Functionals of rational type over the class $S$
HTML articles powered by AMS MathViewer
- by Louis Brickman PDF
- Proc. Amer. Math. Soc. 92 (1984), 372-376 Request permission
Abstract:
Let $L$ be a continuous linear functional on the space of functions holomorphic in the unit disk, and let $f$ be a function in the class $S$ for which Re $L$ achieves its maximum on $S$. Then $L$ is said to be of rational type if the expression $L({f^2}/(f - w))$, which occurs in Schiffer’s differential equation, is a rational function of $w$. Various equivalent formulations of "rational type" are found and an application to the process of arc truncation of support points of $S$ is made.References
- Louis Brickman and Peter Duren, Truncation of support points for univalent functions, Complex Variables Theory Appl. 3 (1984), no. 1-3, 71–83. MR 737473, DOI 10.1080/17476938408814064
- L. Brickman and Y. J. Leung, Exposed points of the set of univalent functions, Bull. London Math. Soc. 16 (1984), no. 2, 157–159. MR 737244, DOI 10.1112/blms/16.2.157 L. Brickman, Y. J. Leung and D. R. Wilken, On extreme points and support points of the class $S$, Ann. Univ. Mariae Curie-Sklodowska (Krzyz special issue) (to appear).
- Louis Brickman and Donald Wilken, Support points of the set of univalent functions, Proc. Amer. Math. Soc. 42 (1974), 523–528. MR 328057, DOI 10.1090/S0002-9939-1974-0328057-1
- P. L. Duren, Y. J. Leung, and M. M. Schiffer, Support points with maximum radial angle, Complex Variables Theory Appl. 1 (1982/83), no. 2-3, 263–277. MR 690498, DOI 10.1080/17476938308814018
- William E. Kirwan and Glenn Schober, New inequalities from old ones, Math. Z. 180 (1982), no. 1, 19–40. MR 656220, DOI 10.1007/BF01214997
- Albert Pfluger, Lineare Extremalprobleme bei schlichten Funktionen, Ann. Acad. Sci. Fenn. Ser. A. I. 489 (1971), 32 (German). MR 296276 M. M. Schiffer, A method of variation within the family of simple functions, Proc. London Math. Soc. (2) 44 (1938), 432-449.
- Otto Toeplitz, Die linearen vollkommenen Räume der Funktionentheorie, Comment. Math. Helv. 23 (1949), 222–242 (German). MR 32952, DOI 10.1007/BF02565600
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 372-376
- MSC: Primary 30C55; Secondary 30C70
- DOI: https://doi.org/10.1090/S0002-9939-1984-0759655-5
- MathSciNet review: 759655