Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
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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
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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.

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Acknowledgement

We are grateful to the referees for their useful suggestions.

References

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