On Ozeki’s inequality for power sums

Alzer, Horst

Czechoslovak Mathematical Journal, Tome 50 (2000) no. 1, pp. 99-102.

Résumé

Let $p\in (0,1)$ be a real number and let $n\ge 2$ be an even integer. We determine the largest value $c_n(p)$ such that the inequality \[ \sum ^n_{i=1} |a_i|^p \ge c_n(p) \] holds for all real numbers $a_1,\ldots ,a_n$ which are pairwise distinct and satisfy $\min _{i\ne j} |a_i-a_j| = 1$. 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 $p\ge 1$ and $n$ even.

Détail

Classification : 26D15