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

Let I-v (x) be the modified Bessel function of the first kind of order v. Motivated by a conjecture on the convexity of the ratio W-v (x) = xI(v) (x) / Iv+1 (x) for v > -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-v (x), W-v (x) - x(2) / (2v +4) and W-v (x(1/theta)) for theta >= 2 on (0, infinity) in different value ranges of v, which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for W-v (x) are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of W-v (x(1/theta)) for theta >= 2 is proposed.