Radii Problems for Normalized Bessel Functions of First Kind
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Radii Problems for Normalized Bessel Functions of First Kind Nisha Bohra1 · V. Ravichandran1
Received: 11 February 2017 / Revised: 13 May 2017 / Accepted: 22 June 2017 © Springer-Verlag GmbH Germany 2017
Abstract For three different normalizations of Bessel functions of first kind, the radius of k-parabolic starlikeness and k-uniform convexity of order α are determined. The radius of strong starlikeness and other related radius are also obtained for these functions. We also find optimal parameters for which these functions are k-parabolic starlike and k-uniformly convex in the open unit disk. Keywords Generalized Bessel function · k-uniformly starlike function of order α · k-uniformly convex function of order α · Radius of k-uniform convexity · Zeros of Bessel functions · Strongly starlike functions Mathematics Subject Classification 33C10 · 33C15 · 30C45
1 Introduction This paper studies the radii problems for normalized Bessel functions of first kind. The Bessel function of first kind of order ν is a particular solution of the second-order homogeneous Bessel differential equation z 2 w (z) + zw (z) + (z 2 − ν 2 )w(z) = 0, ν ∈ C. This function has an infinite series representation given by
Communicated by Stephan Ruscheweyh.
B
Nisha Bohra [email protected] V. Ravichandran [email protected]
1
Department of Mathematics, University of Delhi, Delhi 110 007, India
123
N. Bohra, V. Ravichandran
Jν (z) =
n≥0
z 2n+ν (−1)n , n!(n + ν + 1) 2
where z ∈ C and ν ∈ C such that ν = −1, −2, . . .. We consider the following three normalizations of Bessel functions of first kind: 1 z 3 + · · · , ν = 0, 4ν(ν + 1) 1 z3 + · · · , gν (z) = 2ν (ν + 1)z 1−ν Jν (z) = z − 4(ν + 1) f ν (z) = (2ν (ν + 1)Jν (z))1/ν = z −
and
√ h ν (z) = 2ν (ν + 1)z 1−ν/2 Jν ( z) = z −
1 z2 + · · · . 4(ν + 1)
(1.1) (1.2)
(1.3)
Clearly, the functions f ν , gν , and h ν are normalized and analytic. We also note that f ν (z) = exp ν1 log(2ν (ν + 1)Jν (z)) where log represents the principal branch of the logarithm function and every multivalued function considered in this paper is taken with the principal branch. In 1960s, Brown, Kreyszig and Todd [10,17] considered the univalence and starlikeness of Bessel functions of first kind. They determined the radius of starlikeness of the functions f ν and gν for the case ν > 0. Brown [10] used the fact that the Bessel function of first kind is a particular solution of the Bessel differential equation. Recently, in 2014, Baricz et al. [4], by using a much simpler approach, determined the radius of starlikeness of f ν , gν and h ν for ν > −1. In the same year, Baricz and Szász [5] determined the radius of convexity of the same three functions for ν > −1. Later in [6], they extended it for the case when −2 < ν < −1. The key tools in their proofs were some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivatives, and the fact that the smallest positive zeros of some
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