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 homeomorphismbijection, 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.
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complex convex functionsI 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 homeomorphism, 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.
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