On ratios of certain analytic functions

Let H(z) = f(z) /g(z) and h(z) = f'(z) / g'(z) where bath f and g are in one of the classes whose  members are convex, starlike relative to the origin, or typicallyreal in the open unit disk E. Sharp bounds on   ∝ > 0 and β > 0 such that Re {H(z) } ^∝ > 0,  Re {h (z) } ^β> 0, z ∈ E, are obtained with one exception. In the case where h( z) is considered for f and g typically-real, z is restricted to the precise region in E of univalence of the class of typically-real functions.