Convexity constant of a domain and applications

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⊂C. We show that in the class of simply connected planar domains, K(D)=1 characterizes the convexity of the domain D, and we derive the value of the convexi...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2017-05, Vol.449 (1), p.793-807
Main Authors: Pascu, Nicolae R., Pascu, Mihai N.
Format: Article
Language:English
Subjects:
Convex set
Convexity constant of a domain
Univalence criterion
Univalent function
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Summary: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⊂C. We show that in the class of simply connected planar domains, K(D)=1 characterizes the convexity of the domain D, and we derive the value of the convexity constant for some classes of doubly connected domains of the form DΩ=D−Ω‾, for certain choices of the domains D and Ω. Using the convexity constant of a domain, we derive an extension of the well-known Ozaki–Nunokawa–Krzyz univalence criterion for the case of non-convex domains, and we present some examples, which show that our condition is sharp.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2016.12.024