Fréchet differentiable functionals and support points for families of analytic functions

Cochrane, Paul; MacGregor, Thomas H.

doi:10.1090/S0002-9947-1978-0460611-7

Fréchet differentiable functionals and support points for families of analytic functions
HTML articles powered by AMS MathViewer

by Paul Cochrane and Thomas H. MacGregor

Trans. Amer. Math. Soc. 236 (1978), 75-92

DOI: https://doi.org/10.1090/S0002-9947-1978-0460611-7

PDF | Request permission

Abstract:

Given a closed subset of the family ${S^\ast }(\alpha )$ of functions starlike of order $\alpha$ of a particular form, a continuous Fréchet differentiable functional, J, is constructed with this collection as the solution set to the extremal problem $\max \operatorname {Re} J(f)$ over ${S^\ast }(\alpha )$. Similar results are proved for families which can be put into one-to-one correspondence with ${S^\ast }(\alpha )$. The support points of ${S^\ast }(\alpha )$ and $K(\alpha )$, the functions convex of order $\alpha$, are completely characterized and shown to coincide with the extreme points of their respective convex hulls. Given any finite collection of support points of ${S^\ast }(\alpha )$ (or $K(\alpha )$), a continuous linear functional, J, is constructed with this collection as the solution set to the extremal problem $\max \operatorname {Re} J(f)$ over ${S^\ast }(\alpha )$ (or $K(\alpha )$).

References

N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
D. A. Brannan, J. G. Clunie, and W. E. Kirwan, On the coefficient problem for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A. I. 523 (1973), 18. MR 338343
L. Brickman, T. H. MacGregor, and D. R. Wilken, Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc. 156 (1971), 91–107. MR 274734, DOI 10.1090/S0002-9947-1971-0274734-2
L. Brickman, D. J. Hallenbeck, T. H. MacGregor, and D. R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 (1973), 413–428 (1974). MR 338337, DOI 10.1090/S0002-9947-1973-0338337-5

Über den Variabilitätsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen

32

Funktionentheorie

Theory of functions of a complex variable

12

16

The Taylor series

19

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518, DOI 10.1007/978-1-4615-7904-5
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039, DOI 10.1090/mmono/026
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
J. A. Hummel, Extremal problems in the class of starlike functions, Proc. Amer. Math. Soc. 11 (1960), 741–749. MR 120379, DOI 10.1090/S0002-9939-1960-0120379-2
W. E. Kirwan and G. Schober, On extreme points and support points for some families of univalent functions, Duke Math. J. 42 (1975), 285–296. MR 367174, DOI 10.1215/S0012-7094-75-04226-X
Thomas H. MacGregor, Applications of extreme-point theory to univalent functions, Michigan Math. J. 19 (1972), 361–376. MR 311885
Thomas H. MacGregor, Rotations of the range of an analytic function, Math. Ann. 201 (1973), 113–126. MR 328048, DOI 10.1007/BF01359788
Thomas H. MacGregor, Hull subordination and extremal problems for starlike and spirallike mappings, Trans. Amer. Math. Soc. 183 (1973), 499–510. MR 338339, DOI 10.1090/S0002-9947-1973-0338339-9
A. J. Macintyre and W. W. Rogosinski, Extremum problems in the theory of analytic functions, Acta Math. 82 (1950), 275–325. MR 36314, DOI 10.1007/BF02398280
Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0045823
Ch. Pommerenke, On starlike and convex functions, J. London Math. Soc. 37 (1962), 209–224. MR 137830, DOI 10.1112/jlms/s1-37.1.209
Pasquale Porcelli, Linear spaces of analytic functions, Rand McNally & Co., Chicago, Ill., 1966. MR 0259581
Malcolm I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), no. 2, 374–408. MR 1503286, DOI 10.2307/1968451
Werner Rogosinski, Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), no. 1, 93–121 (German). MR 1545292, DOI 10.1007/BF01186552

Über Potenzreihen

die im Innern des Einheitskreises beschränkt sind

147

Über Potenzreihen

die im Innern des Einheitskreises beschränkt sind

148

Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
Otto Toeplitz, Die linearen vollkommenen Räume der Funktionentheorie, Comment. Math. Helv. 23 (1949), 222–242 (German). MR 32952, DOI 10.1007/BF02565600