Math. Z. 147, 237-247 (1976) Mathematische Zeitschrift 9 by Springer-Verlag 1976 Some Analytic Functions Whose Boundary Values Have Bounded Mean Oscillation Joseph A. Cima and Karl E. Petersen Department of Mathematics, The Universityof North Carolina, Chapel Hill, North Carolina 27514, USA 1. Introduction Fefferman's and Stein's identification of the dual of the Hardy space H 1 with the space of functions of bounded mean oscillation (abbreviated BMO) introduced by John and Nirenberg has led quite naturally to increased study of these functions from the viewpoints both of real variable theory and complex function theory. We take the latter approach and consider the problem of determining which H 1 functions themselves have bounded mean oscillation boundary values. It is not hard to see that, if BMOA denotes the set of functions analytic on the unit disc and having BMO boundary values, then H a c BMOA c (~ H p. The problem p>=l of finding all BMOA functions appears to be quite difficult, but here we are able to exhibit a large subclass of BMOA ",.H ~~ In Section 2 we prove that analytic functions whose boundary values have a certain/~ modulus of continuity are in BMOA. This result allows us to determine a certain linear subspace 5f of BMOA by a condition on asymptotic rate of de- crease of Taylor coefficients: an analytic function f(z)= ~ a,z" is in ~ if and only if ,= o IIJj/=la0J+sup [-1 ~>o I-en=a ]a"rzsinzne] <oo. In particular, any function whose Taylor coefficients are O (~ is in 5r (and hence in BMOA). In Section 3 we show that 5P is a Banach space with respect to the above norm, and that it is nonseparable. Here, also, is given the proof of the surprising fact that for analytic functions the BMO norm is actually dominated by a constant multiple of the ~ norm. Further examples of BMOA functions, namely the loga- rithms of certain schlicht functions, are presented in Section 4, and the paper concludes with several particular examples as well as some open questions. Before proceeding, it may be useful to recall a few fundamental definitions and establish some notation. We are concerned with functions defined on the unit disc