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.



complex convex functions
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 half-plane given that it is convex and has a convex complement??) and then one applies the fundamental theorem of affine geometry. 


