An Univalence Criterion for Analytic Functions Defined in Type $$\varphi $$ φ Convex Domains
| Autor: | Mihai N. Pascu, Nicolae R. Pascu |
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| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Applied Mathematics 010102 general mathematics Mathematical analysis Regular polygon Function (mathematics) Type (model theory) 01 natural sciences Domain (mathematical analysis) Convexity 010101 applied mathematics Computational Mathematics Computational Theory and Mathematics Simply connected space 0101 mathematics Constant (mathematics) Analytic function Mathematics |
| Zdroj: | Complex Analysis and Operator Theory. 11:1781-1787 |
| ISSN: | 1661-8262 1661-8254 |
| Popis: | In the present paper we introduce a new characterization of the convexity of a planar domain, based on the convexity constant K(D) of a domain $$D\subset \mathbb {C}$$ . We show that in the class of simply connected planar domains, $$K(D) =1$$ characterizes the convexity of the domain D. Using the convexity constant of a domain, we derive a sufficient condition for the univalence of an analytic function defined in a type $$\varphi $$ convex domain, similar to the one obtained by Reade (Math Soc Jpn 10:255–259, 1958), but involving the modulus instead of the argument of the derivative of the function. As a corollary we obtain the well-known Ozaki–Nunokawa–Krzyz univalence criterion, and we also show that our condition is sharp. |
| Databáze: | OpenAIRE |
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