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