CLASSES OF FIRST-ORDER DIFFERENTIAL SUPERORDINATIONS

Let ip : C 2 —> C be an analytic function in a domain A C C 2 , let p an analytic function in the unit disc U such that ip(p(z), zp'(z)) is univalent in U and suppose that p satisfies the first-order differential superordination h(z) tp(p(z),zp'(z)). In the case when <p(p(z), zp (z)) = a(p(z)) + P(p(z)) j(zp'(z)) we determine conditions on h, a, 0 and 7 so that the above subordination implies q(z) -< p(z), where q is the largest function so that q(z) -< p(z) for all p functions satisfying the first-order differential superordination. The concept of differential superordination was introduced by S. S. Miller and P. T. Mocanu in [2] like a dual problem of differential subordination [1],