Monotonicity of the ratio of modified Bessel functions of the first kind with applications

Abstract

Abstract Let Wv(x)=xIv(x)/Iv+1(x) $W_{v} ( x ) =xI_{v} ( x ) /I_{v+1} ( x ) $ with Iv $I_{v}$ be the modified Bessel functions of the first kind of order v. In this paper, we prove the monotonicity of the function x↦(Wv(x)−p)2−(2v+2−p)2x2 $$ x\mapsto\frac{ ( W_{v} ( x ) -p ) ^{2}- ( 2v+2-p ) ^{2}}{x^{2}} $$ on (0,∞) $( 0,\infty ) $ for different values of parameter p with v>−2 $v>-2 $. As applications, we deduce some new Simpson–Spector-type inequalities for Wv(x) $W_{v} ( x ) $ and derive a new type of bounds p+rx2+q2 $p+r\sqrt {x^{2}+q^{2}}$ ( r>0 $r>0$) for Wv(x) $W_{v} ( x ) $. In particular, we show that the upper bound Uv−1(2)(x) $U_{v-1}^{ ( 2 ) } ( x ) $ for Wv(x) $W_{v} ( x ) $ is the minimum over all upper bounds {Up(2)(x):p≤v−1,v>−2} $\{ U_{p}^{ ( 2 ) } ( x ) :p\leq v-1,v>-2 \} $, where Up(2)(x)=p+2v+2−pv+2x2+(2v+2−p)2, $$ U_{p}^{ ( 2 ) } ( x ) =p+\sqrt{\frac {2v+2-p}{v+2}x^{2}+ ( 2v+2-p ) ^{2}}, $$ and is not comparable with other sharpest upper bounds. We also find such type of upper bounds for v−1−2 $v>-2$ and for 2v+2<p<v+1/(2v+5) $2v+2< p< v+1/ ( 2v+5 ) $ with −2<v<−3/2 $-2< v<-3/2$.

Keywords

• Modified Bessel functions of the first kind
• Series
• Monotonicity
• Inequality