Stud. Univ. Babe¸s-Bolyai Math. 64(2019), No. 4, 487–496 DOI: 10.24193/subbmath.2019.4.04 Differential superordination for harmonic complex-valued functions Georgia Irina Oros and Gheorghe Oros Abstract. Let Ω and Δ be any sets in C, and let ϕ(r, s, t; z): C 3 × U → C. 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 ) ⊂ Δ. 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) ⇒ Δ ⊂ p(U ). Mathematics Subject Classification (2010): 30C80, 30C46, 30A20, 34A40. Keywords: Differential subordination, harmonic functions, differential superordi- nation, subordinant, best subordinant, analytic function. 1. Introduction and preliminaries The theory of differential subordinations (or the method of admissible functions) for analytic functions was introduced by S.S. Miller and P.T. Mocanu in papers [6] and [7] and later developed in [1], [8], [10], [11], [12], [13]. The theory of differential subordinations has been extended from the analytic functions to the harmonic complex-valued functions in papers [2], [5], [14]. Let U = {z ∈ C : |z| < 1} be the open unit disc of the complex plane with U = {z ∈ C : |z|≤ 1} and ∂U = {z ∈ C : |z| =1}. Denote by H(U ) the class of holomorphic functions in the unit disc U , and A n = {f ∈H(U ): f (z)= z + a n+1 z n+1 + ...},A 1 = A.