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Some properties of functions related to the gamma and psi functions. (English) Zbl 1188.33003

The authors consider complete monotonicity properties of some functions connected with the Euler gamma function. For example, for \(a,b> 0\) put \[ f_{a,b}(x)= x^{a-b}(\Gamma(bx+ 1)/\Gamma(ax+ 1))^{1/x}. \] The reviewer proved in [JIPAM, J. Inequal. Pure Appl. Math. 5, No. 4, Paper No. 114, 4 p., electronic only (2004; Zbl 1182.11007)] that \(\lim_{x\to\infty} f_{a,b}(x)= b^b\cdot a^{-a}\cdot e^{a-b}\). The authors prove that for \(b> a> 0\), the function \(f_{a,b}(x)\) is logarithmically completely monotonic on \((0,+\infty)\).

MSC:

33B15 Gamma, beta and polygamma functions
26A48 Monotonic functions, generalizations
26A51 Convexity of real functions in one variable, generalizations

Citations:

Zbl 1182.11007
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References:

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