On Ozeki's inequality for power sums

Abstract

Let p ∈ (0, 1) be a real number and let n ≥ 2 be an even integer. We determine the largest value c _n(p) such that the inequality

$$\sum\limits_{i = 1}^n {{\text{|}}a_i {\text{|}}^p \geqslant C_n (p)} $$

holds for all real numbers a ₁,...,a _n which are pairwise distinct and satisfy $\mathop {\min }\limits_{i \ne j} {\text{|}}a_i - a_j {\text{|}} = 1$. 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 ≥ 1 and n even.