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A sharp inequality involving the psi function. (English) Zbl 1212.33002

Summary: The aim of this paper is to show that for \(a\in (0,1)\), the function \(f_a(x)=\psi(x+a)-\psi(x)-a/x\) is strictly completely monotonic on \((0,\infty)\). This result improves a previous result of S.-L. Qiu and M. Vuorinen [Math. Comput. 74, No. 250, 723–742 (2005; Zbl 1060.33006)], who proved that \(f_{1/2}\) is strictly decreasing and convex on \((0,\infty)\). As a direct consequence, a sharp inequality involving the psi function is established.

MSC:

33B15 Gamma, beta and polygamma functions
05A16 Asymptotic enumeration

Citations:

Zbl 1060.33006
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