Please Anton, could you remind us what a midconvex function is?

 A function f : I ⊆ R → R is midpoint convex

if

f(a/2 + b/2)≤f (a ) /2+ f (b )/2

for all a, b∈ I.

There is a discussion at math.stackexchange which is relevant to your question. This is the direct link.

http://math.stackexchange.com/questions/71019/example-of-a-function-such-that-varphi-left-fracxy2-right-leq-frac-va

You should pay particular attention to the answer offered by Robert Israel, who explains why an explicit example can not be constructed.