Generalizations of Steffensen's inequality via Fink's identity and related results II

We use Fink's identity to obtain new identities related to generalizations of Steffensen's inequality. Ostrowski-type inequalities related to these generalizations are also given. Using inequalities for the Cebysev functional we obtain bounds for these identities. Further, we use these identities to obtain new generalizations of Steffensen's inequality for n-convex functions. Finally, we use the segeneralizations to construct a linear functional that generates exponentially convex functions.We use Fink’s identity to obtain new identities related to generalizations of Steffensen’s inequality. Ostrowski-type inequalities related to these generalizations are also given. Using inequalities for the Cebysev functional we obtain bounds for these identities. Further, we use these identities to obtain new generalizations of Steffensen’s inequality for n-convex functions. Finally, we use these generalizations to construct a linear functional that aenerates exvonentiallv convex functions

New bounds for the Čebyšev functional

AbstractIn this paper some new inequalities for the Čebyšev functional are presented. They have applications in a variety of branches of applied mathematics

SOME BOUNDS FOR THE COMPLEX µCEBYEV FUNCTIONAL OF ABSOLUTELY CONTINUOUS FUNCTIONS

In this paper we provide several bounds for the modulus of the \textit{%complex \v{C}eby\v{s}ev functional}%\begin{equation*}C\left( f,g\right) :=\frac{1}{b-a}\int_{a}^{b}f\left( t\right) g\left(t\right) dt-\frac{1}{b-a}\int_{a}^{b}f\left( t\right) dt\int_{a}^{b}g\left(t\right) dt\end{equation*}%under various assumptions for the integrable functions $f,$ g:\left[ a,b%\right] \rightarrow \mathbb{C}. We show amongst others that, if $f$ and $g$are absolutely continuous on $\left[ a,b\right]$ with $f^{\prime }\in L_{p}$ $g^{\prime }\in L_{q}\left[ a,b\right] ,$ $p,$ $q>1$and $\frac{1}{p}+\frac{1}{q}=1$, then%\begin{equation*}\max \left\{ \left\vert C\left( f,g\right) \right\vert ,\left\vert C\left(\left\vert f\right\vert ,g\right) \right\vert ,\left\vert C\left(f,\left\vert g\right\vert \right) \right\vert ,\left\vert C\left( \left\vertf\right\vert ,\left\vert g\right\vert \right) \right\vert \right\}\end{equation*}%\begin{equation*}\leq \left[ C\left( \ell ,F_{\left\vert f^{\prime }\right\vert ^{p}}\right) %\right] ^{1/p}\left[ C\left( \ell ,F_{\left\vert g^{\prime }\right\vert^{q}}\right) \right] ^{1/q},\end{equation*}%where $F_{\left\vert h\right\vert }:\left[ a,b\right] \rightarrow \mathbb{[}$ is defined by $F_{\left\vert h\right\vert }\left( t\right):=\int_{a}^{t}$.$\left\vert h\left( t\right) \right\vert dt$ and $\ell :$ $\ell \left( t\right) =t$is the identity function on the interval $\left[ a,b\right] .$ Applicationsfor the trapezoid inequality are also provided

Bivariate Segment Approximation and Splines

The problem to determine partitions of a given rectangle which are optimal for segment approximation (e.g. by bivariate piecewise polynomials) is investigated. We give criteria for optimal partitions and develop algorithms for computing optimal partitions of certain types. It is shown that there is a surprising relationship between various types of optimal partitions. In this way, we obtain good partitions for interpolation by tensor product spline spaces. Our numerical examples show that the methods work efficiently

Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation

In this paper, we provide several bounds for the modulus of the complex Čebyšev functional. Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided