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A Gelfer function f is a holomorphic function in D = {Izl 0 in D. If f is holomorphic in D and if the L2 mean of f' on the circle {IzI = r} is dominated by that of a function of G as r 1 0, then f E BMOA. This has two recent and seemingly different results as corollaries. A core of the proof is the fact that log f E BMOA if f E G. Besides the properties obtained concerning f E G itself, we shall investigate some families of functions where the roles played by P in Univalent Function Theory are replaced by those of G. Some exact estimates are obtained.

Gel′fer functions, integral means, bounded mean oscillation, and univalency