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On functions of bounded boundary rotation I

  • D. A. Brannan (a1)

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Let Vk denote the class of functions

which map conformally onto an image domain ƒ(U) of boundary rotation at most (see (7) for the definition and basic properties of the class ). In this note we discuss the valency of functions in Vk, and also their Maclaurin coefficients.

In (8) it was shown that functions in Vk are close-to-convex in . Here we show that Vk is a subclass of the class K(α) of close-to-convex functions of order α (10) for , and we give an upper bound for the valency of functions in Vk for K>4.

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References

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(1) Brannan, D. A. and Kirwan, W. E.On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431443.
(2) Hayman, W. K.Multivalent functions (Cambridge University Press, 1958).
(3) Kirwan, W. E.A note on the coeflScients of functions with bounded boundary rotation, Michigan Math. J. 15 (1968), 277282.
(4) Littlewood, J. E.Lectures on the Theory of Functions (Oxford University Press, 1944).
(5) Marx, A.Untersuchungen über schlichte Abbildungen, Math. Ann. 107 (1932), 4067.
(6) Macrobert, T. M.Functions of a Complex Variable (MacMillan, London 1958).
(7) Paatero, V.Uber die konforme Abbildungen von Gebieten deren Rander von beschrankter Drehung sind, Ann. Acad. Sci. Fenn. Ser. A, 33 no. 9 (1931).
(8) Pinchuk, B. A variational method for functions of bounded boundary rotation, to appear.
(9) Pommerenke, CH.On the coefficients of close-to-convex functionsy, Michigan Math. J. 9 (1962), 259269.
(10) Pommerenke, CH.On starlike and convex functions, J. London Math. Soc. 37 (1962), 209224.
(11) Prtvalov, I.I.Randeigenschaften analytischer Funktionen (V. E. B. Deutscher Verlag der Wissenschaften, Berlin 1956).

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