A Gelfer function f f is a holomorphic function in D = { | z | > 1 } D = \{ \left | z \right | > 1\} such that f ( 0 ) = 1 f(0) = 1 and f ( z ) ≠ − f ( w ) f(z) \ne - f(w) for all z z , w w in D D . The family G G of Gelfer functions contains the family P P of holomorphic functions f f in D D with f ( 0 ) = 1 f(0) = 1 and Re f > 0 f > 0 in D D . If f f is holomorphic in D D and if the L 2 {L^2} mean of f ′ f’ on the circle { | z | = r } \{ \left | z \right | = r\} is dominated by that of a function of G G as r → 1 − 0 r \to 1 - 0 , then f ∈ B M O A f \in BMOA . This has two recent and seemingly different results as corollaries. A core of the proof is the fact that log f ∈ B M O A {\operatorname {log}}f \in BMOA if f ∈ G f \in G . Besides the properties obtained concerning f ∈ G f \in G itself, we shall investigate some families of functions where the roles played by P P in Univalent Function Theory are replaced by those of G G . Some exact estimates are obtained.