2019
Том 71
№ 11

# Dragomir S. S.

Articles: 3
Article (English)

### Jensen – Ostrowski inequalities and integration schemes via the Darboux expansion

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1123-1140

By using the Darboux formula obtained as a generalization of the Taylor formula, we deduce some Jensen – Ostrowski-type inequalities. The applications to quadrature rules and $f$ -divergence measures (specifically, for higher-order $\chi$ -divergence) are also given.

Article (Russian)

### Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex mappings

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1330-1347

We obtain some inequalities of the Hermite – Hadamard type for $K$-bounded norm convex mappings between two normed spaces. The applications for twice differentiable functions in Banach spaces and functions defined by power series in Banach algebras are presented. Some discrete Jensen-type inequalities are also obtained.

Article (English)

### New Inequalities for the $p$-Angular Distance in Normed Spaces with Applications

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 19–31

For nonzero vectors $x$ and $y$ in the normed linear space $(X, ‖ ⋅ ‖)$, we can define the $p$-angular distance by $${\alpha}_p\left[x,y\right]:=\left\Vert {\left\Vert x\right\Vert}^{p-1}x-{\left\Vert y\right\Vert}^{p-1}y\right\Vert .$$ We show (among other results) that, for $p ≥ 2$, $$\begin{array}{l}{\alpha}_p\left[x,y\right]\le p\left\Vert y-x\right\Vert {\displaystyle \underset{0}{\overset{1}{\int }}{\left\Vert \left(1-t\right)x+ty\right\Vert}^{p-1}dt}\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \left[\frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}+{\left\Vert \frac{x+y}{2}\right\Vert}^{p-1}\right]\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}\le p\left\Vert y-x\right\Vert {\left[ \max \left\{\left\Vert x\right\Vert, \left\Vert y\right\Vert \right\}\right]}^{p-1},\hfill \end{array}$$, for any $x, y ∈ X$. This improves a result of Maligranda from [“Simple norm inequalities,” Amer. Math. Month., 113, 256–260 (2006)] who proved the inequality between the first and last terms in the estimation presented above. The applications to functions f defined by power series in estimating a more general “distance” $‖f(‖x‖)x − f(‖y‖)y‖$ for some $x, y ∈ X$ are also presented.