Let be the class of functions analytic in and be the subclass of consisting of functions of the form . Let and be any sets in the complex plane , let and let . In the present paper, the authors obtain conditions on , and for which the following implication holds: .
When and are simply connected domains with , the above implication becomes , where and are the conformal mappings of onto the domains and respectively. If and are univalent and if satisfies the second order superordination , is the solution of the differential superordination. (If is subordinate to , then is superordinate to .) Subordinant and best subordinant are defined similarly like dominant and best dominant in case of differential subordination.
Denote by , the set of all functions , with , that are analytic and injective on , where , and are such that for . For a set in and with , the class of admissible functions consists of those functions that satisfy the admissibility condition: , whenever , , , where , and .
The principal result proved in the paper for second order differential superordinations is the following:
Theorem. Let be analytic in and . Suppose that has a solution . If , and is univalent in , then and is the best subordinant.
By using the results for first order superordinations together with previously known results for differential subordinations, the authors have obtained several differential “sandwich theorems”. Also a special second order differential superordination is considered. Some applications of the results of this paper was obtained recently by T. Bulboaca [Demonstr. Math. 35, No. 2, 287–292 (2002; Zbl 1010.30020)].