A separation of some Seiffert-type means by power means

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}.\

On Seiffert-like means

We investigate the representation of homogeneous, symmetric means in the form M(x,y)=\frac{x-y}{2f((x-y)/(x+y))}. This allows for a new approach to comparing means. As an example, we provide optimal estimate of the form (1-\mu)min(x,y)+ \mu max(x,y)<= M(x,y)<= (1-\nu)min(x,y)+ \nu max(x,y) and M((x+y)/2-\mu(x-y)/2,(x+y)/2+\mu(x-y)/2)<= N(x,y)<= M((x+y)/2-\nu(x-y)/2,(x+y)/2+\nu(x-y)/2) for some known means

On a Hierarchy of Means

For a class of partially ordered means we introduce a notion of the (nontrivial) cancelling mean. A simple method is given which helps to determine cancelling means for well known classes of Holder and Stolarsky means

Explicit solutions of the invariance equation for means

Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in a recent paper of P. Kahlig and J. Matkowski, and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed

Seiffert means, asymptotic expansions and related inequalites

In this paper we study inequalities of the form (1 - μ) M1(s, t) + μ M3(s, t) ≤ M2(s, t) ≤ (1 - ν) M1(s, t) + ν M3(s, t), which cover some classical bivariate means and Seiffert means. Using techniques of asymptotic expansions detailed analysis was made and the method for obtaining optimal parameters μ and ν was described

On Seiffert-like Means

We investigate the representation of homogeneous, symmetric means in the for