We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of a certain explicitly written measure in the q case and of a certain elementary function in the classical \(q=1\) case. In the latter case we further provide simple new sufficient conditions leading to many new examples of logarithmically completely monotonic gamma ratios. Finally, we apply some of our results to study monotonicity of some gamma ratios and rational functions.

中文翻译：

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基本和普通伽玛函数的完全单调比率

我们研究了参数为变量线性函数的gamma和q -gamma函数乘积对数对数完全单调的条件。对于在q情况下某些显式书写的度量的非负性和在经典\（q = 1 \）情况下的某些基本函数的非负性，我们给出了必要和充分的条件。在后一种情况下，我们进一步提供了简单的新的充分条件，从而产生了许多对数完全单调伽马比的新示例。最后，我们将一些结果应用于研究某些伽玛比和有理函数的单调性。