INTEGRAL ERROR REPRESENTATION OF HERMITE INTERPOLATING POLYNOMIAL AND RELATED INEQUALITIES FOR QUADRATURE FORMULAE

We consider integral error representation related to the Hermite interpolating polynomial and derive some new estimations of the remainder in quadrature formulae of Hermite type, using Holder’s inequality and some inequalities for the Čebyšev functional. As a special case, generalizations for the zeros of orthogonal polynomials are considered.

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Some New Dynamic Inequalities Involving Monotonic Functions on Time Scales

Journal of Function Spaces ◽

10.1155/2019/7584836 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11

Author(s):

S. H. Saker ◽

E. Awwad ◽

A. Saied

Keyword(s):

Time Scales ◽

Hölder’S Inequality ◽

Chain Rule ◽

Integration By Parts ◽

Hölder's Inequality ◽

Monotonic Functions ◽

A Chain ◽

Special Case

In this paper, we prove some new dynamic inequalities involving C- monotonic functions on time scales. The main results will be proved by employing Hölder’s inequality, integration by parts, and a chain rule on time scales. As a special case when T=R, our results contain the continuous inequalities proved by Heinig, Maligranda, Pečarić, Perić, and Persson and when T=N, the results to the best of the authors’ knowledge are essentially new.

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Some Dynamic Inequalities of Hilbert’s Type

Journal of Function Spaces ◽

10.1155/2020/4976050 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13 ◽

Cited By ~ 4

Author(s):

A. M. Ahmed ◽

Ghada AlNemer ◽

M. Zakarya ◽

H. M. Rezk

Keyword(s):

Time Scales ◽

Hölder’S Inequality ◽

Jensen’S Inequality ◽

Jensen's Inequality ◽

Hölder's Inequality ◽

Discrete Inequalities ◽

Special Case ◽

Hilbert Type

This paper is concerned with deriving some new dynamic Hilbert-type inequalities on time scales. The basic idea in proving the results is using some algebraic inequalities, Hölder’s inequality and Jensen’s inequality, on time scales. As a special case of our results, we will obtain some integrals and their corresponding discrete inequalities of Hilbert’s type.

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Refinements of some Hardy–Littlewood–Pólya type inequalities via Green’s functions and Fink’s identity and related results

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02498-3 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Sadia Khalid ◽

Josip Pečarić

Keyword(s):

Shannon Entropy ◽

Convex Functions ◽

Green's Functions ◽

Green’S Functions ◽

Hölder’S Inequality ◽

Hölder's Inequality ◽

Čebyšev Functional ◽

Grüss Type Inequalities ◽

Fink’S Identity

AbstractIn this paper, first we present some interesting identities associated with Green’s functions and Fink’s identity, and further we present some interesting inequalities for r-convex functions. We also present refinements of some Hardy–Littlewood–Pólya type inequalities and give an application to the Shannon entropy. Furthermore, we use the Čebyšev functional and Grüss type inequalities and present the bounds for the remainder in the obtained identities. Finally, we use the obtained identities together with Hölder’s inequality for integrals and present Ostrowski type inequalities.

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Hilbert-type inequalities for time scale nabla calculus

Advances in Difference Equations ◽

10.1186/s13662-020-03079-w ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

H. M. Rezk ◽

Ghada AlNemer ◽

H. A. Abd El-Hamid ◽

Abdel-Haleem Abdel-Aty ◽

Kottakkaran Sooppy Nisar ◽

...

Keyword(s):

Time Scale ◽

Hölder’S Inequality ◽

Hölder's Inequality ◽

Arbitrary Time ◽

Hilbert Type

Abstract This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.

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