Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
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IMPROVEMENT AND GENERALIZATION OF A THEOREM OF T. J. RIVLIN

  • Pritika, Mahajan (Department of Mathematics, National Institute of Technology Manipur) ;
  • Devi, Khangembam Babina (Department of Mathematics, National Institute of Technology Manipur) ;
  • Reingachan, N. (Department of Mathematics, National Institute of Technology Manipur) ;
  • Chanam, Barchand (Department of Mathematics, National Institute of Technology Manipur)
  • Received : 2021.12.21
  • Accepted : 2022.02.21
  • Published : 2022.09.01
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Abstract

Let p(z) be a polynomial of degree n having no zero inside the unit circle. Then for 0 < r ≤ 1, the well-known inequality due to Rivlin [Amer. Math. Monthly., 67 (1960) 251-253] is $$\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}{\geq}{\(\frac{r+1}{2}\)^n}\max\limits_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$. In this paper, we generalize as well as sharpen the above inequality. Also our results not only generalize, but also sharpen some known results proved recently.

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Acknowledgement

The authors are thankful to the referees for their valuable suggestions.

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