EUDML

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.

On an inequality of Gauss.

Horst Alzer — 1991

Revista Matemática de la Universidad Complutense de Madrid

In this note we prove a new extension and a converse of an inequality due to Gauss.

On Ozeki’s inequality for power sums

Horst Alzer — 2000

Czechoslovak Mathematical Journal

Let $p \in (0, 1)$ be a real number and let $n \geq 2$ be an even integer. We determine the largest value $c_{n} (p)$ such that the inequality $\sum_{i = 1}^{n} {| a_{i} |}^{p} \geq c_{n} (p)$ holds for all real numbers $a_{1}, ..., a_{n}$ which are pairwise distinct and satisfy ${min}_{i \neq j} | a_{i} - a_{j} | = 1$ . Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value $c_{n} (p)$ in the case $p > 0$ and $n$ odd, and in the case $p \geq 1$ and $n$ even.

An inequality for the coefficients of a cosine polynomial

Horst Alzer — 1995

Commentationes Mathematicae Universitatis Carolinae

We prove: If $\frac{1}{2} + \sum_{k = 1}^{n} a_{k} (n) cos (k x) \geq 0 for all x \in [0, 2 π),$ then $1 - a_{k} (n) \geq \frac{1}{2} \frac{k^{2}}{n^{2}} for k = 1, \dots, n .$ The constant $1 / 2$ is the best possible.

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An inequality of W.L. Wang and P.F. Wang.

A continuous and a discrete variant of Wirtinger's inequality.

A note on a functional inequality.

On Carleman's inequality

Inequalities for Weierstrass products

Some classes of completely monotonic functions.

The inequality of Milne and its converse. II.

On Ky Fan's inequality.

Verschärfung einer Ungleichung von Ky Fan.

An inequality for arithmetic and harmonic means.

Ein Symmetrisches Mittel in Zwei Veraenderlichen

Über einen Wert, der zwischen dem geometrischen und dem arithmetischen Mittel zweier Zahlen liegt.

Ungleichungen für (e/a)a (b/e)b.

Über Lehmers Mittelwertfamilie.

Über eine Verallgemeinerung der Bernoullischen Ungleichung.

Cyclic mean-value inequalities for the gamma function

The Complete Monotonicity of a Function Studied by Miller and Moskowitz

On an inequality of Gauss.

On Ozeki’s inequality for power sums

An inequality for the coefficients of a cosine polynomial