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Title of Thesis

Exponential Convexity And Cauchy Means Introduced By Inequalities For Monotone And Related Functions

Author(s)

Atiq Ur Rehman

Institute/University/Department Details
Abdus Salam School of Mathematical Sciences / GC University, Lahore
Session
2011
Subject
Mathematics
Number of Pages
101
Keywords (Extracted from title, table of contents and abstract of thesis)
Related, Demonstrate, Erences, Means, Functions, Inequalities, Exponential, Functions, Comparison, Literature, Monotone, Introduced, Convexity, Cauchy

Abstract
There is a lot of literature available on convexity of functions. In contrast, the literature on the exponential convexity is hardly available as there is no operative criteria to recognize exponential convexity.It is not easy to nd and construct exponentially convex functions even-though it is very important sub-class of convex functions in many ways. For example, Laplace transform of a non-negative nite measure is an exponentially convex function. Moreover, one can derive results about positive de nite functions from the properties of exponentially convex functions.
We consider the di erences of Petrovi c and related inequalities, Giaccardi and related inequalities, Chebyshev's inequality, inequality introduced by Lupa s and inequality introduced by Levin-Steckin to construct positive semi-de nite matrices.We derive the classes of exponentially convex functions for the di erences and discuss their properties. We introduce Cauchy means and prove the monotonicity of these means by using the important property of exponentially convex functions.As an application, we establish the mean value theorem of Cauchy type.
In the rst chapter, we organize some basic notions and results.In the second chapter, we use the Jensen-Petrovi c's inequality for star-shaped functions, generalized Petrovi c inequality and inequality introduced by Vasi c and Pe cari c for increasing functions to give results related to power sums.We consider the di erence of these inequalities to construct positive semi-de nite matrices for certain classes of functions to derive families of exponential and logarithmic convex functions. We introduce new means of Cauchy type related to power sums and establish comparison between them.We, also illustrate integral analogs for some results and prove related mean value theorems of Cauchy type.
In the third chapter, we prove the Giaccardi's type inequality for star-shaped type functions and the Giaccardi's inequality for convex-concave antisymmetric functions. We assume the di erences of Giaccardi's type inequality, Giaccardi's inequality for special case and inequality introduced by Vasi c and Stankovi c. By using di erent classes of functions, we formulate families of exponentially convex functions related to these di erences. We introduce new means of Cauchy type and prove monotonicity of these means. We, also exhibit related mean value theorems of Cauchy type. In the fourth chapter, we consider the non-negative di erence of Chebyshev's inequality as Chebyshev functional. We construct symmetric matrices generated by Chebyshev functional for a class of increasing functions and prove positive semide niteness of matrices which implies the exponential and logarithmic convexity of the Chebyshev functional. Moreover, we demonstrate mean value theorems of Cauchy type for the Chebyshev functional and its generalized form.
In the last chapter, we start by considering an inequality related to the Chebyshev's inequality given by A. Lupa s in 1972 but instead of monotone functions there are convex functions. In addition to that we consider the reverse of Chebyshev's inequality without weights introduced by Levin-Steckin; here one function is symmetric increasing and other is continuous convex.By taking the non-negative di erences of each inequality, we construct families of exponentially convex functions. We introduce related Cauchy means and prove related mean value theorems of Cauchy type.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

 
48 KB
2

1

INTRODUCTION

1.1 Monotone functions
1.2 Convex functions
1.3 Exponential convexity

1
192 KB
3 2 EXPONENTIAL CONVEXITY INTRODUCED BY POWER SUMS AND RELATED RESULTS

2.1 Introduction and preliminaries
2.2 Results for star-shaped functions
2.3 Results for convex functions
2.4 Results for increasing functions

8
287 KB
4 3 EXPONENTIAL CONVEXITY INTRODUCED BY GIACCARDI AND RELATED INEQUAL- ITIES

3.1 Introduction and preliminaries
3.2 Giaccardi's type inequality
3.3 Giaccardi's inequality
3.4 Giaccardi's inequality for convex-concave antisymmetric functions
3.5 Petrovi c inequality for convex-concave antisymmetric functions

37
 271 KB
5 4 CAUCHY MEANS INTRODUCED BY THE CHEBYSHEV FUNCTIONAL


4.1 Exponential convexity and related results
4.2 Mean value theorems

64
206 KB
6 5 CAUCHY MEANS INTRODUCED BY AN INEQUALITY OF LUPAS AND LEVIN- STECKIN

5.1 Inequality of Lupa s
5.2 Inequality of Levin-Steckin

76
220 KB
7 6 BIBLIOGRAPHY

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94 KB