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

Abstract

Let I_{nu }left( xright) 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( xright) =xI_{nu }left( xright) /I_{nu +1}left( xright) 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( xright) , W_{nu }left( xright) -x^{2}/left( 2nu +4right) 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( xright) 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.