MATH-8106               Applications of Inequalities                                                            3 (3+0)

The main purpose of this course is the study of some mathematical inequalities. Furthermore, the applications & integral operators of the inequalities involving special functions are to be discussed. Mathematics is not always about “equals”. We compare one thing with the other thing in mathematics & in other branches of science. In mathematics, an inequality is a relation that holds between two values or quantities when they are different. The theory of inequalities has been recognized as one of the central areas of mathematical analysis. During the past several years, the usefulness & applications of mathematical inequalities in various branches of mathematics as well as in other areas of science are well established. It is a fast growing discipline with ever increasing applications in scientific fields such as mathematical economics, game theory, mathematical programming, control theory, variational methods, operational research, probability & statistics. This growth resulted in the appearance of the theory of inequalities as an independent domain of mathematical analysis. Inequalities for the special functions appear infrequently in the literature. Some of these inequalities are closely related to those presented here. Inequalities for the ratio of confluent hypergeometric functions are available in the literature. The formulas are very important, as they include expansions for many transcendent expressions of mathematical physics in series of the classical orthogonal polynomials, Laguerre, Hermite functions.

Contents

1. Grüss type inequalities, Chebychev’s type inequalities
2. Ostrowski’s inequalities
3. Hardy-type Inequalities
4. Jensen’s & related inequalities,
5. Hadamard’s inequalities,
6. Applications of inequalities involving gamma & beta functions
7. Applications of inequalities involving beta functions
8. Introduction to Lp-spacesinvolving special functions
9. Boundedness of integral operators involving some special functions
10. Miscellaneous inequalities

Recommended Texts

1. Pachpatte, B. G. (2005). Mathematical inequalities. The Netherlands: Elsevier.
2. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory & applications of fractional differential equations. The Netherlands: Elsevier.

Suggested Readings

1. Mitrinovic, D. S., Pečarić, J., & Fink, A. M. (1993). Classical & new inequalities in analysis. The Netherl&s: Kluwer Academic Publishers.
2. Pečarić, J., Proschan, F., & Tong, Y. C. (1992). Convex functions, partial orderings & statistical applications. USA: Academic Press.
3. Okikiolu, G. O. (1971). Aspects of the theory of bounded linear operators. Cambridge: Academic Press.
4. Related research papers.

Assessment Criteria

Sessional: 20 (Presentation / Assignment 04, Attendance 08, Result Mid-Term 04, Quiz 04)

Mid-Term Exam:  30

Final-Term Exam: 50

Key Dates and Time of Class Meeting

Monday-Tuesday   (PhD-II-Self Support Evening)                                                           03:30pm-05:00pm

March 15, 2021 (Monday) 

 Mid-Term Examination:

May 17-21, 2021 (Monday to Friday)

Final-Term Examination:

July 12-16, 2021 (Monday to Friday)

Declaration of Result: July 17, 2021 (Friday)