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The invariance equality L◦(M ,N ) = L, where L is the logarithmic mean, and where the unsymmetric compound means M = A ◦ (P1,G) , N = A ◦ (P2,G) are built with the arithmetic A , geometric G , and projective means P1 , P2 , is called “Carlson’s log” and is important in iteration of means. In the present paper we present effective and simple 1-parameter families of unsymmetric means Mt ,Nt : (0,∞) → (0,∞) such that, for all t ∈ (−1,1) , L◦ (Mt ,Nt ) = L and M = M 1 2 , N = N 1 2 . Existence of elementary (simple) symmetric means M and N such that L ◦ (M,N) = L and M = L is posed as an open problem. Mathematics subject classification (2010): Primary: 26A17, 26E60, 39B22.

Logarithmic complementary means and an extension of Carlson's log