A Gelfer function f f is a holomorphic function in the unit disc D = { z : | z | > 1 } D = \{ z:|z| > 1\} such that f ( 0 ) = 1 f(0) = 1 and f ( z ) + f ( w ) ≠ 0 f(z) + f(w) \ne 0 for all z , w z,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 . Yamashita has recently proved that if f f is a Gelfer function then f ∈ H p , 0 > p > 1 f \in {H^p},0 > p > 1 while log ⁡ f ∈ BMOA \log f \in \operatorname {BMOA} and ‖ log ⁡ f ‖ BMO ⁡ A 2 ≤ π / 2 {\left \| {\log f} \right \|_{\operatorname {BMO}{{\text {A}}_2}}} \leq \pi /\sqrt 2 . In this paper we prove that the function λ ( z ) = ( 1 + z ) / ( 1 − z ) \lambda (z) = (1 + z)/(1 - z) is extremal for a very large class of problems about integral means in the class G G . This result in particular implies that G ⊂ H p , 0 > p > 1 G \subset {H^p},0 > p > 1 , and we use it also to obtain a new proof of a generalization of Yamashita’s estimation of the BMOA norm of log ⁡ f , f ∈ G \log f,f \in G .