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The critical points of a typically-real function


Author: A. W. Goodman
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
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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.


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  • [1] E. F. Beckenbach and E. W. Graham, On subordination in complex variable theory. Construction and applications of conformal maps, Proc. Sympos., National Bureau of Standard, Appl. Math. Ser., no. 18, U.S. Government Printing Office, Washington, D.C., 1952, pp. 247-254. MR 14, 632. MR 0052516 (14:632c)
  • [2] S. D. Bernardi, Bibliography of schlicht functions, Courant Institute of Mathematical Sciences, New York University, Technical Report #NR 041-019, IMM 351, Office of Naval Research, New York; New York University, New York, 1966. MR 34 #2849. MR 0202990 (34:2849)
  • [3] W. E. Kirwan, Extremal problems for the typically real functions, Amer. J. Math. 88 (1966), 942-954. MR 34 #2853. MR 0202994 (34:2853)
  • [4] 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.
  • [5] J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. (2) 23 (1925), 481-519.
  • [6] Z. Nehari, On analytic functions possessing certain properties of univalence, Proc. London Math. Soc. (2) 50 (1948), 120-136. MR 9, 576. MR 0024977 (9:576g)
  • [7] M. S. Robertson, On the coefficients of a typically-real function, Bull. Amer. Math. Soc. 41 (1935), 565-572. MR 1563142
  • [8] W. Rogosinski, Über Bildschranken bei Potenzreihen und ihren Abschnitten, Math. Z. 17 (1923), 260-276. MR 1544615
  • [9] -, Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), 93-121. MR 1545292
  • [10] -, Zum Majorantenprinzip der Funktionentheorie, Math. Z. 37 (1933), 210-236. MR 1545391
  • [11] -, On subordinate functions, Proc. Cambridge Philos. Soc. 35 (1939), 1-36.
  • [12] -, On the coefficients of subordinate functions, Proc. London Math. Soc. (2) 48 (1943), 48-82. MR 0008625 (5:36a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0313489-7
Keywords: Typically-real functions, critical points, branch points, subordination, Schwarz lemma, univalent functions, multivalent functions
Article copyright: © Copyright 1973 American Mathematical Society

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