Nonlinear Functional Analysis and Applications

  • 제28권4호
  • /
  • Pages.919-928
  • /
  • 2023
  • /
  • 1229-1595(pISSN)
  • /
  • 2466-0973(eISSN)
경남대학교 수학교육과 (Kyungnam University, Department of Mathematics Eduaction)
권호 표지
DOI QR코드

DOI QR Code

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)
  • 투고 : 2023.02.06
  • 심사 : 2023.04.06
  • 발행 : 2023.12.15
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

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.

키워드

과제정보

We are grateful to the referees for their useful suggestions.

참고문헌

  1. A. Aziz, A refinement of an inequality of S. Bernstein, J. Math. Anal. Appl., 142(1) (1989), 226-235.
  2. S. Bernstein, Lecons Sur Les Proprietes extremales et la meilleure approximation desfunctions analytiques d'une fonctions reele, Gauthier-Villars, Paris, 1926.
  3. C. Frappier, Q.I. Rahman and S. Ruscheweyh, New inequalities for polynomials, Trans. Amer. Math. Soc., 288 (1985), 69-99.
  4. P.D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc., 50 (1944), 509-513.
  5. M.A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115(2) (1992), 337-343.
  6. N.A. Rather, S.H. Ahangar and M.A. Shah, Some inequalities for the derivative of a polynomial, Int. J. Appl. Math., 26(2) (2013), 177-185.
  7. T. B. Singh, K. Krishnadas and B. Chanam, Lr inequalities of generalized Turan-type inequalities of polynomials, Nonlinear Funct. Anal. Appl., 26(4)(2021), 855-868.
  8. R. Soraisam, N. K. Singha and B. Chanam, Improved bounds of polynomial inequalities with restricted zero, Nonlinear Funct. Anal. Appl., 28(2)(2023), 421-437.