leta∈ℝ+; the q-generalized Dini function ϕν,b,caz;q is defined by(10)ϕν,b,caz;q=a−νωv,b,cz;q+zωv,b,c′z;q,=1q;qk−1∑n=0∞−cn1−qb−1/2a+2nq,qk;qnz4n+v/2.Remark 1.
By specializing the value ofa,b,c, and q, we see that(1)
limq⟶1−ωv,b,c1−qz;q=ωv,b,cz is the generalized Bessel function of first kind introduced by Orhan and Yagmur [10](2)
By puttingb=c=1, then ωv,1,1z;q=Jv1z;q is the first kind of q-Bessel function given by Annaby and Mansour [21]. Also, limq⟶1−ωv,1,11−qz;q=Jvz is the familiar Bessel function defined by Baricz [18].(3)
By puttingb=−c=1, then ωv,1,−1z;q=Iv1z;q is the first kind of modified q-Bessel function given by Annaby and Mansour [21]. Also, limq⟶1−ωv,1,−11−qz;q=Ivz is the modified Bessel function defined by Baricz [18].(4)
limq⟶1−ϕν,b,ca1−q2z;q=Dv,a,b,cz is the generalized Dini function investigated by Deniz and Go¨ren [3]. Also, by putting b=c=1 in the last expression, we get dν,az introduced by Aktaş and Orhan [22]. In addition, by putting a=1, we obtain the Dini function dνz which is introduced by Baricz et al. [2].
Because the functionϕν,b,caz;q defined by (10) do not belong to the class A, we consider the following normalized form of the q-generalized Dini function, ψν,b,caz;q:U⟶ℂ, as(11)ψν,b,caz;q=2νq;qk−1a1−qb−1/2z1−v/2ϕν,b,caz;q=z+∑n=1∞ζnzn+1,where(12)ζn=−cna+2n4naq,qk;qn.Definition 3.
(subordination principle, see [23–25]). An analytic function f is said to be subordinate to another analytic function g, written as fz≺gzz∈U, if there exists a Schwarz function ω, which is analytic in U with ω0=0 and ωz<1z∈U, such that fz=gwz. In particular, if the function g is univalent in U, then we have the following equivalence:(13)fz≺gz⇔f0=g0andfU⊂gU.Remark 2.
It is observed that the functionΩz=1+z/1−z maps U conformally into the right half plane such that ℜΩz>0. That function plays a great roll in proving our main results.
Here, we would like to mention the following inequalities,(14)q;qn≥1−qn,and(15)qα;qn≥1−qαn,are valid for q∈0,1, α∈ℝ, and n∈ℕ.Lemma 1.
Let us considera∈ℝ+, b,ν∈ℝ, c∈ℂc<4, and k=ν+b+1/2; the function ψν,b,caz;q, referred by (11), satisfies the following inequalities, for all z∈U:(16)ψν,b,caz;q≤1+4ca+21−q1−qk−ac2a41−q1−qk−c2,ψν,b,caz;q′≤1+32ca+21−q21−qk2−4c23a+21−q1−qk+ac3a41−q1−qk−c3.Proof.
By taking into consideration inequalities (14) and (15), we obtain(17)ψν,b,caz;q=z+∑n=1∞−cna+2n4naq,qk;qnzn+1≤1+∑n=1∞cna+2na4nq;qnqk;qn≤1+∑n=1∞cna+2na4n1−qn1−qkn=1+c41−q1−qk∑n=1∞c41−q1−qkn−1+c2a1−q1−qk∑n=1∞nc41−q1−qkn−11+4ca+21−q1−qk−ac2a41−q1−qk−c2and(18)ψν,b,caz;q′=1+∑n=1∞−cnn+1a+2na4nq,qk;qnzn≤1+∑n=1∞cn2n2+a+2n+aa4nq;qnqk;qn≤1+∑n=1∞2n2+a+2n+aac41−q1−qkn=1+c41−q1−qk∑n=1∞c41−q1−qkn−1+ca+24a1−q1−qk∑n=1∞nc41−q1−qkn−1+c2a1−q1−qk∑n=1∞n2c41−q1−qkn−1=1+32ca+21−q21−qk2−4c23a+21−q1−qka41−q1−qk−c|3.
Thus, we complete the proof.


## 2. Main Results

Theorem 1.
Let us considera∈ℝ+, b,ν∈ℝ, and c∈ℂc<4, k=ν+b+1/2, and the function ψν,b,caz;q be defined by (11) and its partial sum ψν,b,caz;qm=z+∑n=1mζnzn+1. If the inequality,(19)16a1−q21−qk2+2ac2>43a+2c1−q1−qk,is valid, then(20)ℜψν,b,caz;qψν,b,caz;qm>β1and(21)ℜψν,b,caz;qmψν,b,caz;q>β2holds true, for all z∈U, where(22)β1=16a1−q21−qk2+4c2−a1−q1−qk+2ac2a41−q1−qk−c2,β2=a41−q1−qk−c216a1−q21−qk2+4c2−a1−q1−qk.Proof.
From the steps of proving Lemma1, we observe(23)1+∑n=1∞ζn≤1+4ca+21−q1−qk−ac2a41−q1−qk−c2,which is equivalent to(24)μ∑n=1∞ζn≤1,where μ=a41−q1−qk−c2/4ca+21−q1−qk−ac2.
Now, let us consider(25)1+ωz1−ωz=μψν,b,caz;qψν,b,caz;qm−β1=1+∑n=1mζnzn+μ∑n=m+1∞ζnzn1+∑n=1mζnzn.
Therefore,(26)ωz=μ∑n=m+1∞ζnzn2+2∑n=1mζnzn+μ∑n=m+1∞ζnznand(27)ωz≤μ∑n=m+1∞ζn2−2∑n=1mζn−μ∑n=m+1∞ζn.
Inequality (20) holds true if ωz≤1 according to the definition of subordination. Then, the upcoming inequality,(28)∑n=1mζn+μ∑n=m+1∞ζn≤1,implies that ωz≤1. It suffices to show that the left-hand side of (28) is bounded above by(29)μ∑n=1∞ζn,which is equivalent to(30)μ−1∑n=1mζn≥0.
On the contrary, to prove inequality (21), we consider(31)1+pz1−pz=μ+1ψν,b,caz;qmψν,b,caz;q−β2,=1+∑n=1mζnzn−μ∑n=m+1∞ζnzn1+∑n=1mζnzn.
Therefore,(32)pz=−1+μ∑n=m+1∞ζnzn2+2∑n=1mζnzn+1−μ∑n=m+1∞ζnznand(33)pz≤1+μ∑n=m+1∞ζn2−2∑n=1mζn−μ−1∑n=m+1∞ζn.
Inequality (21) holds true if pz≤1 according to the definition of subordination. Then, the upcoming inequality,(34)∑n=1mζn+μ∑n=m+1∞ζn≤1,implies that pz≤1. Since the left-hand side of (34) is bounded above μ∑n=1∞ζn, thus, we complete the proof.Theorem 2.
Let us considera∈ℝ+, b,ν∈ℝ, c∈ℂc<4, and k=ν+b+1/2, the function ψν,b,caz;q:U⟶ℂ, be defined by (11), and its partial sum be ψν,b,caz;qm=z+∑n=1mζnzn+1. If the inequalities,(35)32a+2c1−q21−qk2+ac3>43a+2c21−q1−qk,32a1−q31−qk3+43a+1c21−q1−qk>85a+4c1−q21−qk2+ac3,are valid, then(36)ℜψν,b,caz;q′ψν,b,caz;qm′>γ1and(37)ℜψν,b,caz;qm′ψν,b,caz;q′>γ2,holds true, for all z∈U, where(38)γ1=64a1−q31−qk3−16c5a+41−q21−qk2+53a+1c21−q1−qk−2ac3a41−q1−qk−c3and(39)γ2=a41−q1−qk−c364a1−q31−qk3+16c4−a1−q21−qk2−8c1−q1−qk.Proof.
From Lemma1, we observe that(40)1+∑n=1∞ηn≤1+32ca+21−q21−qk2−4c23a+21−q1−qk+ac3a41−q1−qk−c3,which is equivalent to(41)λ∑n=1∞ηn≤1,where ηn=−cnn+1a+2n/4naq,qk;qn and(42)λ=a41−q1−qk−c332ca+21−q21−qk2−4c23a+21−q1−qk+ac3.
Now, let us consider(43)1+rz1−rz=λψν,b,caz;q′ψν,b,caz;qm′−γ1=1+∑n=1mηnzn+λ∑n=m+1∞ηnzn1+∑n=1mηnzn.
Therefore,(44)rz=λ∑n=m+1∞ηnzn2+2∑n=1mηnzn+λ∑n=m+1∞ηnznand(45)rz≤λ∑n=m+1∞ηn2−2∑n=1mηn−λ∑n=m+1∞ηn.
Inequality (36) holds true if rz≤1 according to subordination principle. Then, the upcoming inequality,(46)∑n=1mηn+λ∑n=m+1∞ηn≤1,implies that rz≤1. It suffices to show that the left-hand side of (47) is bounded above by(47)λ∑n=1∞ηn,which is equivalent to(48)λ−1∑n=1mηn≥0.
On the contrary, to prove inequality (37), we consider(49)1+sz1−sz=λ+1ψν,b,caz;qm′ψν,b,caz;q′−γ2,=1+∑n=1mηnzn−λ∑n=m+1∞ηnzn1+∑n=1mηnzn.
Therefore,(50)sz=−1+λ∑n=m+1∞ηnzn2+2∑n=1mηnzn−λ−1∑n=m+1∞ηnznand(51)sz≤1+λ∑n=m+1∞ηn2−2∑n=1mηn−λ−1∑n=m+1∞ηn.
Inequality (37) holds true if sz≤1 according to subordination principle. Then, the upcoming inequality,(52)∑n=1mηn+λ∑n=m+1∞ηn≤1,implies that sz≤1. Since the left-hand side of (52) is bounded above by λ∑n=1∞ηn; thus, the proof is now completed.



---

*Source: 8496249-2022-02-21.xml*