prove that f is convex on (a,b) if and only if f satisfies $ f( \frac{x+y}{2} ) \leq \frac{f(x)+f(y)}{2} $ for all $ x,y \in (a,b) $

Asked May 28 '18 at 12:49

Active May 28 '18 at 12:49

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Problem

Let $f$ be continuous on (a,b). prove that $f$ is convex on (a,b) if and only if $f$ satisfies $ f( \frac{x+y}{2} ) \leq \frac{f(x)+f(y)}{2} $ for all $x,y \in (a,b) $

I thought that it is easy by using the definition of convex

a function f on (a,b) is said to be convex if $f(tx+(1-t)y)$ $ \leq $ $tf(x) + (1-t)f(y)$ for all $ x,y \in (a,b) $ and $ t \in [0,1] $

But my professor said that I was wrong. "you should use induction"
I didn't understand what he said, but he doesn't say anymore.
How can I prove this?

asked May 28 '18 at 12:49

alryosha

2

See https://math.stackexchange.com/questions/1002248/if-f-is-continuous-and-f-big-frac12xy-big-le-frac12-big-fx and https://math.stackexchange.com/questions/83383/midpoint-convex-and-continuous-implies-convex?noredirect=1&lq=1 – A. Goodier May 28 '18 at 12:57
You're definitely not wrong, but (putting t=1/2) you have only shown the "only if" part of the statement. You need to check the "if" part. – TSU May 28 '18 at 13:06

prove that f is convex on (a,b) if and only if f satisfies $ f( \frac{x+y}{2} ) \leq \frac{f(x)+f(y)}{2} $ for all $ x,y \in (a,b) $

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