Inequalities for the median of the gamma distribution

We provide several new inequalities involving λ n , the median of the gamma distribution of order n + 1 with parameter 1. Among others, we present sharp upper and lower bounds for the arithmetic mean of λ 1 , λ 2 , … , λ n . For all integers n ⩾ 1 we have α n + 7 6 + n 2 + 8 405 H n n < 1 n ∑ k = 1 n λ k ⩽ β n + 7 6 + n 2 + 8 405 H n n with the best possible constants α = ∑ k = 1 ∞ ( λ k − k − 2 3 − 8 405 k ) = − 0.0150 … and β = λ 1 − 683 405 = − 0.0080 … . Here, H n denotes the n th harmonic number.