Differential superordination for harmonic complex-valued functions
DOI:
https://doi.org/10.24193/subbmath.2019.4.04Keywords:
Differential subordination, harmonic functions, differential superordination, subordinant, best subordinant, analytic function.Abstract
Let Ω and ∆ be any sets in C, and let ϕ(r, s, t; z) : C3 × 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), D2p(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), D2p(z); z) ⇒ ∆ ⊂ p(U ).
Mathematics Subject Classification (2010): 30C80, 30C46, 30A20, 34A40.
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