Convexity constant of a domain and applications

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• 作者：Nicolae R. Pascu^a ; ^{npascu@kennesaw.edu} ; Mihai N. Pascu^b ; ^{mihai.pascu@unitbv.ro}
• 关键词：Convex set ; Convexity constant of a domain ; Univalent function ; Univalence criterion
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 May 2017
• 年：2017
• 卷：449
• 期：1
• 页码：793-807
• 全文大小：376 K
• 卷排序：449

文摘

In the present paper we introduce a new characterization of the convexity of a planar domain, based on the convexity constant K(D)K(D) of a domain D⊂CD⊂C. We show that in the class of simply connected planar domains, K(D)=1K(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.