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In the paper, the authors review origins, motivations, and generalizations ofa series of inequalities involving several exponential functions and sums,establish three new inequalities involving finite exponential functions andsums by finding convexity of a function related to the generating function ofthe Bernoulli numbers, survey the history, backgrounds, generalizations,logarithmically complete monotonicity, and applications of a series of ratiosof finite gamma functions, present complete monotonicity of a linearcombination of finite trigamma functions, construct a new ratio of finite gammafunctions, derives monotonicity, logarithmic convexity, concavity, completemonotonicity, and the Bernstein function property of the newly constructedratio of finite gamma functions, and suggest two linear combinations of finitetrigamma functions and two ratios of finite gamma functions to be investigated.

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions