In this paper, we answer the question: for any q > 0 with q ≠ 1, what are the greatest value p1= p1(q) and the least value p2= p2(q) such that the double inequality Lp1(a, b) < [L(aq, bq)]1/q < Lp2(a, b) holds for all a, b > 0 with a ≠ b? Here L(a, b) and Lp(a, b) are the logarithmic and pth generalized logarithmic means of a and b, respectively.
In this paper, we answer the question: What are the greatest value p = p(α) and least value q = q(α)such that the double inequality Jp(a,b) < αA(a,b) + (1 - α)G(a,b) < Jq(a,b) holds for any α ∈ (0,1) and all a,b > 0 with a ≠ b? Here, A(a,b) = (a+b):2, G(a,b) = √ab and Jp (a,b) denote the arithmetic, geometric and p-th one-parameter means of a and b, respectively.
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