• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.105-110
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• 2010

In this paper, we introduce the concept of $\beta$-preconvex sets on preconvexity spaces. We study some properties for $\beta$-preconvex sets by using the co-convexity hull and the convexity hull. Also we introduce and study the concepts of ${\beta}c$-convex function and $\beta^*c$-convex function.

• Honam Mathematical Journal
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• v.37 no.3
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• pp.317-323
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• 2015

The purpose of the present paper is to study certain radii problems for the function $$f(z)=\[{\frac{z^{1-{\gamma}}}{{\gamma}+{\beta}}}\(z^{\gamma}[D^nF(z)]^{\beta}\)^{\prime}\]^{1/{\beta}}$$, where ${\beta}$ is a positive real number, ${\gamma}$ is a complex number such that ${\gamma}+{\beta}{\neq}0$ and the function F(z) varies various subclasses of analytic functions with fixed second coefficients. Relevant connections of the results presented herewith various well-known results are briefly indicated.