Inequalities for the median of the gamma distribution

We provide several new inequalities involving λnλn, the median of the gamma distribution of order n+1n+1 with parameter 1. Among others, we present sharp upper and lower bounds for the arithmetic mean of λ1,λ2,…,λnλ1,λ2,…,λn. For all integers n⩾1n⩾1 we have αn+76+n2+8405Hnn<1n∑k=1nλk⩽βn+76+n2+8405Hnn with the best possible constants α=∑k=1∞(λk−k−23−8405k)=−0.0150…andβ=λ1−683405=−0.0080…. Here, HnHn denotes the nnth harmonic number.