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Monotonicity results for the polygamma functions. (English) Zbl 1152.33002
The psi(or digamma) function is defined for all positive real numbers $$x$$ as the logarithmic derivative of the gamma function $$\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}.$$ Its derivatives $$\psi^{(n)}(x)$$ are called polygamma functions. In the paper under review, the authors investigate monotonicity of the functions $x^c| \psi^{(n)}(x+\beta)| , \qquad x^{k-n}\frac{\psi^{(k)}(x+\beta)}{\psi^{(n)}(x+\beta)}$ on $$(0, \infty),$$ where $$k>n\geq 1$$ are integers and $$b,c \in {\mathbb R}.$$ This extends several results of H. Alzer [Forum Math., 16, 181–221 (2004; Zbl 1048.33001)]; H. Alzer and O. G. Ruehr, [J. Comput. Appl. Math., 101, 53–60 (1999; Zbl 0943.33001)] and H. Alzer [Aequationes Math. 61, 151–161 (2001; Zbl 0968.33003)]. The authors also establish completely monotonicity of the function $$\frac{d\psi(\ln x)}{dx}$$ on $$(1, \infty).$$
##### MSC:
 33B15 Gamma, beta and polygamma functions 26A48 Monotonic functions, generalizations
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