The field of inequalities is like a dense jungle. It is also a field in which numerous literature is available. Starting with Hardy, Littlewood and Polya’s book Inequalities published in the year 1934, there are several books focusing on the topic of inequalities. Given this, is there a space for a new book on Inequalities? The answer is a definite YES. In the recently concluded IMO 2021, the problem P2 was a problem on inequality that turned out to be a tough one with only 16 students getting full score. Even the official solution used advanced techniques. Thus books giving newer techniques for proving inequalities are always welcome.
The book New, Newer and Newest Inequalities has two chapters. The first chapter presents classical inequalities as well as some newer ones, such as Newton’s inequality, Maclaurin’s inequality, Blundon’s inequality. Each of these are illustrated with several applications. The book also discusses two powerful techniques – Mixing Variable method and Stronger Mixing Variables method. The method of Lagrange multipliers for constrained optimization is also discussed but this method uses Calculus. There are several interesting problems – many of these have been shown to have simpler solutions using the above inequalities. For example, the combination of Newton’s, Maclaurin’s and AM – GM inequalities is a powerful technique as showcased in the examples 49 to 60 in section 1.4.
The second chapter presents 65 introductory problems and 65 advanced problems with full solutions.
A nice feature of this book is that along with the first and second principle of Mathematical Induction, the charming Cauchy Principle of Mathematical Induction (a forward and backward induction used by Cauchy to prove Arithmetic Mean Geometric Mean inequality) is also presented.
Many of the problems were created by the authors and these vary in difficulty levels. A thorough study of this book will positively enhance the reader’s knowledge on inequalities. The book is a must have for any olympiad student.
