About: In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums, establish three new inequalities involving finitely many 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 finitely many gamma functions, present complete monotonicity of a linear combination of finitely many trigamma functions, construct a new ratio of finitely many gamma functions, derive monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finitely many gamma functions, and suggest two linear combinations of finitely many trigamma functions and two ratios of finitely many gamma functions to be investigated.   Goto Sponge  NotDistinct  Permalink

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  • In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums, establish three new inequalities involving finitely many 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 finitely many gamma functions, present complete monotonicity of a linear combination of finitely many trigamma functions, construct a new ratio of finitely many gamma functions, derive monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finitely many gamma functions, and suggest two linear combinations of finitely many trigamma functions and two ratios of finitely many gamma functions to be investigated.
subject
  • Functional analysis
  • Convex analysis
  • Exponentials
  • Linear algebra
  • Order theory
  • Types of functions
  • Gamma and related functions
  • Binary operations
  • Real analysis
  • Unary operations
  • Generalized convexity
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