A Nice Separation of Some Seiffert-Type Means by Power Means

Seiffert has defined two well-known trigonometric means denoted by ð ’« and ð ’¯ . In a similar way it was defined by Carlson the logarithmic mean â„’ as a hyperbolic mean. Neuman and Sándor completed the list of such means by another hyperbolic mean ℳ . There are more known inequalities between the means ð ’« , ð ’¯ , and â„’ and some power means ð ’œ ð ‘ . We add to these inequalities two new results obtaining the following nice chain of inequalities ð ’œ 0 < â„’ < ð ’œ 1 / 2 < ð ’« < ð ’œ 1 < ℳ < ð ’œ 3 / 2 < ð ’¯ < ð ’œ 2 , where the power means are evenly spaced with respect to their order.