Geodesic
A covering theorem for typically real functions
Glasgow mathematical journal, Tome 10 (1969) no. 2, pp. 153-155

Voir la notice de l'article provenant de la source Cambridge University Press

Let T denote the class of functionsf(z) = z+a2z2+...that are analytic in U = {|z| <1}, and satisfy the conditionImf(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 functionsg(z) = z + b2z2...that are analytic and univalent in U is the Koebe-Bieberbach covering theorem which states that {|w| <1⁄4} ⊂ 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]. 
Brannan, D. A.; Kirwan, W. E. A covering theorem for typically real functions. Glasgow mathematical journal, Tome 10 (1969) no. 2, pp. 153-155. doi: 10.1017/S0017089500000719
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[1] 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] 2.Kirwan, W. E., Extremal problems for the typically real functions, Amer. J. Math. 88 (1966), 942–954. Google Scholar | DOI

[3] 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] 4.Robertson, M. S., On the coefficients of typically real functions, Bull. Amer. Math. Soc. 41 (1936), 565–572. Google Scholar | DOI

[5] 5.Rogosinski, W. W., Über positive harmonische Entwicklungen und typische-reelle Potenzreihen, Math.Z. 35 (1932), 93–121. Google Scholar | DOI

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