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Optimal one-parameter mean bounds for the convex combination of arithmetic and geometric means. (English) Zbl 1276.26063
Summary: In this paper, we answer the question: What are the greatest value \(p = p(\alpha)\) and least value \(q = q(\alpha)\) such that the double inequality \[ J_p(a, b) < \alpha A(a, b) + (1 - \alpha)G(a, b) < J_q(a, b) \] holds for any \(\alpha \in (0, 1)\) and all \(a, b > 0\) with \(a \neq b\)? Here, \(A(a, b) = \frac{a + b}{2}\), \(G(a, b) = \sqrt{ab}\) and \(J_p(a, b)\) denote the arithmetic, geometric and \(p\)-th one-parameter means of \(a\) and \(b\), respectively.

MSC:
26E60 Means
26D20 Other analytical inequalities
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