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

摘要

Let Wv(x) = xIv(x)/Iv+1(x) with Iv 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 on (0, ∞) for different values of parameter p with v  −2. As applications, we deduce some new Simpson–Spector-type inequalities for Wv(x) and derive a new type of bounds p+rx2+q2 (r  0) for Wv(x). In particular, we show that the upper bound Uv−1(2)(x) for Wv(x) is the minimum over all upper bounds {Up(2)(x):p≤v−1,v>−2}, where Up(2)(x)=p+2v+2−pv+2x2+(2v+2−p)2, and is not comparable with other sharpest upper bounds. We also find such type of upper bounds for v − 1  p  min {v + 1/2, 2v + 2} with v  −2 and for 2v + 2  p  v + 1/(2v + 5) with −2  v  −3/2.