A covering theorem for convex mappings

Thomas H. MacGregor
1964 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
more » ... 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.
doi:10.1090/s0002-9939-1964-0158984-3 fatcat:ipq4h5ck2jd35kh6hw3rdvluhq