Monotonicity properties of some Dini functions

Abstract

Abstract-In this note our aim is to deduce some new monotonicity properties for a special combination of Bessel functions of the first kind by using a recently developed Mittag-Leffler expansion for the derivative of a normalized Bessel function of the first kind. These monotonicity properties are used to obtain some new inequalities for Bessel functions of the first kind.Index Terms-Bessel functions of the first kind, infinite product, absolute monotonicity, Dini functions, monotonicity properties, inequalities.

Close‐to‐convexity of normalized Dini functions

“…Proof of Theorem Let us start with the following result, which was proved quite recently in [, Theorem 1]: Let and consider the Dini function defined by If denotes the n th positive zero of the Dini function then the following Weierstrassian factorization is valid where the infinite product is uniformly convergent on each compact subset of the complex plane. By using this result we obtain On the other hand, by using the Mittag–Leffler expansion [, Lemma 2.4] (see also [, Theorem 3]) …”

Modified Dini functions: monotonicity patterns and functional inequalities

“…Now from part c and using the fact that product of log-concave function is log-concave, the conclusion follows, as x → x ν is log-concave on (0, ∞) for all ν ≥ 0. Another proof can be seen in [7,Theorem 3]. e. We note that this part has been proved in [7,Theorem 6] for ν ∈ (−1, ∞) and x ∈ (0, α ν,1 ) but because of the following expression…”

“…Another proof can be seen in [7,Theorem 3]. e. We note that this part has been proved in [7,Theorem 6] for ν ∈ (−1, ∞) and x ∈ (0, α ν,1 ) but because of the following expression…”

“…Thus we conclude that the function x → D ν (x) is strictly log-concave on R \ △. Note that this part has been proved also in [7,Theorem 4] but only for x ∈ (0, ∞) \ △ 2 . d. From (2.18) we have…”

“…, which in view of I ν (z) = i −ν J ν (iz) gives the modified Dini function ξ ν : C → C, defined by ξ ν (z) = i −ν d ν (iz) = (1 − ν)I ν (z) + zI ′ ν (z) = I ν (z) + zI ν+1 (z). In view of the Weierstrassian factorization of d ν (z) (see [7]),…”