Families of real and symmetric analytic functions

Abu-Muhanna, Yusuf; MacGregor, Thomas H.

doi:10.1090/S0002-9947-1981-0590411-9

Families of real and symmetric analytic functions
HTML articles powered by AMS MathViewer

by Yusuf Abu-Muhanna and Thomas H. MacGregor PDF

Trans. Amer. Math. Soc. 263 (1981), 59-74 Request permission

Abstract:

We introduce families of functions analytic in the unit disk and having rotational symmetries. The families include the $k$-fold symmetric univalent functions which have real coefficients. We relate the families to special classes of functions with a positive real part and then determine their extreme points. The case $k = 2$ corresponds to the odd functions which "preserve quadrants" and the extreme points of this set are characterized by having a radial limit which is real or imaginary almost everywhere. We also find estimates on the initial coefficients of functions in the families.

References

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
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655

Eine Bemerkung über ungerade schlichte Funktionen

8

David J. Hallenbeck, Convex hulls and extreme points of families of starlike and close-to-convex mappings, Pacific J. Math. 57 (1975), no. 1, 167–176. MR 379820
John G. Milcetich, On the extreme points of some sets of analytic functions, Proc. Amer. Math. Soc. 45 (1974), 223–228. MR 352470, DOI 10.1090/S0002-9939-1974-0352470-X
Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768
M. S. Robertson, On the coefficients of a typically-real function, Bull. Amer. Math. Soc. 41 (1935), no. 8, 565–572. MR 1563142, DOI 10.1090/S0002-9904-1935-06147-6
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
A. C. Schaeffer and D. C. Spencer, The coefficients of schlicht functions, Duke Math. J. 10 (1943), 611–635. MR 9631
M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898