A nice separation of some Seiffert-type means by power means (Q456501)

Summary: Seiffert has defined two well-known trigonometric means denoted by \(\mathcal P\) and \(\mathcal T\). In a similar way it was defined by Carlson the logarithmic mean \(\mathcal L\) as a hyperbolic mean. Neuman and Sándor completed the list of such means by another hyperbolic mean \(\mathcal M\). There are more known inequalities between the means \(\mathcal{P, T}\), and \(\mathcal L\) and some power means \(\mathcal A_p\). We add to these inequalities two new results obtaining the following nice chain of inequalities \(\mathcal A_0 < \mathcal L < \mathcal A_{1/2} < \mathcal P < \mathcal A_1 < \mathcal M < \mathcal A_{3/2} < \mathcal T < \mathcal A_2\), where the power means are evenly spaced with respect to their order.