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

Let $I_{\nu }\left(x\right)$ be the modified Bessel function of the first kind of order $\nu$ . Motivated by a conjecture on the convexity of the ratio $W_{\nu }\left(x\right)=x{I}_{\nu }\left(x\right)/I_{\nu +1}\left(x\right)$ for $\nu >-2$ , using the monotonicity rules for a ratio of two power series and an elementary technique, we present fully the convexity of the functions $W_{\nu }\left(x\right)$ , $W_{\nu }\left(x\right)-x^2/\left(2,\nu ,+,4\right)$ and $W_{\nu }\left(x^{1/\theta }\right)$ for $\theta \ge 2$ on $\left(0,,,\infty \right)$ in different value ranges of $\nu$ , which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for $W_{\nu }\left(x\right)$ are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of $W_{\nu }\left(x^{1/\theta }\right)$ for $\theta \ge 2$ is proposed.