Functions whose derivative has a positive real part 论文

摘要

Introduction.Let P denote the class of functions which are regular and satisfy Re /' (z) > 0 for | z \ < 1 and are normalized by /(0) = 0 and /' (0) = 1.This paper develops some properties of functions in P.An early consideration of functions satisfying the condition Re/'(z)>0 can be found in a paper by J. W. Alexander [l, p. 18].He proves: if f(z) is regular in \z\ <1 and f'(z) "maps \z\ <1 upon a region contained within a half-plane bounded by a straight line through the origin" then/(2) is schlicht for \z\ <1.J. Wolff [ll] showed that/(z) is schlicht in Re z>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| <1 providing there is a function g(z) analytic, schlicht and convex in \z\ <1 for which Re{/'(z)/g'(z)} >0.Each function close-toconvex in \z\ <1 is schlicht there.Because of Alexander's result each function in P is schlicht in \z\ <1.Functions in Pean be constructed by integrating functions g(z) = I+&1Z+ which satisfy Reg(z)>0for \z\ <1.The function/(z) = -z -2 log (1-z) is in P for its derivative if f'(z) = (l+z)/(l-z) and w=(l+z)/(l-z) maps \z\ <1 onto Re w>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 \z\ <1 and Re g(z)>0 then