We prove sharp inequalities for the two-variable versions of the sum Sigma(n)(k=1) sin(kx)/k.

This paper gives an overview about particular quasi-entropies, generalized quantum covariances, quantum Fisher informations, skew-informations and their relations. The point is the dependence on operator monotone functions. It is proven that a skew-information is the Hessian of a quasi-entropy. The skewinformation and some inequalities are extended to a von Neumann algebra setting. 2000Mathematics Subject Classification. Primary 15A63, 94A17; Secondary 47N50, 46L60.

In 1954 it was proved if f is infinitely differentiable in the interval I and some derivative (of order depending on x) vanishes at each x, then f is a polynomial. Later it was generalized for multi-variable case. In this paper we give an extension for distributions.

This paper gives an overview about particular quasi-entropies, generalized quantum covariances, quantum Fisher informations, skew informations and their relations. The point is the dependence on operator monotone functions. It is proven that a skew information is the Hessian of a quasi-entropy. The skew information and some inequalities are extended to a von Neumann algebra setting.

In this paper we show that a class of functions is matrix (operator) monotone and give some consequences of it.

In this paper we deal with weighted interpolation on the real line. We show that the Hermite-Fejér interpolation process is not optimal, but the Grünwald operator is the ideal case.

The purpose of the paper is to investigate weighted l(p) convergence of Lagrange interpolation taken at the zeros of Hermite polynomials. It is shown that if a continuous function satisfies some conditions, then the corresponding Lagrange interpolation process converges in every L(p) (1 < p < infinity) and the rate of convergence is estimated.

In this paper we prove some known and new inequalities using an elementary inequality and some basic facts from differential calculus.