The critical points of a typically-real function
HTML articles powered by AMS MathViewer
- by A. W. Goodman PDF
- Proc. Amer. Math. Soc. 38 (1973), 95-102 Request permission
Abstract:
The critical points of a typically-real function cannot lie too close to the real axis. By adding a mild restriction, we determine ${D_k}$ the domain of variability of a $k$th order critical point. Similar results are obtained for a $k$th order branch point. We determine the domain of univalence for typically-real functions and propose a reasonable conjecture for the domain of $k$-valence.References
- E. F. Beckenbach and E. W. Graham, On subordination in complex variable theory, Construction and applications of conformal maps. Proceedings of a symposium, National Bureau of Standards Applied Mathematics Series, No. 18, U.S. Government Printing Office, Washington, D.C., 1952, pp. 247–254. MR 0052516
- S. D. Bernardi, Bibliography of Schlicht functions, Office of Naval Research, New York; New York University, New York, 1966. Courant Institute of Mathematical Sciences, New York University, Tech. Report No. NR 041-019, IMM 351. MR 0202990
- William E. Kirwan, Extremal problems for the typically real functions, Amer. J. Math. 88 (1966), 942–954. MR 202994, DOI 10.2307/2373090 E. Lindelöf, Mémoire sur certaines inégalités dans la théorie des fonctions monogènes et sur quelques propriétés nouvelles de ces fonctions dans le voisinage d’un point singulier essentiel, Acta Soc. Sci. Fenn. 35 (1908), no. 7. J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. (2) 23 (1925), 481-519.
- Zeev Nehari, On analytic functions possessing certain properties of univalence, Proc. London Math. Soc. (2) 50 (1948), 120–136. MR 24977, DOI 10.1112/plms/s2-50.2.120
- 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 Bildschranken bei Potenzreihen und ihren Abschnitten, Math. Z. 17 (1923), no. 1, 260–276 (German). MR 1544615, DOI 10.1007/BF01504347
- 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
- Werner Rogosinski, Zum Majorantenprinzip der Funktionentheorie, Math. Z. 37 (1933), no. 1, 210–236 (German). MR 1545391, DOI 10.1007/BF01474571 —, On subordinate functions, Proc. Cambridge Philos. Soc. 35 (1939), 1-36.
- Werner Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (2) 48 (1943), 48–82. MR 8625, DOI 10.1112/plms/s2-48.1.48
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 38 (1973), 95-102
- MSC: Primary 30A32
- DOI: https://doi.org/10.1090/S0002-9939-1973-0313489-7
- MathSciNet review: 0313489