On power deformations of univalent functions |
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Authors: | Yong Chan Kim Toshiyuki Sugawa |
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Affiliation: | 1.Department of Mathematics Education,Yeungnam University,Gyongsan,Korea;2.Division of Mathematics, Graduate School of Information Sciences,Tohoku University,Aoba-ku, Sendai,Japan |
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Abstract: | For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) ? 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f c (z) = z(f (z)/z) c for a complex number c. We determine those values c for which the operator ({f mapsto f_c}) maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions. |
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