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On Coefficient Means of Certain Subclasses of Univalent Functions

F. Holland, J. B. Twomey
1973 Transactions of the American Mathematical Society  
Let *R denote the class of regular functions whose derivatives have positive real part in the unit disc y and let S denote the class of functions starlike in y. In this paper we investigate the rates of growth of the means s"(\) = n"1 2" |a*|* (00) as n -» +oo for bounded f(z) = 2r«áz'e¿?U¿ It is proved, for example, that the estimate r"(X) = o(l)(log 7i)~°w (n -» +oo), where a(X) = X/2 for 0 < X < 2 and a(X) = 1 for X > 2, holds for such functions/, and that it is best possible for each fixed
more » ... ble for each fixed X > 0 within the class J? and for each fixed X > 2 within the class S. It is also shown that the inequality i"(l) = o(l)n~'(log n)1^, which holds for all bounded univalent functions, cannot be improved for bounded / e <=/?. The behavior of r"(X) as n -* +00 when ak > 0 (k > 1) and X > 1 is also examined.
doi:10.2307/1996431 fatcat:i2bm2cdhgvfmpju4safxo2bckm