New Perspectives on the Theory of Inequalities for Integral and Sum

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
			Mean Value Theorems
			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