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A covering theorem for typically real functions

Published online by Cambridge University Press:  18 May 2009

D. A. Brannan  and
W. E. Kirwan
D. A. Brannan
Affiliation:
University of MarylandCollege Park, MD., U.S.A.
W. E. Kirwan
Affiliation:
University of MarylandCollege Park, MD., U.S.A.
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Let T denote the class of functions

f(z) = z+a2z2+…

that are analytic in U = {|z| <1}, and satisfy the condition

Imf(z). Imz≧ 0 (zεU).

Thus T denotes the class of typically real functions introduced by W. Rogosinski [5].

One of the most striking results in the theory of functions

g(z) = z + b2z2…

that are analytic and univalent in U is the Koebe-Bieberbach covering theorem which states that {|w| <¼} ⊂ g(U). In this note we point out that the same result holds for functions in the class T, a fact which seems to have been overlooked previously. We also determine the largest subdomain of U in which every f(z) in T is univalent, extending previous results in [1] and [2].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 10 , Issue 2 , July 1969 , pp. 153 - 155
Copyright
Copyright © Glasgow Mathematical Journal Trust 1969

References

REFERENCES

1.Čakalov, L., Maximal domains of univalency of some classes of analytic functions, Ukr. Math. J. 11, no. 4 (1959), 408–412 (Russian).Google Scholar
2.Kirwan, W. E., Extremal problems for the typically real functions, Amer. J. Math. 88 (1966), 942–954.CrossRefGoogle Scholar
3.Remizova, M. P., Extremal problems in the class of typically-real functions, Izv. Vyss. Uc. Zaved. Mat. (1963), no. 1 (32), 135–144.Google Scholar
4.Robertson, M. S., On the coefficients of typically real functions, Bull. Amer. Math. Soc. 41 (1936), 565–572.CrossRefGoogle Scholar
5.Rogosinski, W. W., Über positive harmonische Entwicklungen und typische-reelle Potenzreihen, Math.Z. 35 (1932), 93–121.CrossRefGoogle Scholar