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IMPROVED VERSION ON SOME INEQUALITIES OF A POLYNOMIAL

  • Rashmi Rekha Sahoo (Department of Mathematics, National Institute of Technology Manipur) ;
  • N. Reingachan (Department of Mathematics, National Institute of Technology Manipur) ;
  • Robinson Soraisam (Department of Mathematics, National Institute of Technology Manipur) ;
  • Khangembam Babina Devi (Department of Mathematics, National Institute of Technology Manipur) ;
  • Barchand Chanam (Department of Mathematics, National Institute of Technology Manipur)
  • Received : 2023.02.06
  • Accepted : 2023.04.06
  • Published : 2023.12.15

Abstract

Let P(z) be a polynomial of degree n and P(z) ≠ 0 in |z| < 1. Then for every real α and R > 1, Aziz [1] proved that $$\max\limits_{{\mid}z{\mid}=1}{\mid}P(Rz)-P(z){\mid}{\leq}{\frac{R^n-1}{2}}(M^2_{\alpha}+M^2_{{\alpha}+{\pi}})^{\frac{1}{2}}{\mid},$$ where $$M{\alpha}={\max\limits_{1{\leq}k{\leq}n}}{\mid}P(e^{i({\alpha}+2k{\pi})n}){\mid}.$$ In this paper, we establish some improvements and generalizations of the above inequality concerning the polynomials and their ordinary derivatives.

Keywords

Acknowledgement

We are grateful to the referees for their useful suggestions.

References

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