标题： 单调性规则用于两个函数级数和两个积分变换的比值 显示英文标题

标题： Monotonicity rules for the ratio of two function series and two integral transforms

摘要： 在本文中，我们研究函数$t \mapsto \frac{\sum_{k=0}^\infty a_k w_k(t)}{\sum_{k=0}^\infty b_k w_k(t)}$和$x \mapsto \frac{\int_\alpha^\beta f(t) w(t,x) \textrm{d} t}{\int_\alpha^\beta g(t) w(t,x) \textrm{d} t}$的单调性，重点关注$a_k/b_k$和$f(t)/g(t)$的单调性仅改变一次的情况。 所呈现的结果还为两个幂级数的比值、两个$\mathcal{Z}$-变换、两个离散拉普拉斯变换、两个离散梅林变换、两个拉普拉斯变换和两个梅林变换的单调性提供了见解。 最后，我们将这些单调性规则应用于特殊函数和随机序领域的几个应用中。 显示英文摘要

摘要： In this paper, we investigate the monotonicity of the functions $t \mapsto \frac{\sum_{k=0}^\infty a_k w_k(t)}{\sum_{k=0}^\infty b_k w_k(t)}$ and $x \mapsto \frac{\int_\alpha^\beta f(t) w(t,x) \textrm{d} t}{\int_\alpha^\beta g(t) w(t,x) \textrm{d} t}$, focusing on case where the monotonicity of $a_k/b_k$ and $f(t)/g(t)$ change once. The results presented also provide insights into the monotonicity of the ratios of two power series, two $\mathcal{Z}$-transforms, two discrete Laplace transforms, two discrete Mellin transforms, two Laplace transforms, and two Mellin transforms. Finally, we employ these monotonicity rules to present several applications in the realm of special functions and stochastic orders.