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

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