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This question already has an answer here:

Julián Aguirre gave a simple-amazing answer for this question: How to prove this property using convexity?

Unfortunately, i don't understand how to deduce the last inequality $a+b-c<(a+b)/2$.

Somebody help me. I don't know what I'm missing.

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marked as duplicate by Did real-analysis Jan 11 at 10:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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He is trying to show that if the result is true for all convex functions (with no assumption on monotonicity if $f''$ then there is another convex function $g$ for which the new $c$ (which is $a+b-c$) is less than $\frac {a+b} 2$. Hnec the result cannot be true with just convexity.

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