TRANSACTIONS OF THE AMERICANMATHEMATICAL SOCIETY Volume 185, November 1973 ON COEFFICIENT MEANS OF CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS BY F. HOLLAND AND J. B. TWOMEY ABSTRACT. 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*|* (0<X<1) and /„(X) = n-'2rftxklA (a>0) as n -» +oo for bounded f(z) = 2r«áz'e¿?U¿ It is proved,for example, that the estimater„(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 possiblefor each fixedX > 0 within the class J? and for each fixedX > 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. 1. Introduction and statement of results. Let J? denote the class of regular functions of the form OD m = ïaHz" (\z\<\) i which satisfy Re/'(z) > 0 in y = [z: \z\ < 1},let S denote the class of regular functions (1.1) which are starlike in y, and let -¿"denote the class of functions/ for which f(z) = \og(g(z)/z) (\z\ < 1), where g G <£. It is well known that if / G J? U <£, then/is univalent in y. It is also well known that/ G JOS h G <=/?, where h'(z) = zf'(z)/f(z) and h(0) = 0, and it follows easily that we also have /G -¿"iff* G y?, where (1.2) k(z)= JT [?/'(H + 1R - z + j¡tt;^LTfl»z',+,• If / G <=/?, then, by a classical result for functions with positive real part, n\a„\ < 2 Re ax (n > 2). Applying this result to the function k defined in (1.2), we deduce that if/ G X, then n\an\ < 2 (n > 1). For/ G <£,on the other hand, we have [2] n\a„\ = 0(1) (n -» +oo) provided/is also bounded in y. Combining these results we get (1.3) \a„\ = 0(1)/« (n -+ +oo) Received by the editorsMarch7, 1972. AMS(MOS) subject classifications (1970). Primary 30A32, 30A34; Secondary 30A76. Key words and phrases. Coefficient means, bounded univalent functions, starlike functions, derivatives with positive real part. Copyright O 1974,American Mathematical Society 151 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use