A separation of some Seiffert-type means by power means

Abstract

Consider the identric mean \(\mathcal{I}\), the logarithmic mean \(\mathcal{L,}\) two trigonometric means defined by H. J. Seiffert and denoted by \(\mathcal{P}\) and \(\mathcal{T,}\) and the hyperbolic mean \(\mathcal{M}\) defined by E. Neuman and J. Sándor. There are a number of known inequalities between these means and some power means \(\mathcal{A}_{p}.\) We add to these inequalities some new results obtaining the following chain of inequalities\[\mathcal{A}_{0}<\mathcal{L}<\mathcal{A}_{1/3}<\mathcal{P<A}_{2/3}<\mathcal{I}<\mathcal{A}_{3/3}<\mathcal{M}<\mathcal{A}_{4/3}<\mathcal{T}<\mathcal{A}_{5/3}.\]

Keywords

• Seiffert type means
• power means
• logarithmic mean
• identric mean
• inequalities of means