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23 papers found.
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Levinson’s type generalization of the Jensen inequality and its converse for real Stieltjes measure

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi’s inequality. The obtained results are then applied for ...

On an upper bound for Sherman’s inequality

Considering a weighted relation of majorization, Sherman obtained a useful generalization of the classical majorization inequality. The aim of this paper is to extend Sherman’s inequality to convex functions of higher order. An upper bound for Sherman’s inequality, as well as for generalized Sherman’s inequality, is given with some applications. As easy consequences, some new ...

Exponential convexity for Jensen’s inequality for norms

In this paper, we investigate n-exponential convexity and log-convexity using the positive functional defined as the difference of the left-hand side and right-hand side of the inequality from (Pečarić and Janić in Facta Univ., Ser. Math. Inform. 3:39-42, 1988 ). We also give mean value theorems of Lagrange and Cauchy types. Finally, we construct means with Stolarsky property. MSC: ...

On Levinson’s operator inequality and its converses

We give new results on Levinson’s operator inequality and its converse for normalized positive linear mappings and some large class of ‘3-convex functions at a point c’. MSC: 47A63, 47B15.

Generalizations of Sherman’s inequality by Lidstone’s interpolating polynomial

In majorization theory, the well-known majorization theorem plays a very important role. A more general result was obtained by Sherman. In this paper, concerning 2n-convex functions, we get generalizations of these results applying Lidstone’s interpolating polynomials and the Čebyšev functional. Using the obtained results, we generate a new family of exponentially convex functions. ...

A monotonic refinement of Levinson’s inequality

In this paper we give a monotonic refinement of the probabilistic version of Levinson’s inequality based on the monotonic refinement of Jensen’s inequality obtained by Cho et al. (Panam. Math. J. 12:43-50, 2002). MSC: 26D15.

Generalizations of Steffensen’s inequality via Taylor’s formula

We generalize Steffensen’s inequality to the class of n-convex functions using Taylor’s formula. Further, we use inequalities for the Čebyšev functional to obtain bounds for identities related to generalizations of Steffensen’s inequality, and we give Ostrowski-type inequalities related to obtained generalizations. Finally, we apply our results to obtain new Stolarsky-type means. ...

Reverse Poincaré-type inequalities for the difference of superharmonic functions

In this paper, we develop the weighted square integral inequalities for the difference of two smooth superharmonic functions. Then we prove the existence and integrability of the Sobolev derivative for superharmonic functions. The inequalities are generalized for the difference of two weak superharmonic functions. We also establish that the superharmonic approximation is indeed the ...

Some Grüss type inequalities and corrected three-point quadrature formulae of Euler type

We obtain some new Grüss type inequalities for the general corrected three-point quadrature formulae of Euler type. As special cases, we derive some new bounds for the corrected Euler Simpson formula, the corrected dual Euler Simpson formula and the corrected Euler Maclaurin formula. Also, applications for the corrected Euler Bullen-Simpson formula are considered. MSC: 26D15, ...

On the refinements of the integral Jensen-Steffensen inequality

In this paper, we present integral versions of some recently proved results which refine the Jensen-Steffensen inequality. We prove the n-exponential convexity and log-convexity of the functions associated with the linear functionals constructed from the refined inequalities and also prove the monotonicity property of the generalized Cauchy means. Finally, we give several examples ...

On the refinements of the Hermite-Hadamard inequality

Abstract In this paper, we present some refinements of the classical Hermite-Hadamard integral inequality for convex functions. Further, we give the concept of n-exponential convexity and log-convexity of the functions associated with the linear functionals defined by these inequalities and prove monotonicity property of the generalized Cauchy means obtained via these functionals. ...

Refined converses of Jensen’s inequality for operators

In this paper converses of a generalized Jensen’s inequality for a continuous field of self-adjoint operators, a unital field of positive linear mappings and real-valued continuous convex functions are studied. New refined converses are presented by using the Mond-Pečarić method improvement. Obtained results are applied to refine selected inequalities with power functions.MSC: ...

On some inequalities for functions with nondecreasing increments of higher order

We investigate a class of functions with nondecreasing increments of higher order. A generalization of Brunk’s theorem is proved for that class of functions. Also, we consider functions with nondecreasing increments of order three, we obtain the Levinson-type inequality, a generalization of Burkill-Mirsky-Pečarić’s results, and a result for the integral mean of a function with ...

Refinement of integral inequalities for monotone functions

In this paper, we give refinements of some inequalities for generalized monotone functions by using log-convexity of some functionals.

Exponential convexity for majorization

In this article, we give more generalized results than in Anwar et al. (2010) and Latif and Pečarić (2010) in new direction by using second-order divided difference. We investigate the exponential convexity and logarithmic convexity for majorization type results by using class of continuous functions in linear functionals. We also construct positive semi-definite matrices for ...

Hardy-Hilbert-Type Inequalities with a Homogeneous Kernel in Discrete Case

The main objective of this paper is a study of some new generalizations of Hilbert's and Hardy-Hilbert's type inequalities. We apply our general results to homogeneous functions. We shall obtain, in a similar way as Yang did in(2009), that the constant factors are the best possible when the parameters satisfy appropriate conditions.

On the refinements of the Jensen-Steffensen inequality

Science, Education and Sports, under the Research Grants 0581170889-1050 (Iva Franjić) and 117-1170889-0888 (Josip Pečarić). Authors’ contributions JP made the main contribution in conceiving the presented

On an Inequality of H. G. Hardy

We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and ...

Exponential convexity of Petrović and related functional

We consider functionals due to the difference in Petrović and related inequalities and prove the log-convexity and exponential convexity of these functionals by using different families of functions. We construct positive semi-definite matrices generated by these functionals and give some related results. At the end, we give some examples.