Complete monotonicity of a function involving the divided difference of digamma functions. (English) Zbl 1286.26007
Summary: In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presented, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions are derived, and two double inequalities for bounding the ratio of two gamma functions are discovered.
MSC:
| 26A48 | Monotonic functions, generalizations |
| 26A51 | Convexity of real functions in one variable, generalizations |
| 26D15 | Inequalities for sums, series and integrals |
| 33B15 | Gamma, beta and polygamma functions |
| 44A10 | Laplace transform |
Keywords:
necessary and sufficient condition; completely monotonic function; logarithmically completely monotonic function; divided difference; psi function; inequality for sums; bound for the ratio of two gamma functionsReferences:
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