The inequalities play an important role in almost all branches of mathematics as well as in other areas of science. The basic Inequalities and Pólya appeared in 1934 and the Inequalities Analytic Inequalities Mitrinovi´c published in 1970 made considerable contributions to this field and supplied motivations, ideas, techniques and applications. Since 1934 an enormous amount of effort has been devoted to the discovery of new types of inequalities and to the application of inequalities in many parts of analysis. The usefulness of mathematical inequalities is felt from the very beginning and is now widely acknowledged as one of the major driving forces behind the development of modern real analysis. The theory of inequalities is in a process of continuous development state and inequalities have become very effective and powerful tools for studying a wide range of problems in various branches of mathematics. This theory in recent years has attracted the attention of many researchers, stimulated new research directions, and influenced various aspects of mathematical analysis and applications. Among the many types of inequalities, those associated with the names of Jensen, Hadamard and Hermite have deep roots and made a great impact on various branches of mathematics. The last few decades have witnessed important advances related to these inequalities that remain active areas of research and have grown into substantial fields of research with many important applications. The development of the theory related to these inequalities resulted in a renewal of interest in the field and has attracted interest from many researchers. A host of new results have appeared in the literature. The present course provides a systematic study of some of the most famous and fundamental inequalities originated by the above-mentioned mathematicians and brings together the latest, interesting developments in this important research area under a unified framework.

**Contents**

- Jensen's and related inequalities
- Some general inequalities involving convex functions
- Hadamard's qualities
- Inequalities of Hadamard type I
- Inequalities of Hadamard type II
- Some inequalities involving concave functions
- Miscellaneous inequalities

**Recommended Texts**

- Pachpatte, B. G. (2005). Mathematical inequalities. New York: Elsevier.
- Convex functions, partial orderings and statistical applications (Vol. 187). Cambridge: Academic Press, Boston, Mass.

**Suggested Readings**

- Classical and new inequalities in analysis. Netherlands: Kluwer Academic Publishers.
- Recent research articles.

1. Applications of Inequalities in mathematical analysis.

2. Use of ideas from the proofs of basics inequalities for new inequalities.

3. Applications to different type of means.

*ASSESSMENT CRITERIA*

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Term Paper: 30

Final-Term Exam: 50

Monday 02:00 PM - 03:30 PM

Tuesday 02:00 PM - 03:30 PM

Commencement of Classes March 15, 2021

Mid Term May 17-21, 2021

Final Term Examination July 12-16, 2021

Declaration of Result July 27, 2021