On some geometric results for generalized k-Bessel functions

Abstract: The main aim of this article is to present some novel geometric properties for three distinct normalizations of the generalized k-Bessel functions, such as the radii of uniform convexity and of α-convexity. In addition, we show that the radii of α-convexity remain in between the radii of starlikeness and convexity, in the case when [ ] ∈ α 0, 1 , and they are decreasing with respect to the parameter α. The key tools in the proof of our main results are infinite product representations for normalized k-Bessel functions and some properties of real zeros of these functions.


Introduction
For more details about convex and starlike functions, one can refer to [1][2][3] and the references therein. A function ∈ f is said to be uniformly convex in , if ( ) f z is in class of convex function in and has the property that for every circular arc γ contained in , with center ℓ also in , the arc ( ) f γ is a convex arc. In 1993, Rønning [4] determined the necessary and sufficient conditions for analytic functions to be uniformly convex in the open unit disk, while in 2002, Ravichandran [5] also came up with a simpler criterion for uniform convexity. Analytically, the function f is uniformly convex in r if and only if The radius of uniform convexity is given by the real number Moreover, a function f is said to be in the class of β-uniformly convex function of order α, denoted by It is obvious that these classes give a unified presentation of various subclasses. The class ( ) β,0 is the class of β-uniformly convex functions [7] (see also [8,9]) and ( ) 1, 0 is the class of uniformly convex functions defined by Goodman [1] and Rønning [4]. The real number is called the radius of β-uniform convexity of order α of the function f .
Finally, by ( ) α β , , we mean the subclasses of consisting of functions that are α-convex of order β in , The radius of α-convexity of order β of the function f is given by the real number It is clear that radius of α-convexity of order β of the function f give a unified presentation of the radius of starlikeness of order β and of the radius of convexity of order β. That is, we have the relations ( ) For more details on starlike, convex, and α-convex functions, one can refer to [2,3,10,11] and the references therein.
As is well known, certain recent extensions of the gamma functions, such as the q-gamma functions and its generalizations (( ) p q , -gamma functions) [12,13] and also k-gamma functions, have been of interest to a wide audience in recent years. The main reason is that these topics stand for a meeting point of today's fast-developing areas in mathematics and physics, like the theory of quantum orthogonal polynomials and special functions, quantum groups, conformal field theories, and statistics. Diaz and Pariguan [14] introduced and investigated k-gamma functions when they were evaluating Feynman integrals. These integrals play a significant role in high-energy physics because they offer a general integral representation of the involved functions [15]. Since then, k-gamma functions have played an important role in the theory of special functions, are closely related to factorial, fractional differential equations, and mathematical physics, and have cropped up in many unexpected places in analysis. For more comprehensive and detailed studies of related works, we can refer the interested reader to [14][15][16][17][18][19] and the associated references therein. From the above series of articles, many generalizations about special functions arise. Hence, the authors from geometric function theory field continued the study of this family of generalized functions and suggested that many geometric properties of classical special functions have a counterpart in this more general setting. For the studies concerning special functions seen in geometric function theory, one may refer to the works [19][20][21][22][23][24][25][26][27] and the references therein.
By taking inspiration from the above series of articles, in our current investigation, our main aim is to determine the radii of uniform convexity and α-convexity for three different kinds of normalized k-Bessel functions.

The generalized k-Bessel function
We shall focus on a generalization of the k-Bessel function of order ν defined by the series , and Γ k stands for the k-gamma functions studied in [14] and defined as For several intriguing properties of k-gamma and k-Bessel functions, we refer the readers to [18,28,29].
Observe that as → k 1, the k-Bessel function W ν,1 1 is reduced to the classical Bessel function J v , whereas − W ν, 1 1 coincides with the modified Bessel function I ν . It is easy to check that the function ↦ z W ν c k , does not belong to class . Thus, first we shall perform some natural normalization. We define three functions originating from where ∈ y and ≠ y 0, and from (2), we conclude that , the zeros of ( ) which implies that ∑ ∑ . Moreover, we assume that inside Q r there are m positive and m negative roots of ( ) and according to Lemma 2.1, the zeros of ( ) are simple and real. Furthermore, Q r is not passing through any zeros. Then, F is uniformly bounded on these circles.
The above series will converge uniformly, and we can take the limit. It follows that since ( ) = F 0 0, Then, the following equality holds for all ≠ ∈ z τ n , Proof. For the sake of convenience, we shall first prove the following formula: By making use of the recurrence formula (see [28]) ). Moreover, we assume that inside Q s there are m positive and m negative roots of ( ) Of course, it is possible to make this assumption since ( ) , and according to Lemma 2.1, the zeros of ( ) are simple and real. Furthermore, Q s is not passing through any zeros. Then, F is uniformly bounded on these circles.
where R s denotes the radius of Q . s Hence, The above series will converge uniformly, and we can take the limit. It follows that since ( ) = F 0 0, which implies that and finally, we obtain  [24,32]) If | | > > ≥ a b r z and ≤ λ 1, then The following results can be obtained as a natural consequence of this inequality: and It is important to note here that in [32, Lemma 2.1], it was assumed that [ ] ∈ λ 0, 1 , but following the proof of [32, Lemma 2.1], it is obvious that we do not need the assumption ≥ λ 0. This means that the inequality given by (6) remains valid for ≤ λ 1.

Main results
We are now in a position to build up our main results. We will prove the theorem in two steps. First, suppose > 1.
With the aid of equations (12) and (13), we obtain In light of equation (7) With the help of equation (14), we obtain From equations (15) and (16)