STRONG DIFFERENTIAL SUBORDINATION TO BRIOT-BOUQUET DIFFERENTIAL-EQUATIONS

Let p(z) be analytic in the unit disc D, let g(z) be convex in D, let f(z) be analytic in D such that z(f'(z)/f(z)) is analytic and different from zero in D, and let alpha and beta be complex numbers. The authors show that if p(z)+zp'(z)/xi(f'(xi)/f(xi))[alpha p(z)+beta]<g(z), z is an element of D, xi is an element of D, where < denotes subordination and Re{f'(xi)/f(xi)[alpha g(z)+beta]}>0 z is an element of D, xi is an element of D is satisfied, then p(z)<g(z). Further, if the differential equation q(z)+ zq'(z)/z(f'(z)/f(z))[alpha q(z)+beta]=g(z) verifying (1), has a univalent solution g(z), then sharp subordination p(z)<q(z) holds. (C) 1994 Academic Press, Inc.