Note on Alzer's inequality

Abstract

If the sequence $ \{a_i \}_{i=1}^{\infty} $ satisfies $ \triangle a_i=a_{i+1}-a_i<0$, $ \triangle^2 a_i=\triangle(\triangle a_{i})=a_{i+2}-2a_{i+1}+a_{i}\geqslant0 $, $ i=0,1,2,\ldots $, $ a_0=0 $. Then

$$ \frac{a_n}{a_{n+1}}< \left(\frac{1}{a_n}\sum_{i=1}^{n}a_{i}^{r}\bigg / \frac{1}{a_{n+1}}\sum_{i=1}^{n+1}a_{i}^{r}\right)^{1/r} $$

for all natural numbers $ n $, and all real $ r>0$.