If f is continuous function on $\displaystyle (a,b)$is convex,show that
for $\displaystyle x_{1},x_{2},....,x_{m}\in(a,b)$,
that
$\displaystyle f(\frac{x_{1}+x_{2}+...+x_{m}}{m})\leq\frac{1}{m}( f(x_{1})+f(x_{2})+...+f(x_{m}))$
thanks very much
If f is continuous function on $\displaystyle (a,b)$is convex,show that
for $\displaystyle x_{1},x_{2},....,x_{m}\in(a,b)$,
that
$\displaystyle f(\frac{x_{1}+x_{2}+...+x_{m}}{m})\leq\frac{1}{m}( f(x_{1})+f(x_{2})+...+f(x_{m}))$
thanks very much
That's a particular case of Jensen's Inequality
Here's a proof I particularly enjoyed reading Art of Problem Solving Forum
And here you have the traditional proof: http://en.wikipedia.org/wiki/Jensen's_inequality
It's a very useful inequality.