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Journals / Turkish Journal of Mathematics / 2007 / Cilt: 31 - Sayı: 1

A generalization of Ankeny and Rivlin's result on the maximum modulus of polynomials not vanishing in the interior of the unit circle

V. K. JAIN
Pages
89–94
Publication Date
2022-11-05
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

By considering the polynomial p(z) with no zeros in |z| < k, $(kgeq 1)$ and simultaneously thinking of $s^{th}$ derivative $(0leq s < n)$ of the polynomial, we have obtained the generalization M(p^(s),R)leq..................................... ( (1/2){ ds dRs (Rn + kn)}(2/(1 + k))nM(p, 1), R≥ k, (1/(Rs + ks))[{ ds dxs (1 + xn)}x=1]((R + k)/(1 + k))nM(p, 1), 1 ≤ R ≤ k, of Ankeny and Rivlin’s result.

📌 3 Key Takeaways

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Objective & Hypothesis:For an arbitrary entire function f(z), let M(f, r) = $max_{|z|}=r |f(z)|$.
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Key Findings & Data:For a polynomial p(z) of degree n, it is known that $M(p,R)leq R^n M(p, 1)$, R>1.
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Conclusion & Impact:By considering the polynomial p(z) with no zeros in |z| < k, $(kgeq 1)$ and simultaneously thinking of $s^{th}$ derivative $(0leq s < n)$ of the polynomial, we have obtained the generalization M(p^(s),R)leq..................................... ( (1/2){ ds dRs (Rn + kn)}(2/(1 + k))nM(p, 1), R≥ k, (1/(Rs + ks))[{ ds dxs (1 + xn)}x=1]((R + k)/(1 + k))nM(p, 1), 1 ≤ R ≤ k, of Ankeny and Rivlin’s result.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

For an arbitrary entire function f(z), let M(f, r) = $max_{|z|}=r |f(z)|$. For a polynomial p(z) of degree n, it is known that $M(p,R)leq R^n M(p, 1)$, R>1. By considering the polynomial p(z) with no zeros in |z| 1. By considering the polynomial p(z) with no zeros in |z| < k, $(kgeq 1)$ and simultaneously thinking of $s^{th}$ derivative $(0leq s < n)$ of the polynomial, we have obtained the generalization M(p^(s),R)leq..................................... ( (1/2){ ds dRs (Rn + kn)}(2/(1 + k))nM(p, 1), R≥ k, (1/(Rs + ks))[{ ds dxs (1 + xn)}x=1]((R + k)/(1 + k))nM(p, 1), 1 ≤ R ≤ k, of Ankeny and Rivlin’s result.

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Cite This Article

JAIN, V. K. (2022). A generalization of Ankeny and Rivlin's result on the maximum modulus of polynomials not vanishing in the interior of the unit circle. Turkish Journal of Mathematics, Cilt: 31 - Sayı: 1, 89–94. https://ajindex.com/articles/608652

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References (3)

  1. Ankeny, N. C. and Rivlin, T. J.: On a theorem of S. Bernstein, Pac. J. Maths. 5, 849–852 1955
  2. Aziz, A. and Mohammad, Q. G.: Growth of polynomials with zeros outside a circle, Proc. Amer. Math. Soc. 81, 549–553 1981
  3. Govil, N. K. and Rahman, Q. I.: Functions of exponential type not vanishing in a half plane and related polynomials, Trans. Amer. Math. Soc. 137, 501–517 1969.