You are here

Functional Equations and Inequalities with Applications

Pl. Kannappan
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Michael Berg
, on

As a number theorist, I have a great, if not unnatural, fondness for functional equations. But they is not just a parochial matter: their utility (and ubiquity) is clearly visible from a dozen mathematical perspectives. Goodness, even physicists appreciate these constructs (see p. 682 — yes, p. 682 — of the book under review), and their presence in analysis proper is known to any one who has ever opened either Blue or Green Rudin. To get back to number theory, however, the sheer magic worked with functional equations by Euler, Riemann, and Dirichlet is a widespread source of enchantment and inspiration for nigh on every mathematical sensibility. And the fact that the functional equation for Riemann’s zeta function drives the greatest open problem in all of mathematics underscores the importance of the present topic even more emphatically.

This said, I was nonetheless stunned by the sheer size and scope of Kannappan’s Functional Equations and Inequalities with Applications. In seventeen chapters and on the order of 800 pages (yikes!) we are presented with what must surely be something of an encyclopaedic treatment of the according topics. Starting with “Basic equations: Cauchy and Pexider equations,” the author proceeds to a long discussion of trigonometric functional equations, quadratic functional equations, connections with information theory, Abel’s work on functional equations, difference equations (a most welcome inclusion), special functions (Yeah! Gamma, Beta, Zeta, &c. — fun, fun, fun!), material on miscellaneous and general equations, and then a truly titanic chapter on “Applications.” The latter is something of an opus in its own right, covering the role of functional equations and inequalities in economics, the mathematics of finance, physics (e.g., QM), topology, digital filtering, geometry, statistics, information theory (again), and the behavioral sciences. Truly an amazing list!

To be sure, Kannappan’s book presents us with a most useful vade mecum for many mathematicians (not just arithmeticians) as well as various users of mathematics, including the “engineers and applied scientists” Kannappan specifically targets (see p.x of the book). It is an impressive treatment of an important subject not often given the airplay it deserves.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Preface.- 1. Basic Equations. Cauchy and Pexider Equations.- 2. Matrix Equations.- 3. Trigonometric Functional Equations.- 4. Quadratic Functional Equations.- 5. Characterization of Inner Product Spaces.- 6. Stability.- 7. Characterization of Polynomials.- 8. Nondifferentiable Functions.- 9. Characterization of Groups, Loops and Closure Conditions.- 10. Functional Equations from Information Theory.- 11. Abel Equations and Generalizations.- 12. Regularity Conditions|Christensen Measurability.- 13. Dierence Equations.- 14. Characterization of Special Functions.- 15. Miscellaneous Equations.- 16. General Inequalities.- 17. Applications.- Symbols.- Bibliography.- Author Index.- Subject Index.