Mean value theorems : generalizations and associated functional equations

Abstract

The Lagrange’s Mean Value Theorem is a very important result in Analysis. It originated from Rolle’s theorem, which was proved by the French mathematician Michel Rolle (1652-1719) for polynomials in 1691. This theorem appeared for the first time in the book “M´ethode pour r´esoudre les ´egalit´es” without a proof and without any special emphasis. Rolle’s Theorem got its recogni- tion when Joseph Lagrange (1736-1813) presented his mean value theorem in his book “Th´eorie des functions analytiques” in 1797. It received further recognition when Augustin Louis Cauchy (1789- 1857) proved his mean value theorem in his book entitled “E´quations diff´erentielles ordinaires”. Most of the results in Cauchy’s book were established using the Mean Value Theorem or indirectly Rolle’s Theorem. Since their discovery, many papers have appeared deal- ing (directly or indirectly) with of the Rolle’s Theorem and La- grange’s Mean Value Theorem. Recently, many functional equa- tions, motivated by various Mean Value Theorems, were studied. The main goal of this thesis is to prove several Mean Value Theorems, present some of their applications and general- izations and study their Associated Functional Equations. In Chapter 1, we prove the Lagrange’s Mean Value Theorem and present some of its applications. Many examples are given to il- lustrate its importance in Analysis. Then, we discuss and solve functional equations associated to it. After proving the Topologi- cal and Weak Topological Mean Value Theorems and deduce some regularity results, we finish Chapter 1 by establishing the Flett’s Mean Value Theorem.

In Chapter 2, we prove the Cauchy’s Mean Value Theorem which generalizes Lagrange’s Mean Value Theorem. We then study a functional equation associated to the Cauchy’s Mean Value The- orem. Mainly, we characterize all pairs of smooth functions for which the mean value is taken at a point having a well-determined position in the interval. As an application, a partial answer to a question, posed by Sahoo and Riedel, is obtained. As we did in Chapter 1, we finish Chapter 2 by establishing the Cauchy’s Mean Value Theorem for divided differences. Chapter 3 introduces the Ostrowski’s inequality via the Cauchy’s Mean Value Theorem, with some applications and generalizations. Chapter 4 deals with a variation of the Lagrange’s mean value theorem due to Dimitri Pompeiu. It is called the Pompeiu’s Mean Value Theorem. This theorem has been the source of motivations for many Stamate type functional equations. We finish Chapter 4 by studying some functional equations motivated by the Simpson’s rule for numerical integration.