New perspectives on the theory of inequalities for integral and sum / Nazia Irshad [and three others].

Format:

Book

Author/Creator:

Irshad, Nazia, author.

Language:

English

Subjects (All):

Inequalities (Mathematics).

Inequalities (Mathematics)--Data processing.

Physical Description:

1 online resource (319 pages)

Place of Publication:

Cham, Switzerland : Springer International Publishing, [2022]

Summary:

This book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green's functions, are presented. The second chapter is dedicated to Ostrowski's inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order.

Contents:

Intro

Preface

Contents

Notations and Terminologies

1 Linear Inequalities via Interpolation Polynomials and Green Functions

1.1 Linear Inequalities and the Extension of Montgomery Identity with New Green Functions

1.1.1 Results Obtained by the Extension of Montgomery Identity and New Green Functions

1.1.2 Inequalities for n-Convex Functions at a Point

1.1.3 Bounds for Remainders and Functionals

1.1.4 Mean Value Theorems

1.2 Linear Inequalities and the Taylor Formula with New Green Functions

1.2.1 Results Obtained by the Taylor Formula and New Green Functions

1.2.2 Inequalities for n-Convex Functions at a Point

1.2.3 Bounds for Remainders and Functionals

1.2.4 Mean Value Theorems and Exponential Convexity

Mean Value Theorems

Logarithmically Convex Functions

n-Exponentially Convex Functions

1.2.5 Examples with Applications

1.3 Linear Inequalities and Hermite Interpolation with New Green Functions

1.3.1 Results Obtained by the Hermite Interpolation Polynomial and Green Functions

1.3.2 Inequalities for n-Convex Functions at a Point

1.3.3 Bounds for Remainders and Functionals

1.4 Linear Inequalities and the Fink Identity with New Green Functions

1.4.1 Results Obtained by the Fink identity and New Green functions

1.4.2 Inequalities for n-Convex Functions at a Point

1.4.3 Bounds for Remainders and Functionals

1.5 Linear Inequalities and the Abel-Gontscharoff's Interpolation Polynomial

1.5.1 Results Obtained by the Abel-Gontscharoff's Interpolation

1.5.2 Results Obtained by the Abel-Gontscharoff's Interpolation Polynomial and Green Functions

1.5.3 Inequalities for n-Convex Functions at a Point

1.5.4 Bounds for Remainders and Functionals

2 Ostrowski Inequality

2.1 Generalized Ostrowski Type Inequalities with Parameter.

2.1.1 Ostrowski Type Inequality for Bounded Differentiable Functions

2.1.2 Ostrowski Type Inequalities for Bounded Below Only and Bounded Above Only Differentiable Functions

2.1.3 Applications to Numerical Integration

2.2 Generalized Ostrowski Type Inequalities for Functions of Lp Spaces and Bounded Variation

2.2.1 Ostrowski Type Inequality for Functions of Lp Spaces

2.2.2 Ostrowski Type Inequality for Functions of Bounded Variation

2.2.3 Applications to Numerical Integration

2.3 Generalized Weighted Ostrowski Type Inequality with Parameter

2.3.1 Weighted Ostrowski Type Inequality with Parameter

2.3.2 Applications to Numerical Integration

2.4 Generalized Weighted Ostrowski-Grüss Type Inequality with Parameter

2.4.1 Weighted Ostrowski-Grüss Type Inequality with Parameter by Using Korkine's Identity

2.4.2 Applications to Probability Theory

2.4.3 Applications to Numerical Integration

2.5 Generalized Fractional Ostrwoski Type Inequality with Parameter

2.5.1 Fractional Ostrowski Type Inequality Involving Parameter

2.6 Generalized Inequalities for Functions of Lp Spaces via Montgomery Identity with Parameters

2.6.1 Montgomery Identity for Functions of Two Variables involving Parameters

2.6.2 Generalized Ostrowski Type Inequality

2.6.3 Generalized Grüss Type Inequalities

3 Functions with Nondecreasing Increments

3.1 Functions with Nondecreasing Increments in Real Life

3.2 Relationship Among Functions with Nondecreasing Increments and Many Others

3.3 Functions with Nondecreasing Increments of Order 3

3.3.1 On Levinson Type Inequalities

3.3.2 On Jensen-Mercer Type Inequalities

4 Popoviciu and Čebyšev-Popoviciu Type Identities and Inequalities

4.1 Linear Inequalities for Higher Order -Convex and Completely Monotonic Functions.

4.1.1 Discrete Identity for Two Dimensional Sequences

4.1.2 Discrete Identity and Inequality for Functions of Two Variables

4.1.3 Integral Identity and Inequality for Functions of One Variable

4.1.4 Integral Identity and Inequality for Functions of Two Variables

4.1.5 Mean Value Theorems and Exponential Convexity

Exponential Convexity

Examples of Completely Monotonic and Exponentially Convex Functions

4.2 Generalized Čebyšev and Ky Fan Identities and Inequalities for -Convex Functions

4.2.1 Generalized Discrete Čebyšev Identity and Inequality

4.2.2 Generalized Integral Čebyšev Identity and Inequality

4.2.3 Generalized Integral Ky Fan Identity and Inequality

4.3 Weighted Montgomery Identities for Higher Order Differentiable Function of Two Variables and Related Inequalities

4.3.1 Montgomery Identities for Double Weighted Integrals of Higher Order Differentiable Functions

Special Cases

4.3.2 Ostrowski Type Inequalities for Double Weighted Integrals of Higher Order Differentiable Functions

4.3.3 Grüss Type Inequalities for Double Weighted Integrals of Higher Order Differentiable Functions

Bibliography

Index.

Notes:

Description based on print version record.

Other Format:

Print version: Irshad, Nazia New Perspectives on the Theory of Inequalities for Integral and Sum

ISBN:

3-030-90563-2