A mean-value inequality for the gamma function Original Research Article

Let α = 0.21609…. If wi 0 (i = 1,…, n; n ≥ 2) are positive real numbers with Σi=1nwi = 1, then we have for all real numbers xiϵ (0,α] (i = 1,…,n), View the MathML source where w = min1≤i≤nwi. The lower bound is sharp. Inequality (∗) completes a result of Lucht, who proved that the best possible upper bound in (∗) is 1.