Differential superordination for harmonic complex-valued functions

Abstract

Let p be a complex-valued harmonic function in the unit disc U of the form p(z) = p1(z) + p2(z), where p1 and p2 are analytic in U. In [5] the authors have determined properties of the function p such that p satisfies the differential subordination ϕ(p(z), Dp(z), D 2 p(z); z) ⊂ Ω ⇒ p(U) ⊂ ∆. In this article, we consider the dual problem of determining properties of the function p, such that p satisfies the second-order differential superordination Ω ⊂ ϕ(p(z), Dp(z), D 2 p(z); z) ⇒ ∆ ⊂ p(U).

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