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 Post subject: Midpoint convex => rational convexPosted: Sun, 20 Feb 2011 15:40:02 UTC
 S.O.S. Oldtimer

Joined: Sun, 4 Nov 2007 12:08:30 UTC
Posts: 246
Location: Bratislava, Slovakia
Probably too easy for experienced members, but at least for someone this might be a good exercise to get acquainted with a technique.

Let be a midpoint convex function, i.e.

for any .
Show that then this function fulfills

for any and any rational number .

Hint
Spoiler:
Cauchy induction: wikipedia
AoPS

Sketch of the solution:
Spoiler:
It is relatively easy to see, that it suffices to show for any integer k (and any choice of ).

The case is a straightforward induction.

Now, if then we denote . Now from

we get by a simple algebraic manipulation.

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