• 제목/요약/키워드: $L^r$-norm inequalities

검색결과 2건 처리시간 0.016초

TURÁN-TYPE Lr-INEQUALITIES FOR POLAR DERIVATIVE OF A POLYNOMIAL

  • Robinson Soraisam;Mayanglambam Singhajit Singh;Barchand Chanam
    • Nonlinear Functional Analysis and Applications
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    • 제28권3호
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    • pp.731-751
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    • 2023
  • If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into Lr-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

SOME Lq INEQUALITIES FOR POLYNOMIAL

  • Chanam, Barchand;Reingachan, N.;Devi, Khangembam Babina;Devi, Maisnam Triveni;Krishnadas, Kshetrimayum
    • Nonlinear Functional Analysis and Applications
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    • 제26권2호
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    • pp.331-345
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    • 2021
  • Let p(z)be a polynomial of degree n. Then Bernstein's inequality [12,18] is $${\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;n\;{\max_{{\mid}z{\mid}=1}{\mid}(z){\mid}}$$. For q > 0, we denote $${\parallel}p{\parallel}_q=\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}$$, and a well-known fact from analysis [17] gives $${{\lim_{q{\rightarrow}{{\infty}}}}\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}={\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p(z){\mid}$$. Above Bernstein's inequality was extended by Zygmund [19] into Lq norm by proving ║p'║q ≤ n║p║q, q ≥ 1. Let p(z) = a0 + ∑n𝜈=𝜇 a𝜈z𝜈, 1 ≤ 𝜇 ≤ n, be a polynomial of degree n having no zero in |z| < k, k ≥ 1. Then for 0 < r ≤ R ≤ k, Aziz and Zargar [4] proved $${\max\limits_{{\mid}z{\mid}=R}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;{\frac{nR^{{\mu}-1}(R^{\mu}+k^{\mu})^{{\frac{n}{\mu}}-1}}{(r^{\mu}+k^{\mu})^{\frac{n}{\mu}}}\;{\max\limits_{{\mid}z{\mid}=r}}\;{\mid}p(z){\mid}}$$. In this paper, we obtain the Lq version of the above inequality for q > 0. Further, we extend a result of Aziz and Shah [3] into Lq analogue for q > 0. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.