A covering theorem for convex mappings
Proceedings of the American Mathematical Society
References 1. Richard Arens, Extension of functions on fully normal spaces, Pacifie J. Math. 2(1952), 11-22. 2. O. Hanner, Retraction and extension of mappings of metric and non-metric spaces, The following theorems are classical. Proofs can be found in [l, pp. 214, 223]. Theorem 1. If f(z) is regular and univalent in \z\ <l, f(0) = 0 and /'(0) = 1 then the image domain covers the circle \ w\ < 1/4. Theorem 2. // f(z) is regular and univalent in \z\ <1, /(0) = 0, /'(0) = 1 and the image D is
... d the image D is convex, then D covers the circle \w\ <l/2. The purpose of this note is to show that Theorem 2 can be proven as a simple consequence of Theorem 1. Suppose then that/(z) satisfies the hypotheses of Theorem 2 and f(z)^c. Let g(z) = (f(z) -c)2. Suppose that zx and z2 are distinct points in the unit circle and g(zi)=g(zi). Then either/(zx) =/(z2) or 2-l(f(zi)+f(z2))=c. The first equation cannot hold since/(z) is univalent in |z| <1. Neither can the second equation hold, for D is convex and therefore the average of every two points in D is also in D. This proves that g(z) is univalent in |z| <1. The function h(z) = (c2 -g(z))/2c satisfies the hypotheses of Theorem 1, and h(z)¿éc/2 since f(z)^c. Therefore, |c/2|¡gl/4, |c| =51/2.