Differential superordination for harmonic complex-valued functions

Let Omega and Delta be any sets in C, and let phi(r, s, t, z) : C-3 x U -> C. Let p be a complex-valued harmonic function in the unit disc U of the form p(z) = p(1)(z) + <(p(2)(z))over bar>, where p(1) and p(2) are analytic in U. In [5] the authors have determined properties of the function p such that p satisfies the differential subordination phi(p(z), Dp(z), D(2)p(z), z) subset of Omega double right arrow p(U) subset of Delta. In this article, we consider the dual problem of determining properties of the function p, such that p satisfies the second-order differential superordination Omega subset of phi(p(z), Dp(z), D(2)p(z); z) double right arrow Delta subset of p(U).