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.
1. Introduction and Basic Concepts
Let denote the class of functions that have the following Maclaurin’s form,which is analytic and univalent in the open unit disc and satisfies the normalization conditions .
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 of q-generalized Dini function to its sequences of partial sums, for , . We will investigate the following:
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]):
The function is known as the generalized Bessel function of the first kind of order , which is a particular solution of equation (3). The function has the following series representation:where stands for the Gamma function.
Now, let and ; the q-shifted factorial is defined by
The limit of as tends to infinity exists and is denoted by :
The multiple q-shifted factorial for complex numbers is defined by
If , for all , we define to be
Definition 1. Let and ; the q-generalized Bessel function is defined bywhere , and .
Now, we introduce the q-generalized Dini function in terms of .
Definition 2. Let ; the q-generalized Dini function is defined by
Remark 1. By specializing the value of , and , we see that(1) is the generalized Bessel function of first kind introduced by Orhan and Yagmur [10](2)By putting , then is the first kind of q-Bessel function given by Annaby and Mansour [21]. Also, is the familiar Bessel function defined by Baricz [18].(3)By putting , then is the first kind of modified q-Bessel function given by Annaby and Mansour [21]. Also, is the modified Bessel function defined by Baricz [18].(4) is the generalized Dini function investigated by Deniz and Gren [3]. Also, by putting in the last expression, we get introduced by Aktaş and Orhan [22]. In addition, by putting , we obtain the Dini function which is introduced by Baricz et al. [2].Because the function defined by (10) do not belong to the class , we consider the following normalized form of the q-generalized Dini function, , aswhere
Definition 3. (subordination principle, see [23–25]). An analytic function is said to be subordinate to another analytic function , written as , if there exists a Schwarz function , which is analytic in with and , such that . In particular, if the function is univalent in , then we have the following equivalence:
Remark 2. It is observed that the function maps conformally into the right half plane such that . That function plays a great roll in proving our main results.
Here, we would like to mention the following inequalities,andare valid for , , and .
Lemma 1. Let us consider , , , and ; the function , referred by (11), satisfies the following inequalities, for all :
Proof. By taking into consideration inequalities (14) and (15), we obtainandThus, we complete the proof.
2. Main Results
Theorem 1. Let us consider , , and , , and the function be defined by (11) and its partial sum . If the inequality,is valid, thenandholds true, for all , where
Proof. From the steps of proving Lemma 1, we observewhich is equivalent towhere .
Now, let us considerTherefore,andInequality (20) holds true if according to the definition of subordination. Then, the upcoming inequality,implies that . It suffices to show that the left-hand side of (28) is bounded above bywhich is equivalent toOn the contrary, to prove inequality (21), we considerTherefore,andInequality (21) holds true if according to the definition of subordination. Then, the upcoming inequality,implies that . Since the left-hand side of (34) is bounded above , thus, we complete the proof.
Theorem 2. Let us consider , , , and , the function , be defined by (11), and its partial sum be . If the inequalities,are valid, thenandholds true, for all , whereand
Proof. From Lemma 1, we observe thatwhich is equivalent towhere andNow, let us considerTherefore,andInequality (36) holds true if according to subordination principle. Then, the upcoming inequality,implies that . It suffices to show that the left-hand side of (47) is bounded above bywhich is equivalent toOn the contrary, to prove inequality (37), we considerTherefore,andInequality (37) holds true if according to subordination principle. Then, the upcoming inequality,implies that . Since the left-hand side of (52) is bounded above by ; thus, the proof is now completed.
Data Availability
No data were used in this paper.
Ethical Approval
This study does not contain any studies with human participants or animals performed by any of the authors.
Conflicts of Interest
The authors confirm no conflicts of interest.
Authors’ Contributions
The authors contributed to the draft of the manuscript; they have read and approved the final manuscript.