Convexity of a ratio of the modified Bessel functions of the second kind with applications

Abstract

Let K ν K_{\nu } be the modified Bessel functions of the second kind of order ν \nu . The ratio Q ν ( x ) = x K ν − 1 ( x ) / K ν ( x ) Q_{\nu }\left ( x\right ) =xK_{\nu -1}\left ( x\right ) /K_{\nu }\left ( x\right ) appeared in physics and probability. In this paper, we collate properties of this ratio, and prove the conjecture that ( − 1 ) n Q ν ( n ) ( x ) > ( > ) 0 \left ( -1\right ) ^{n}Q_{\nu }^{\left ( n\right ) }\left ( x\right ) >\left ( >\right ) 0 for x > 0 x>0 and n = 2 , 3 n=2,3 if | ν | > ( > ) 1 / 2 \left \vert \nu \right \vert >\left ( >\right ) 1/2 holds for n = 2 n=2 . This yields several new consequences and improves some known results. Finally, two conjectures are proposed.

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

“…, which will give a simple proof of Theorem 1 in [22]. Moreover, the validity of the guess (1.5) for n = 3 can yield more monotonicity and convexity or concavity results related to the ratio Q ν (x).…”

“…or equivalently, is Q ′ ν (x) a completely monotonic function of x? The authors [22] pointed out that the answer is negative since the fourth derivative of Q 5/2 (x) changes sign on (0, ∞) (the details can be seen in Section 5 in this paper). From this we see that Q ′ ν (x) for |ν| ≥ 0 is an incompletely monotonic function of x on (0, ∞).…”

“…Let h ν (x) be defined by (3.3). Differentiation yields The following corollary is a consequence of Corollary 3.6, which gives an answer to [22,Conjecture 2].…”

“…From this we see that Q ′ ν (x) for |ν| ≥ 0 is an incompletely monotonic function of x on (0, ∞). The authors [22,Conjecture 1] further guessed that…”

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

“…In a recent paper [ 17 ], Yang and Tian pointed out that Baricz’s conjecture may be modified as follows: is strictly convex on for all and concave on if .…”