Functions whose derivatives take values in a half-plane

Fernando Gray, Stephan Ruscheweyh
1988 Proceedings of the American Mathematical Society  
We derive sharp upper and lower bounds for \zf'(z)/f(z)\ where / € 31, i.e. / analytic in D with /(0) = 0, /'(0) = 1 and Reeiaf'(z) > 0 in D for a certain a = a(f) G R. The extremal function is k(z) = -z -21og(l -z). This result improves an earlier one of D. K. Thomas. Let 3er¡ denote the class of analytic functions / in the unit disk D with /(0) = 0, /'(0) = 1, and Re/'(z) > 0 in D. In a recent paper D. K. Thomas [4] proved that
doi:10.1090/s0002-9939-1988-0958069-6 fatcat:6zg4anr4hraclkmlv5s2xjbple