Yang Zhen-Hang, Zheng Shen-Zhou
1Department of Mathematics, Beijing Jiaotong University, Beijing, P.R. China.
Present Address: Department of Science and Technology, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, P.R. China.
J Inequal Appl. 2018;2018(1):57. doi: 10.1186/s13660-018-1648-4. Epub 2018 Mar 9.
Let [Formula: see text] with [Formula: see text] be the modified Bessel functions of the first kind of order . In this paper, we prove the monotonicity of the function [Formula: see text] on [Formula: see text] for different values of parameter with [Formula: see text]. As applications, we deduce some new Simpson-Spector-type inequalities for [Formula: see text] and derive a new type of bounds [Formula: see text] ([Formula: see text]) for [Formula: see text]. In particular, we show that the upper bound [Formula: see text] for [Formula: see text] is the minimum over all upper bounds [Formula: see text], where [Formula: see text] and is not comparable with other sharpest upper bounds. We also find such type of upper bounds for [Formula: see text] with [Formula: see text] and for [Formula: see text] with [Formula: see text].
设([公式：见正文])，其中([公式：见正文])为第一类修正贝塞尔函数，阶数为(。)在本文中，我们证明了对于参数([公式：见正文])的不同值，函数([公式：见正文])在([公式：见正文])上的单调性。作为应用，我们推导了一些关于([公式：见正文])的新的辛普森 - 斯佩克特型不等式，并得到了([公式：见正文])（([公式：见正文])）的一种新型界。特别地，我们表明([公式：见正文])的上界([公式：见正文])是所有上界([公式：见正文])中的最小值，其中([公式：见正文])且与其他最尖锐上界不可比。我们还找到了([公式：见正文])（([公式：见正文])）和([公式：见正文])（([公式：见正文])）的此类上界。
