Gruss Type Inequalities Research Articles

A Survey on Gruss-Type Inequalities by Means of Different Generalized Fractional Integrals

In this survey paper, we review a generalization of Gruss - type inequality by means of two different operators, one through K - fractional integral and other through Katugampola fractional integral. We state some related theorems, corollaries and their proofs. We study how results are interrelated?

Čebyšev–Grüss inequalities for $$\alpha $$-partial derivatives

The main purpose of the present paper is to introduce a new concept by calculating first order variation of multivariate function, and call it $$\alpha $$ -multivariate partial derivatives, which is a generalization of the well-known $$\alpha $$ -conformable derivatives. For the $$\alpha $$ -multivariate partial derivatives, we prove Lagrange’s, Rolle’s and Cauchy’s mean value theorems in n-dimensional space. As applications, we establish some new Cebysev–Gruss type inequalities for the $$\alpha $$ -multivariate partial derivatives, which in special cases yield Milovanovic, Pecaric and Fink’s, and Pachpatte’s inequalities.

Generalization of weighted Ostrowski-Gruss type inequality by using Korkine's identity

We obtained generalized weighted Ostrowski Gruss type inequality with parameters, for differentiable functions by using the weighted Korkine’s identity and then we applied these obtain inequalities to probability density functions. Also, we discussed some applications of numerical quadrature rules.

Some Grüss-type inequalities using generalized Katugampola fractional integral

The main objective of this paper is to obtain a generalization of some Gruss-type inequalities in case of functional bounds by using a generalized Katugampola fractional integral. We obtained new Gruss type inequalitys with functional bounds via the generalized fractional integral operators having same and different parameters. Results obtained are more generalized in nature.

Berezin Number, Grüss-Type Inequalities and Their Applications

In this paper, we study the Berezin number inequalities by using the transform $$C_{\alpha ,\beta }\left( A\right) $$ on reproducing kernel Hilbert spaces (RKHS). Moreover, we give Gruss-type inequalities for selfadjoint operators in RKHS.

Generalizations of Sherman’s Inequality Via Fink’s Identity and Green’s Function

New generalizations of Sherman’s inequality for n-convex functions are obtained with the help of Fink’s identity and Green’s function. By using inequalities for the Chebyshev functional, we establish some new Ostrowski- and Gruss-type inequalities related to these generalizations.

Grüss type and related integral inequalities in probability spaces

In this paper we study Gruss type inequalities for real and complex valued functions in probability spaces. Some earlier Gruss type inequalities are extended and refined. Our approach leads to new integral inequalities which are interesting in their own right. As an application, we give a Gruss type inequality for normal operators in a Hilbert space. Similar results are obtained only for self-adjoint operators in earlier papers.

Chebyshev-Grüss Type Inequalities for Hadamard \(k\)-Fractional Integrals

Integral inequalities are taken up to be important as they are useful in the study of different classes of differential and integral equations. During the past several years, many researchers have obtained various fractional integral inequalities comprising the different fractional differential and integral operators. A considerable work is done associated with classical and variants of Gruss type inequality, which actually connects the integral of the product of two functions with the product of their integrals. In this paper, we present the Chebyshev-Gruss type inequalities for Hadamard fractional integrals in the framework of parameter \(k > 0\).

CERTAIN INEQUALITIES INVOLVING THE (k, \rho)-FRACTIONAL INTEGRAL OPERATOR

Since Gruss presented an interesting integral inequality in [9], its various generalizations and variants, which are called Gruss type inequalities, have been investigated. Very recently, certain Gruss type inequalities involving ? ? s , k-fractional integral operator have been established. Motivated by the above mentioned works, we aim to present a Young's type inequality and a weighted AM-GM type inequality involving the ? ? ? , k-fractional integral operator. Some special cases of the main results here are also considered.

A Perturbed Version of General Weighted Ostrowski Type Inequality and Applications

The main purpose of this paper is to derive some new generalizations of weighted Ostrowski type inequalities. The new established inequalities are carried out for a twice differentiable mapping in different L p spaces. Applications throught considering Gruss type inequality and numerical integration are also provided.

Properties of discrete non-multiplicative operators

The paper is focused on general sequences of discrete linear operators, say $$(L_n)_{n\ge 1}$$ . The special case of positive operators is also to our attention. Concerning the quantity $${\Delta } (L_n,f,g):=L_n(fg)-(L_n f)(L_n g), f$$ and g belonging to some certain spaces, we propose different estimates. Firstly, we study its asymptotic behavior in Voronovskaja’s sense. Examples are presented. Secondly, we prove an extension of Chebyshev–Gruss type inequality for the above quantity. Special cases are investigated separately. Finally we establish sufficient conditions that ensure statistical convergence of the sequence $${\Delta }(L_n,f,g)$$ .

Generalization of Jensen-type linear functional on time scales via lidstone polynomial

In this paper, we prove Jensen’s inequality for diamond integrals on time scales and generalize this inequality for 2n-convex functions via Lidstone’s polynomials. Bounds for Chebyshev functional, Gruss-type inequality and Ostrowaski-type inequality related to Jensen’s type functional are also given in the paper. Moreover, our results are valid if we replace diamond integrals with delta integrals, nabla integrals or α-diamond integrals on time scales as well (being special cases).

(p, q)-Variant of Szász–Beta operators

In the present paper, we introduce certain (p, q) analogue of the Szasz–Beta operators using (p, q)-variant of Beta function of second kind. We present direct theorem in weighted spaces in terms of suitable weighted modulus of smoothness and Gruss-type inequality for mentioned operator. A Voronovskaya type theorem is also given.

MAJORIZATION TYPE INEQUALITIES VIA GREEN FUNCTION AND HERMITE'S POLYNOMIAL

The Hermite polynomial and Green function are used to constructthe identities related to majorization type inequalities for convex function. By using Cebysev functional the bounds for the new identities are found to develop the Gruss and Ostrowski type inequalities. Further more exponential convexity together with Cauchy means is presented for linear functionals associated with the obtained inequalities.DOI : http://dx.doi.org/10.22342/jims.22.1.251.1-25

GR\"{U}SS TYPE INEQUALITIES INVOLVING THE GENERALIZED GAUSS HYPERGEOMETRIC FUNCTIONS

In this paper, we establish certain generalized Gruss type inequality for general- ized fractional integral inequalities involving the generalized Gauss hypergeometric function. Moreover, we also consider their relevances for other related known results.

New Results on Rational Approximation

First asymptotic relations of Voronovskaya-type for rational operators of Shepard-type are shown. A positive answer in some senses to a problem on the pointwise approximation power of linear operators on equidistant nodes posed by Gavrea, Gonska and Kacso is given. Direct and converse results, computational aspects and Gruss-type inequalities are also proved. Finally an application to images compression is discussed, showing the outperformance of such operators in some senses.

A generalized form of Grüss type inequality and other integral inequalities

The aim of this presentation is to show several integral inequalities. Among these inequalities we have the inequality , where denotes the h-variance of f, which is a bounded function defined on with , and , are two constants. This inequality is important because it proves a generalized form of the Gruss type inequality. This improvement is given by the inequality Using the integral arithmetic mean and h-integral arithmetic mean for a Riemann-integrable function f we can also rewrite several integral inequalities. In addition, we will give a generalization of inequality of Gruss for normalized isotonic linear functionals. MSC:26D15, 26D10.

IDENTITIES OF SONIN-TYPE AND APPLICATIONS

Some weighted generalizations of recently obtained identities for perturbed Cebysev functionals are given by using Sonin's identity. They are used to obtain the new bounds for perturbed weighted Cebysev functionals. In a special cases they are reduced to Gruss type inequalities.

Reverse Cauchy-Schwarz type inequalities in pre-inner product C*-modules

In the framework of a pre-inner product C*-module over a unital C*-algebra, we show several reverse Cauchy-Schwarz type inequalities of additive and multiplicative types, by using some ideas in N. Elezovic et al. [Math. Inequal. Appl., 8 (2005), no.2, 223-231]. We apply our results to give Klamkin-Mclenaghan, Shisha-Mond and Cassels type inequalities. We also present a Gruss type inequality.

Some bounds for integrals with refinements of the Grüss inequality

Some integral bounds are obtained which provide refinements and extensions of the Gruss type inequalities. We also get an alternative formulation of the Gruss inequality.