ISBN 978-93-90768-76-9 (Print)
ISBN 978-93-90768-80-6 (eBook)
DOI: 10.9734/bpi/mono/978-93-90768-76-9

The branch of mathematical inequalities has continued to extend rapidly since the beginning of 20th  century. Jensen, Cauchy, Schwartz, Hölder, Hadamard, Minkowski and Hardy of that era are mathematicians known for their work in inequalities. Inequalities are essential to study Mathematics and many related fields. Several applications of inequalities have been established in the field of calculus, differential and integral equations, optimization theory, control theory, game theory, spectral theory, functional analysis, harmonic analysis, probability theory, interpolation theory, physics, geometry and economics etc. A number of researchers use integral inequalities in studying existence, uniqueness, boundedness, stability and asymptotic behavior of solutions of ordinary and partial differential equations. There are numerous known inequalities and the list is ongoing. The database of MathSciNet contains over 23,000 references of inequalities and their applications. Now we discuss topics chapter-wise as follows:

In the first chapter, we focus the importance of mathematical inequalities due to the huge number of applications of inequalities in various fields of mathematics and further domains of physical sciences, social sciences and management sciences etc., it also describes real life examples, history, literature, relationship among chapters including objectives of the book.

Second chapter collects and presents several types of basic definitions, preliminary results and notations related to convex functions, generalized convex functions and convex functions of different types. This chapter divides into three sections, the first section is for convex functions that provide the historical background, applications and definitions of convex functions. The second section is for generalized convex functions and other different type of convex functions. This chapter covers all the related material which help in sequential manner in every chapter, without these impossible to start and complete of this book.

The third chapter deals with functions with non-decreasing increments (FWNDI) and related results. The functions with non-decreasing increments, was introduced by Brunk in 1964. We extend this for mth order by using finite difference operator with equally spaced interval. With the help of this special definition of function with non-decreasing increments, we get relationship among functions with non-decreasing increments and 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 as different cases of \(\Delta_{h}^{m}f\). We also discuss some examples in each above stated relation. Generalizations of the Levinson’s-type inequality and Jensen-Mercer’s-type inequality by using Jensen-Boas inequality for function with non-decreasing increments of third order are also deduced.

In the fourth chapter, we would give few definitions and preliminaries before starting of main sections, in the first section we obtain discrete identity for sequence \(\displaystyle \sum_{j=1}^{M}\sum_{l=1}^{N}p_{jl}a_{jl}\) for two dimension in which involves higher order \(\nabla\) divided difference and also get the similar result as given above identity for two sequences \(a_{j}\) and \(b_{l}\). Similarly in the second section, we get discrete identity for function \(\displaystyle \sum_{j=1}^{M}\sum_{l=1}^{N}p_{jl}f(y_{jt}z_{l})\) for two independent variables in the interval  \(J_{1}\times L_{1}\subset \mathbb{R}\times \mathbb{R}\) in which involves higher order \(\nabla\) divided difference and provide result for function \(\displaystyle \sum_{j=1}^{M}\sum_{l=1}^{M}p_{jl}f(y_{jt}z_{l})\) for two independent variables in the interval \(J_{1}^{2}\subset \mathbb{R}\times \mathbb{R}\). Moreover, we discuss the similar result as given above discrete identity of function for two \(f\) and \(g\) functions and also get the result about the necessary and sufficient conditions of result of discrete identity of above function for Popoviciu type characterisation of positivity of sums for discrete function \(\displaystyle \sum_{j=1}^{M}\sum_{l=1}^{N}p_{jl}f(y_{jt}z_{l})\geq 0\), by using higher order \(\nabla\)-convex functions.

Fifth chapter starts by recalling of higher order completely monotonic function of one and two variables then we use variety of classes of completely monotonic functions and make examples and applications of completely monotonic functions. Our main work to get general integral identity for two independent variables in the interval \(J\times L\)  by three several methods, first method is: interchanging technique, second: Two-dimensional Induction and third: Taylor expansion for higher order differentiable function of two independent variables of Popoviciu type and give result for integral case for two independent variables in the interval \(J^{2}\)  and also obtain the similar result as given above general integral identity for \(f\) and \(g\) functions. After that we get necessary and sufficient conditions by results of integral identity using linear \(\Lambda(f)\) for higher order \(\nabla\) -convex function for two variables and find positivity of result of integral identity using linear functional \(\Lambda(f)\) for higher order completely monotonic function for two independent variables. We also discuss about some generalized Lagrange and Cauchy-type’s mean value theorems for \(\nabla\)-convex functions of higher order for two independent variables. Further, we discuss nonnegative functional on exponentially convex functions of special type and give some properties and we present examples and applications of completely monotonic, exponentially convex functions by using various classes of functions.

Sixth chapter consists on three sections, in the first section by applying the results of chapter 4, we provide the generalization of discrete \(\check{C}\)eby\(\check{s}\)ev identity for function in the interval \(J^{2}\) for \(\nabla\) divided difference of higher order with two independent variables and also discuss similar result as given above discrete \(\check{C}\)eby\(\check{s}\)ev identity of function for sequence of higher order \(\nabla\) divided difference for two dimension if apply substitution \(y_{j}=j_{t}z_{l}=l\) and \(f(y_{jt}z_{l})=f(j_{t}l)=a_{jl}\) in the above result of discrete \(\check{C}\)eby\(\check{s}\)ev identity of function and find results of discrete inequality of \(\check{C}\)eby\(\check{s}\)ev using \(\nabla\)-convex function of higher order with two variables in the interval \(J^{2}\). In the second and third sections, by applying the results of chapter 5, we provide the generalization of integral \(\check{C}\)eby\(\check{s}\)ev and Ky Fan’s identities for differentiable function of higher order for two independent variables and also find results of integral inequalities of \(\check{C}\)eby\(\check{s}\)ev and Ky Fan using \(\nabla\)-convex function of higher order with two variables.

In the last chapter, we obtain the more generalized Montgomery’s identities for differentiable function of higher order with two independent variables. Generalized Montgomery’s identities support us for contribution in the generalized Ostrowski- and Grüss-type inequalities for double weighted integrals. We also provide more generalized Ostrowski and Grüss type inequalities for differentiable functions of higher order with two variables as compare to the existing results related to the subject.

 

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Functions with Non-decreasing Increments and Popoviciu Type Identities and Inequalities for Sums and Integrals

Faraz Mehmood

Functions with Non-decreasing Increments and Popoviciu Type Identities and Inequalities for Sums and Integrals, 12 February 2021, Page 1-107
https://doi.org/10.9734/bpi/mono/978-93-90768-76-9

In past few decades, the topic of functions with non-decreasing increments has gained popularity in several branches of Mathematics and this topic got attention of many mathematicians due to its importance. We would like to establish the relationship(s) among functions with non-decreasing increments and 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 as different cases of \(\Delta_{h}^{m}f\). We also consider function with nondecreasing increments of third order and obtain the generalizations of the Levinson’s-type inequality and Jensen-Mercer’s-type inequality by using Jensen-Boas inequality. We will deduce some general identities of Popoviciu type for discrete case for sums for function and sequence in two dimensions using higher order \(\nabla\) divided difference, positivity of these expressions are characterized for higher order \(\nabla\)-convex functions. We will also obtain some general identities of Popoviciu type for integral \(\displaystyle \iint P(y_{t}z)f(y_{t}z)dy~dz\) of higher order differentiable function and positivity of these expressions are characterized for higher order \(\nabla\)-convex and completely monotonic functions. We would discuss some applications in terms of generalized Cauchy-type means and exponential convexity as well. We would get the generalization of discrete identity and inequality of \(\check{C}\)eby\(\check{s}\)ev-type and discuss generalization of integral identities and inequalities of \(\check{C}\)eby\(\check{s}\)ev-type and Ky Fan-type for higher order \(\nabla\)-convex functions with two variables. Moreover, we will obtain double weighted integrals Montgomery’s identities with two variables and using obtained identities we deduce generalization of Ostrowski-type and Grüss-type inequalities for higher order differentiable functions.