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Minimal C_a so that inequality holds

Source: KöMaL A. 709

December 14, 2017
inequalitiesalgebra

Problem Statement

Let a>0a>0 be a real number. Find the minimal constant CaC_a for which the inequalityCa∑k=1n1xk−xk−1>∑k=1nk+axk\displaystyle C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}holds for any positive integer nn and any sequence 0=x0<x1<⋯<xn0=x_0<x_1<\cdots <x_n of real numbers.
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