Convexity of ratios of the modified Bessel functions of the first kind with applications

Abstract

Let be the modified Bessel function of the first kind of order . Motivated by a conjecture on the convexity of the ratio for , using the monotonicity rules for a ratio of two power series and an elementary technique, we present fully the convexity of the functions , and for on in different value ranges of , which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of for is proposed.

High order monotonicity of a ratio of the modified Bessel function with applications

“…[3, p. 591] conjectured the function W ν (x) is strictly convex on (0, ∞) for all ν > −1. Very recently, Yang and Tian[23, Theorem 3] have proved that W ν (x) is strictly convex on (0, ∞) if and only if ν ≥ −1/2. Due to the relationsy ν (x) = xI ′ ν (x) I ν (x) = W ν−1 (x) − ν(see[28, Eq.…”