# convex function

• Apr 30th 2008, 06:53 PM
Xingyuan
convex function
If f is continuous function on $(a,b)$is convex,show that
for $x_{1},x_{2},....,x_{m}\in(a,b)$,
that

$f(\frac{x_{1}+x_{2}+...+x_{m}}{m})\leq\frac{1}{m}( f(x_{1})+f(x_{2})+...+f(x_{m}))$

thanks very much (Surprised)
• Apr 30th 2008, 07:21 PM
PaulRS
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. (Nod)