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

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