The Complete Monotonicity of a Function Studied by Miller and Moskowitz

Alzer, Horst

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 2, pp. 449-452

Résumé

Let $$S(x) = log(1+x) + \int_{0}^{1} \left[ 1 - \left( \frac{1+t}{2} \right) ^{x} \right] \frac{dt}{\log t} \quad \text{and} \quad F(x) = \log 2 - S(x) \,\, (0 x \in \mathbb{R}).$$ We prove that $F$ is completely monotonic on $(0,\infty)$. This complements a result of Miller and Moskowitz (2006), who proved that $F$ is positive and strictly decreasing on $(0,\infty)$. The sequence $\{ S(k)\}$$(k=1,2,\dots)$ plays a role in information theory.