zbMATH — the first resource for mathematics

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.)
Keywords:
$$k$$-uniformly convex
Citations:
Zbl 0744.30010; Zbl 0805.30012
Full Text:
References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.