Systems of simultaneous differential inequalities, inclusions and subordinations in the complex plane
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Systems of simultaneous differential inequalities, inclusions and subordinations in the complex plane José A. Antonino1 · Sanford S. Miller2 Received: 22 October 2019 / Accepted: 18 May 2020 © Springer Nature Switzerland AG 2020
Abstract There are many articles in the literature dealing with a first, second and third-order differential inequalities, inclusions or subordinations in the complex plane, none of which deals with systems of such topics. This article investigates systems of two second-order simultaneous differential inequalities, inclusions and subordinations in two complex functions p and q in the complex plane. A typical example of a system of differential inequalities is ⎧ ⎨ Re[2 p(z) + zp (z) + z 2 p (z) − q(z)] > 0, ⎩ Re[ p(z) + 7zq (z)] > 4. The authors determine properties of the functions p and q satisfying some special systems of differential inequalities, and extend their results to differential inclusions and subordinations. Keywords Differential inequalities · Differential inclusion · Differential subordination · Dominant · Analytic functions Mathematics Subject Classification Primary 34A40 · 34A60; Secondary 30C80
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Sanford S. Miller [email protected] José A. Antonino [email protected]
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Departamento de Matemática Aplicada, ETSICCP, Universidad Politécnica de Valencia, 46071 Valencia, Spain
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Department of Mathematics, SUNY Brockport, Brockport, NY 14420, USA 0123456789().: V,-vol
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J. A. Antonino, S. S. Miller
1 Introduction We begin by introducing the important classes of functions considered in this article. Let H H [U] denote the class of functions analytic in the unit disk U, and let H [a, n] f ∈ H : f (z) a + an z n + · · · . A common problem in differential inequalities in the complex plane is to determine the range of a function p from an inclusion relation involving several of the derivatives of p. Let Ω and be sets in C and let p ∈ H [a, n]. Let D be a differential operator such that D[ p] is an analytic function defined on U. A natural question is to ask what conditions on D, Ω and are needed so that D[ p](U) ⊂ Ω ⇒ p(U ) ⊂ .
(1)
Many inclusion results such as (1) can be written very neatly in terms of subordination. Let f and F be members of H . The function f is said to be subordinate to F, or F is superordinate to f, written f ≺ F, if there exists a function w analytic in U with w(0) 0 and |w(z)|< 1, and such that f (z) F(w(z)). If, in addition, F is univalent, then f ≺ F if and only if f (0) F(0) and f (U) ⊂ F(U). If in (1) is a simply connected domain, then it may be possible to rephrase (1) in terms of subordination. If is a simply connected domain containing the point p(0) a and C, then there is a conformal mapping g of U onto such that g(0) a. In this case (1) can be rewritten as D[ p](U) ⊂ Ω ⇒ p(z) ≺ g(z).
(2)
If in addition Ω is a simply connected domain and Ω C, then there is a conformal mapping h of U onto Ω such that h(0) D[ p](0). In this case (2) can be rewritten in terms of subordinat
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