×

On functions convex in the direction of the real axis with a fixed second coefficient. (English) Zbl 1211.30020

Summary: We consider the class \(\Gamma \) of analytic and univalent functions \(f\) in the unit disk \(\Delta\) normalized by \(f(0) = f^{\prime}(0) - 1 = 0\) which have real coefficients and such that \(f(Delta)\) is convex in the direction of the real axis. We are especially interested in some subclasses of \(\Gamma \). The most important of them is \(\Gamma (c)\) consisting of the functions whose second Taylor coefficient is equal to \(c\). We obtain the Koebe set for this class as well as for the classes \(\Gamma ^{+}(c)\) and \(\varGamma ^{ - }(c)\) of functions which are in some sense convex in the direction of the positive and the negative axis, respectively.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Royster, W. C.; Ziegler, M., Univalent functions convex in one direction, Publ. Math., Debrecen., 23, 340-345 (1976) · Zbl 0365.30005
[2] Hengartner, W.; Schober, G., On schlicht mappings to domains convex in one direction, Comment. Math. Helv., 45, 303-314 (1970) · Zbl 0203.07604
[3] Koczan, L., Typically real functions convex in the direction of the real axis, Ann. Univ. Mariae Curie-Skodowska, Sect. A, 52, 2, 103-112 (1998)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.