Note on Alzer's inequality
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If the sequence $ \{a_i \}_{i=1}^{\infty} $ satisfies $ \triangle a_i=a_{i+1}-a_i $$ \frac{a_n}{a_{n+1}} for all natural numbers $ n $, and all real $ r>0$.
[1] József Sándor. On an Inequality of Alzer , 1995 .
[2] Horst Alzer,et al. On an Inequality of H. Minc and L. Sathre , 1993 .
[3] R. Webster. AN ELEMENTARY PROOF OF HORN'S THEOREM , 1996 .