Yang Zhen-Hang, Tian Jing-Feng
Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, 071003 Baoding, People's Republic of China.
Department of Science and Technology, Zhejiang Society for Electric Power, 310014 Hangzhou, Zhejiang People's Republic of China.
Rev Mat Complut. 2022 Aug 9:1-27. doi: 10.1007/s13163-022-00439-w.
Let be the modified Bessel function of the first kind of order . Motivated by a conjecture on the convexity of the ratio for , using the monotonicity rules for a ratio of two power series and an elementary technique, we present fully the convexity of the functions , and for on in different value ranges of , which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of for is proposed.
设 为第一类 阶修正贝塞尔函数。受关于 时 比值凸性的一个猜想的启发，利用两个幂级数比值的单调性规则和一种基本技巧，我们在 的不同取值范围内，全面给出了函数 、 和 在 上的凸性，这回答了该猜想并扩展了已知结果。作为推论，建立了一些关于 的单调性结果和新的函数不等式。作为应用，解决了一个开放问题并验证了一个猜想。最后，提出了一个关于 时 的完全单调性的猜想。
