Remarks on Convexity in Dimension (2,2)

Wojciech Kryński

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An integral operator on the classes $𝒮^{*} (α)$ and $𝒞𝒱ℋ (β)$

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The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $α$ and from the class $𝒞𝒱ℋ (β)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $𝒞𝒱ℋ (β)$ .

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We give a complete characterization of closed sets $F \subset ℝ^{2}$ whose distance function $d_{F} : = dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $ℝ^{2}$ ). Using this characterization, a number of properties of such sets is proved.