Functions whose derivative has a positive real part

Introduction. Let P denote the class of functions which are regular and satisfy Re /' (z) > 0 for | z \ 0 can be found in a paper by J. W. Alexander l, p. 18. He proves: if f (z) is regular in \ 0 if it is regular there and satisfies Re/' (z) >0. K. Noshiro 6, p. 151 and S. Warschawski 10, p. 312 each demonstrated that Re/' (z) >0 is a sufficient condition for the schlichtness of f (z) in any convex domain. Conversely, S. R. Tims 9 proved that for each simply connected nonconvex domain D there is a function/ (z) regular in D such that Re/' (z) >0 and f (z) is not schlicht in D. This result is a particular consequence of some more general theorems contained in a paper by F. Herzog and G. Piranian 2. They determine both necessary and sufficient conditions for a domain D-not necessarily simply connected-to have the property that every function regular and satisfying Re/' (z) >0 in D is schlicht there. A more general class of functions than those satisfying Re/' (z) >0 is the class of close-to-convex functions. W. Kaplan 3 calls a function f (z) close-to-convex in |z| 0. Each function close-toconvex in \ 0for \ 0. 2. Distortion theorems. The following lemma contains results due to C. Caratheodory. A proof can be found in 7, Vol. 1, Problem 235, p. 129, and Vol. 1, Problem 287, p. 140. Lemma. If g (z) = l+ Sn-i onzn is regular in \ 0 then