Some Mean-Value Theorems of the Cauchy Type

2000 Mathematics Subject Classification: Primary 26A24, 26D15; Secondary 41A05Some mean-value theorems of the Cauchy type, which are connected with Jensen’s inequality, are given in [2] in discrete form and in [5] in integral form. Several further generalizations and applications of these results are presented here

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

On generalizations of Ostrowski inequality via Euler harmonic identities

Copyright © 2002 L. J. Dedić et al. This work is licensed under a Creative Commons License.Some generalizations of Ostrowski inequality are given, by using some Euler identities involving harmonic sequences of polynomials.L. J. Dedić, M. Matić, J. Pečarić, and A. Vukeli

On Bicheng-Debnath's generalizations of Hardy's integral inequality

We consider Hardy's integral inequality and we obtain some new generalizations of Bicheng-Debnath's recent results. We derive two distinguished classes of inequalities covering all admissible choices of parameter k from Hardy's original relation. Moreover, we prove the constant factors involved in the right-hand sides of some particular inequalities from both classes to be the best possible, that is, none of them can be replaced with a smaller constant

Opial Type Integral Inequalities for Widder Derivatives and Linear Differential Operators

In this paper we establish Opial type integral inequalities for Widder derivatives and linear di_erential operator. Also, for applications we construct some related inequalities as special cases

Generalization of Jensen's and Jensen-Steffensen's inequalities and their converses by Lidstone's polynomial and majorization theorem

In this paper, using majorization theorems and Lidstone's interpolating polynomials we obtain results concerning Jensen's and Jensen-Steffensen's inequalities and their converses in both the integral and the discrete case. We give bounds for identities related to these inequalities by using Chebyshev functionals. We also give Grüss type inequalities and Ostrowsky type inequalities for these functionals. Also we use these generalizations to construct a linear functionals and we present mean value theorems and n-exponential convexity which leads to exponential convexity and then log-convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give Stolarsky type means with their monotonicity

New refinements of Hölder and Minkowski inequalities with weights

In this paper, we present on new refinements of the discrete Jensen’s inequality given in [3] and [4]. Our results are more general than the refinement results given in [5]. Also the parameter dependent results correspond to some new refinements of H¨older’s and Minkowski’s inequalities