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Extremal Problems in Polynomials and Entire Functions

In: Handbook of Functional Equations

Author

Listed:
  • N. K. Govil

    (Auburn University)

  • Q. M. Tariq

    (Virginia State University)

Abstract

The subject of extremal problems for polynomials and related classes of functions plays an important and crucial role in obtaining inverse theorems in approximation theory. Frequently, the further progress in inverse theorems has depended upon first obtaining the corresponding analogue or generalization of Markov’s and Bernstein’s inequalities, and these inequalities have been the starting point of a considerable literature in Mathematics. In this chapter, we begin with the earliest results in the subject (Markov’s and Bernstein’s inequalities), and present some of their generalizations and refinements. In the process, some of the problems that are still open have also been mentioned. Since there are many results in this subject, we have concentrated here mainly on results concerning Bernstein’s inequality. The chapter contains four sections, with Sect. 1 dealing with introduction to Bernstein’s and Markov’s inequalities along with some of their generalizations. In Sect. 2, we discuss some constrained Bernstein type inequalities, that is Bernstein type inequalities for some classes of polynomials, while in Sect. 3 the extension to entire functions of exponential type for some of the results of Sect. 2 has been discussed. Finally, Sect. 4 contains some of the open problems, discussed in the text of this chapter, that could be of interest to some of the readers.

Suggested Citation

  • N. K. Govil & Q. M. Tariq, 2014. "Extremal Problems in Polynomials and Entire Functions," Springer Optimization and Its Applications, in: Themistocles M. Rassias (ed.), Handbook of Functional Equations, edition 127, pages 209-245, Springer.
    • Handle: RePEc:spr:spochp:978-1-4939-1246-9_10
    DOI: 10.1007/978-1-4939-1246-9_10
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