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We find the greatest value p1 = p1(α) and the least value p2 = p2(α) such that the double inequality Jp1 (a,b) < αA(a,b)+(1−α)L(a,b) < Jp2 (a,b) holds for any α ∈ (0,1) and all a,b > 0 with a = b . Here, A(a,b) , L(a,b) and Jp(a,b) denote the arithmetic, logarithmic and p -th one-parameter means of two positive numbers a and b , respectively.

Optimal one-parameter mean bounds for the convex combination of arithmetic and logarithmic means