Generalizations of Alzer's and Kuang's inequality

Abstract

Let $ f$ be a strictly increasing convex (or concave) functions on $ (0,1]$, then, for $ k$ being a nonnegative integer and $ n$ a natural number, the sequence $ {1\over n}\sum_{i=k+1}^{n+k} f({i\over n+k})$ is decreasing in $ n$ and $ k$ and has a lower bound $ \int_0^1 f(t) dt$. Form this, some new inequalities involving $ {\root n\of {(n+k)!/k!}}$ are deduced. By the Hermie-Hadamard inequality, several inequalities are obtained.