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Optimal evaluations of some Seiffert-type means by power means. (English) Zbl 06447280
Summary: Let us consider the second trigonometric mean \(\mathcal{T}\) defined by Seiffert and the hyperbolic mean \(\mathcal{M}\) defined by Neuman and Sándor. There are some known inequalities between these means and some power means \(\mathcal{A}_p\). We prove that the evaluations \[ \mathcal{A}_{\ln 2/\ln(\ln(3+2\sqrt{2}))}<\mathcal{M}<\mathcal{A}_{4/3} \text{and} \mathcal{A}_{\ln 2/\ln (\pi/2)}<\mathcal{T}<\mathcal{A}_{5/3} \] are optimal. In some details of the proofs we have used the computer algebra Mathematica.

MSC:
26E60 Means
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