48 research outputs found
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 some inequality of Hermite-Hadamard type
It is well-known that the left term of the classical Hermite-Hadamard
inequality is closer to the integral mean value than the right one. We show
that in the multivariate case it is not true. Moreover, we introduce some
related inequality comparing the methods of the approximate integration, which
is optimal. We also present its counterpart of Fejer type.Comment: Submitted to Opuscula Mat
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
On Seiffert-like Means
We investigate the representation of homogeneous, symmetric means in the for
On Schur-Nonconvexity of Gini Means
We complete the results of Shi et al. by finding areas where Gini means are
not Schur-concave nor Schur-convex