Note on Alzer's inequality

Authors

  • Chao-Ping Chen
  • Feng Qi

DOI:

https://doi.org/10.5556/j.tkjm.37.2006.175

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$.

Author Biographies

Chao-Ping Chen

Department of Applied Mathematics and Informatics, Henan Polytechnic University, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, China.

Feng Qi

Department of Applied Mathematics and Informatics, Henan Polytechnic University, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, China.

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Published

2006-03-31

How to Cite

Chen, C.-P., & Qi, F. (2006). Note on Alzer’s inequality. Tamkang Journal of Mathematics, 37(1), 11–14. https://doi.org/10.5556/j.tkjm.37.2006.175

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Section

Papers