Radii of Uniform Convexity of Lommel and Struve Functions
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Radii of Uniform Convexity of Lommel and Struve Functions Erhan Deniz1 · Sercan Kazımoglu ˘ 1 · Murat Çaglar ˘ 1 Received: 22 January 2020 / Revised: 17 August 2020 / Accepted: 27 August 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we determine the radii of β-uniform convexity of order α for six kinds of normalized Lommel and Struve functions of the first kind. One of the most important things which we have learned in this study is that the radii of uniform convexity are obtained as solutions of some transcendental equations. Keywords Lommel functions · Struve functions · Univalent functions · β-uniformly convex functions of order α · Zeros of Lommel functions of the first kind · Zeros of Struve functions of the first kind Mathematics Subject Classification 30C45 · 30C80 · 33C10
1 Introduction and Preliminaries It is well known that the concepts of convexity, starlikeness, close-to-convexity and uniform convexity including necessary and sufficient conditions, have a long history as a part of geometric function theory. It is known that special functions, like Bessel, Struve and Lommel functions of the first kind have some beautiful geometric properties. Recently, the above geometric properties of the Bessel functions were investigated in some earlier results (see [1–5,12]). On the other hand, the radii of convexity and starlikeness of the Struve and Lommel functions were studied by Baricz et al. [8,10]. Motivated by the above developments, in this paper, our aim was
Communicated by Fereshteh Sady.
B
Erhan Deniz [email protected] Sercan Kazımo˘glu [email protected] Murat Ça˘glar [email protected]
1
Department of Mathematics, Faculty of Science and Letters, Kafkas University, Campus, 36100 Kars, Turkey
123
Bulletin of the Iranian Mathematical Society
to give some new results for the radius of β-uniformly convex functions of order α of the normalized Struve and Lommel functions of the first kind. In the special cases of the parameters α and β, we can obtain some earlier results. The key tools were some Mittag–Leffler expansions of Lommel and Struve functions of the first kind and special properties of the zeros of these functions and their derivatives. Let U (z 0 , r ) = {z ∈ C : |z − z 0 | < r } denote the disk of radius r and center z 0 . We use U (r ) = U (0, r ) and U = U (0, 1) = {z ∈ C : |z| < 1}. Let (an )n≥2 be a sequence of complex numbers with 1
d = lim sup |an | n ≥ 0, and r f = n→∞
1 . d
If d = 0, then r f = +∞. As usual, we denote by A the class of all analytic functions f : U (r f ) → C of the form f (z) = z +
∞
an z n .
(1.1)
n=2
We say that a function f of the form (1.1) is convex if f is univalent and f (U (r )) is a convex domain in C. An analytic description of this definition is that
z f (z) f ∈ A is convex if and only if 1 + f (z)
> 0, z ∈ U (r ).
The radius of convexity of the function f is defined by
z f (z) > 0, z ∈ U (r ) . r cf = sup r ∈ (0, r f ) : 1 + f (z)
In the following, we deal with the class of the uniformly
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