×

The support points of several classes of analytic functions with fixed coefficients. (English) Zbl 1131.30021

Summary: Let \(B\) denote the set of functions \(\varphi(z)\) that are analytic in the unit disk \(D\) and satisfy \(|\varphi(z)|\leq1\) \((|z|<1)\). Let \({\mathcal P}\) denote the set of functions \(p(z)\) that are analytic in \(D\) and satisfy \(p(0)=1\) and \(\operatorname{Re}\, p(z)>0\) \((|z|<1)\). Let \(T\) denote the set of functions \(f(z)\) that are analytic in \(D\), normalized by \(f(0)=0\) and \(f'(0)=1\) and satisfy that \(f(z)\) is real if and only if z is real \((|z|<1)\). In this article we investigate the support points of the subclasses of \(B\), \({\mathcal P}\) and \(T\) of functions with fixed coefficients.

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abu-Muhanna, Y., Variability regions and support points of subordinate families, J. London Math. Soc., 29, 477-484 (1984) · Zbl 0569.30020
[2] Boyd, D., Schur’s algorithm for bounded holomorphic functions, Bull. London Math. Soc., 11, 145-150 (1979) · Zbl 0421.30002
[3] Cochrane, P. C.; MacGregor, T. H., Fréchet differentiable functionals and support points for families of analytic functions, Trans. Amer. Math. Soc., 236, 75-92 (1978) · Zbl 0377.30010
[4] Foias, C.; Frazho, A., The Commutant Lifting Approach to Interpolation Problems, Oper. Theory Adv. Appl., vol. 44 (1990), Birkhäuser: Birkhäuser Basel · Zbl 0718.47010
[5] Garnett, J. B., Bounded Analytic Functions (1981), Academic Press: Academic Press New York, pp. 180-181
[6] Hallenbeck, D. J.; MacGregor, T. H., Support points of families of analytic functions described by subordination, Trans. Amer. Math. Soc., 278, 523-546 (1983) · Zbl 0521.30018
[7] Hallenbeck, D. J.; MacGregor, T. H., Subordinations and extreme point theory, Pacific J. Math., 50, 455-468 (1974) · Zbl 0258.30015
[8] Hallenbeck, D. J.; MacGregor, T. H., Linear Problems and Convexity Techniques in Geometric Function Theory (1984), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston · Zbl 0581.30001
[9] Kuznetsov, V. O., Typically real functions with singularities on the real axis, J. Math. Sci., 59, 1173-1180 (1992)
[10] Kiepiela, K.; Naraniecka, I.; Szynal, J., The Gegenbauer polynomials and typically real functions, J. Comput. Appl. Math., 153, 273-282 (2003) · Zbl 1019.30013
[11] Ma, William., Generalized Zalcman conjecture for starlike and typically real functions, J. Math. Anal. Appl., 234, 328-339 (1999) · Zbl 0936.30012
[12] Peng, Zhigang., The extreme points of several classes of analytic functions, Acta Math. Sci., 19, 457-462 (1999) · Zbl 0947.30007
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.