Inegalitate cu medii armonice

[Cezar Lupu]

Pentru orice numere reale strict pozitive, \( a_{1}, a_{2}, \ldots, a_{n} \), notam cu \( H(a_{1}, a_{2}, \ldots, a_{n}) \) media lor armonica. Sa se arate ca

\( H(1+a_{1}, 1+a_{2}, \ldots, 1+a_{n})\geq 1+H(a_{1}, a_{2}, \ldots, a_{n}) \).

[mihai++]

Aducand la numitor obtinem inegaliteatea echivalenta:
\( n(\sum\frac{1}{a_i}\cdot\frac{1}{a_i+1})\geq\sum \frac{1}{a_i}\cdot\sum \frac{1}{a_i+1} \) care exact Cebisev aplicata n-uplelor la fel ordonate \( (\frac{1}{a_1},\dots,\frac{1}{a_n}) \) si \( (\frac{1}{a_1+1},\dots,\frac{1}{a_n+1}) \)