Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
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ON AN INEQUALITY OF S. BERNSTEIN

  • Received : 2020.09.29
  • Accepted : 2021.02.09
  • Published : 2021.06.15
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Abstract

If $p(z)={\sum\limits_{{\nu}=0}^{n}}a_{\nu}z^{\nu}$ is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then Govil [3] proved that $${\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;{\frac{n}{k^n+k^{n-1}}}\;{\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p(z){\mid}$$. In this paper, by involving certain coefficients of p(z), we not only improve the above inequality but also improve a result proved by Dewan and Mir [2].

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Acknowledgement

The authors are extremely grateful to the referee for his valuable comments and suggestions about the paper.

References

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