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Summary:
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.
References:
[1] D.S. Mitrinović and G. Kalajdžić: On an inequality. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678–715 (1980), 3–9. MR 0623215
[2] N. Ozeki: On the estimation of inequalities by maximum and minimum values. J. College Arts Sci. Chiba Univ. 5 (1968), 199–203. (Japanese) MR 0254198
[3] D.C. Russell: Remark on an inequality of N. Ozeki. General Inequalities 4, W. Walter (ed.), Birkhäuser, Basel, 1984, pp. 83–86. MR 0821787
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