Ostrowski Type Research Articles

Novel Ostrowski–Type Inequalities for Generalized Fractional Integrals and Diverse Function Classes

Novel Ostrowski–Type Inequalities for Generalized Fractional Integrals and Diverse Function Classes

Generalized discrete Grüss and related results with applications

Abstract Grüss inequality is subject of interest for many authors due to its effectiveness in predicting bounds in several quadrature problems. In the present article, we give weighted treatment of the discrete Čebyšev and Grüss type inequalities pertaining two n-tuples of real numbers in which the bounding constants are mobilised with bounding sequences of real numbers. As an application estimations of discrete Ostrowski type inequalities are provided. Finally, by practicing obtained results along with Jensen’s difference, a wide range of estimations are formalised by considering Jensen-Grüss differences.

Cauchy-Schwarz inequality for shifted quantum integral operator

We show that Cauchy-Schwarz inequality for shifted quantum integral operator does not hold in general and we prove the conditions under which it is valid. We apply it to Ostrowski type inequalities for shifted quantum integral operator.

Better Approximations for Quasi-Convex Functions

In this paper, by using Hölder-İşcan, Hölder integral inequality and a general identity for differentiable functions we can get new estimates on generalization of Hadamard, Ostrowski and Simpson type integral inequalities for functions whose derivatives in absolute value at certain power are quasi-convex functions. It is proved that the result obtained Hölder-İşcan integral inequality is better than the result obtained Hölder inequality. Keywords: Hölder-İşcan scan inequality, Hermite-Hadamard inequality, Simpson and Ostrowski type inequality, midpoint and trapezoid type inequality, quasi-convex functions.

Ostrowski Type Inequalities Via ψ − (α, β, γ, δ)−Convex Function

In this paper, we are introducing very first time the class of ψ − (α, β, γ, δ)−convex function in mixed kind, which is the generalization of many classes of convex functions. We would like to state well-known Ostrowski inequality via Montgomery identity for ψ−(α, β, γ, δ)−convex function in mixed kind. In addition, we establish some Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are ψ − (α, β, γ, δ)-convex functions in mixed kind by using different techniques including Hölder's inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, some applications in terms of special means would also be given. Keywords: Ostrowski inequality, Montgomery identity, convex functions, special means.

Updated Ostrowski inequalities over a spherical shell

Here we present general multivariate mixed Ostrowski type inequalities over spherical shells and balls. We cover the radial and not necessarily radial cases. The proofs derive by the use of some estimates coming out of some new trigonometric and hyperbolic Taylor’s formulae ([2]) and reducing the multivariate problem to a univariate one via general polar coordinates.

Integral inequalities of Ostrowski type for two kinds of s-logarithmically convex functions

Abstract In the paper, the authors establish several integral inequalities of the Ostrowski type for s-logarithmically convex functions. These integral inequalities modify the conditions and correct errors in two main theorems of the paper [A. O. Akdemir and M. Tunç, Ostrowski type inequalities for s-logarithmically convex functions in the second sense with applications, Georgian Math. J. 22 (2015), no. 1, 1–7].

Reverse Ostrowski's type weighted inequalities for convex functions on linear spaces with applications

Reverse Ostrowski's type weighted inequalities for convex functions on linear spaces with applications

Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes

<abstract><p>Mathematical inequalities supporting interval-valued stochastic processes are rarely addressed. Recently, Afzal et al. introduced the notion of $ \mathtt{h} $-Godunova-Levin stochastic processes and developed Hermite-Hadamard and Jensen type inequalities in the setting of interval-valued functions. This note introduces a more generalized class of Godunova-Levin stochastic process that unifies several previously published results through the use of Kulisch-Miranker type order relations that are rarely discussed in relation to stochastic processes. Further, it is the first time that fractional version of Hermite-Hadamard inequality has been developed by using interval-valued stochastic processes in conjunction with a classical operator. Moreover, we give new modified forms for Ostrowski type results and present a new way to treat Jensen type inclusions under interval stochastic processes by using a discrete sequential form. We end with an open problem regarding Milne type results and discuss the importance of different types of order relations related to inequality terms in interval-valued settings.</p></abstract>

K-Fractional Ostrowski Type Inequalities via (s, r)−Convex

We introduce the generalized class named it the class of (s,r)−convex in mixed kind, this class includes s−convex in 1st and 2nd kind, P−convex, quasi convex and the class of ordinary convex. Also, we state the generalization of the classical Ostrowski inequality via k−fractional integrals, which is obtained for functions whose first derivative in absolute values is (s,r)−convex in mixed kind. Moreover, we establish some Ostrowski type inequalities via k−fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are (s,r)−convex in mixed kind by using different techniques including Hölder’s inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, some applications in terms of special means are given.

Some new results of ostrowski type inequalities using 4-step quadratic kernel and their applications

This work is a generalization of Ostrowski type integral inequalities using a special 4-step quadratic kernel. Some new and useful results are obtained. Applications to Quadrature Rules and special Probability distribution are also evaluated.

On a general approach to some problems of approximation of operators

In the article we propose a general scheme for solutions of some approximation problems under a rather general setting. We illustrate the application of the proposed scheme by a series of examples, in particular we show that many results in the area of Ostrowski type inequalities can be obtained by standard arguments.

Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces

In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.

Opial and Ostrowski type inequalities based on trigonometric and hyperbolic type Taylor formulae

Based on [4], we produce a collection of interesting trigonometric and hyperbolic Taylor formulae with integral remainders. Using these we derive Opial and Ostrowski type corresponding inequalities of various kinds and norms.

A Note on the New Ostrowski and Hadamard Type Inequalities via the Hölder–İşcan Inequality

For all convex functions, the Hermite–Hadamard inequality is already well known in convex analysis. In this regard, Hermite–Hadamard and Ostrowski type inequalities were obtained using exponential type convex functions in this work. In addition, new generalizations were found for different values of θ. In conclusion, we believe that our work’s technique will inspire more study in this field.

Korneichuk-Stechkin Lemma, Ostrowski and Landau Inequalities, and Optimal Recovery Problems for L-Space Valued Functions

We prove an analogue of the Korneichuk–Stechkin lemma for functions with values in L-spaces. As applications, we obtain sharp Ostrowski type inequalities and solve problems of optimal recovery of identity and convexifying operators, as well as the problem of integral recovery on the classes of L-space valued functions with given majorant of modulus of continuity. The recovery is done based on n mean values of the functions over intervals. Moreover, on the classes of functions with given majorant of modulus of continuity of their Hukuhara type derivative, we solve the problem of optimal recovery of the function and the Hukuhara type derivative. The recovery is done based on n values of the function. We obtain sharp Landau type inequalities and solve an analogue of the Stechkin problem about approximation of unbounded operators by bounded ones and the problem of optimal recovery of an unbounded operator on a class of elements, known with error. Consideration of L-space valued functions gives a unified approach to solution of the mentioned above extremal problems for the classes of multi- and fuzzy-valued functions, and for the classes of functions with values in Banach spaces, in particular random processes, and many other classes of functions.

Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

\( \newcommand\norm[1]{\left\lVert#1\right\rVert}\newcommand\normx[1]{\left\Vert#1\right\Vert} \)In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping $f$$$\norm{(1\otimes B-A\otimes 1)^{-1}[\operatorname{exp}(1\otimes B)-\operatorname{exp}(A\otimes 1)]- \operatorname{exp}\left(\frac{A\otimes 1+1\otimes B}{2}\right)}$$$$\leqslant \norm{1\otimes B-A\otimes 1}^{2}\frac{\norm{f''}_{I,+\infty}}{24}.$$

Generalized Ostrowski type results for twice differentiable functions on fractal sets

We first obtain a new auxiliary identity by utilizing twice differentiable functions on α−type sets. Afterwards, two generalized Ostrowski type estimations for mappings whose second local fractional derivatives are bounded are derived, and special cases of these comprehensive inequalities are observed. Finally, some related applications such as inequalities including special means and generalized quadrature rules are presented.

Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

K-fractional Ostrowski type inequalities via (s, r)-conex

We are introducing very first time a generalized class named it the class of (s, r)-convex in mixed kind, this class includes s-convex in 1st and 2nd kind, P-convex, quasi-convex, and the class of ordinary convex. Also, we would like to state the generalization of the classical Ostrowski inequality via k-fractional integrals, which is obtained for functions whose first derivative in absolute values is (s, r)-convex in mixed kind. Moreover, we establish some Ostrowski type inequalities via k-fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are (s, r)-convex in mixed kind by using different techniques including Holder's inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, some applications in terms of special means would also be given.