I call a function f defined and valued on a domain A in the plane convex if it maps convex areas to convex areas. Some obvious example of convex functions. If f is also a bijection, what can we say more about it? I guessed that if f is a diffeomorphism(C2) of the complex plane then it is linear, say az+b. This is related to a quadratic form of its differentials fxx,fyy,fxy. However, I cannot strictly confirm this. Is there any classical analysis addressing with this question? If any, please help show the sources.
I think that A bijection from the plane to itself that sends convex sets into convex sets is an affine transformation.
Indeed, it must not be hard to see that such bijection sends lines to lines (into what would it send a halfplane given that it is convex and has a convex complement??) and then one applies the fundamental theorem of affine geometry.

$\begingroup$ So this quesion is indeed trivial. Thanks for your tips. $\endgroup$ – Zhipeng Lu Mar 20 '12 at 17:45

2$\begingroup$ nice! it seems it's a rule: when dealing with some problem about convex objects, never forget to start by the linear or affine ones... $\endgroup$ – Pietro Majer Mar 20 '12 at 17:54

$\begingroup$ Well, I think that there are little details to be checked, but if you assume the map is a homeomorphism, then its easy. $\endgroup$ – alvarezpaiva Mar 20 '12 at 20:10