in this paper, a generalization of trapezoid inequality for functions of two independent variables with bounded variation and some applications are given.

A series of physically motivated operations appearing in the study composite materials are interpreted terms elementary continued fraction transforms matrix valued, rational Stieltjes functions.

The rational meromorphic functions on $\mathbb{C}\backslash\mathbb{R}$ are studied. We consider the some classes of one, as generalized Nevanlinna $\mathbf{N}_{\kappa}$ and Stieltjes $\mathbf{N}_{\kappa}^{k}$ classes. By Euclidean algorithm, we can find indices $\kappa$ $k$, i.e. determine which class function belongs to $\mathbf{N}_{\kappa}^{k}$.

Two new inequalities for Riemann--Stieltjes integral are introduced functions of bounded $p$-variation and H\"{o}lder continuous integrators.