TY - JOUR AU - Chen, Chao-Ping AU - Qi, Feng PY - 2006/03/31 Y2 - 2024/03/28 TI - Note on Alzer's inequality JF - Tamkang Journal of Mathematics JA - Tamkang J. Math. VL - 37 IS - 1 SE - Papers DO - 10.5556/j.tkjm.37.2006.175 UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/175 SP - 11-14 AB - <p>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</p><p>$$ \frac{a_n}{a_{n+1}}< \left(\frac{1}{a_n}\sum_{i=1}^{n}a_{i}^{r}\bigg / \frac{1}{a_{n+1}}\sum_{i=1}^{n+1}a_{i}^{r}\right)^{1/r} $$</p><p>for all natural numbers $ n $, and all real $ r>0$.</p> ER -