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Conic regions and \(k\)-uniform convexity. (English) Zbl 0944.30008
Let \(f(z)=z+a_2z^2+\dots\) be analytic and univalent in the unit disk \(D\). A function \(f\) is said to be \(k\)-uniformly convex in \(U\) if the image of every circular arc \(\gamma\), \(\gamma\subset D\) with its center at \(a\), \(|a|\leq k\), \(0\leq k<+\infty\), is convex. The authors establish a necessary and sufficient condition for \(f\) to be \(k\)-uniformly convex, study their properties and solve several extremal problems. Their results generalise some earlier ones due to a A. W. Goodman [Ann. Pol. Math. 56, No. 1, 87-92 (1991; Zbl 0744.30010)] and F. Rønning [Proc. Am. Math. Soc. 118, No. 1, 189-196 (1993; Zbl 0805.30012)].

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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[1] A.W. Goodman, Univalent Functions, Polygonal Publishing House, Washington, NJ, 1983.
[2] Goodman, A.W., On uniformly convex functions, Ann. polon. math., 56, 87-92, (1991) · Zbl 0744.30010
[3] J. Krzyż, J. Ławrynowicz, Elementy Analizy Zespolonej, WNT, Warszawa, 1981.
[4] J. Gerretsen, G. Sansone, Lectures on the Theory of Functions of a Complex Variable, Wolters-Noordhoff Publishing, Groningen, Netherlands, 1969. · Zbl 0188.38104
[5] W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, Proc. Int. Conf. on Complex Anal. at the Nankai Inst. of Math., 1992, pp. 157-169. · Zbl 0823.30007
[6] Ma, W.; Minda, D., Uniformly convex functions, Ann. polon. math., 57, 2, 165-175, (1992) · Zbl 0760.30004
[7] Rønning, F., Uniformly convex functions and a corresponding class of starlike functions, Proc. amer. math. soc., 118, 189-196, (1993) · Zbl 0805.30012
[8] Rønning, F., Some radius results for univalent functions, J. math. anal. appl., 194, 319-327, (1995) · Zbl 0834.30011
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