In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.

中文翻译：

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有限多个伽马函数的比率及其与应用的性质

在论文中，作者建立了一个涉及指数函数和和的不等式，引入了许多伽马函数的比率，讨论了新引入的比率的性质，包括单调性、对数凸性、（对数）完全单调性和伯恩斯坦函数性质，并构造多项系数和多元贝塔函数的两个不等式。