Chapter 2 - Different types of Bernstein inequalities

Abstract

The second chapter is devoted to different types of Bernstein inequalities. The Bernstein-type inequalities and extremal problems have been studied under different norms and for various classes of functions associated with polynomials, such as rational functions and entire functions of exponential type. In this chapter are introduced analogues of Markov and Bernstein inequalities in these different situations. Some Bernstein-type inequalities concerning growth of polynomials are studied in Section 2.1. The Section 2.2 deals with extensions of these inequalities to rational functions. In Section 2.3, are presented certain Bernstein type inequalities in Lr norm, a topic that has been studied in more detail in Chapter 5. In Section 2.4, are discussed in brief, Bernstein-type inequalities in other environments, such as in normed linear spaces, and for wavelets. Finally, Section 2.5 deals with Bernstein type inequalities for entire functions of exponential type.