V. K. JAIN

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

ISSN: 1300-0098
Yayın Aralığı: 6
Yayıncı: TÜBİTAK

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, Ankeny and Rivlin obtained the refinement $M(p,R)leq {(R^n +1)/2}M(p, 1)$, R>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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[1] Ankeny, N. C. and Rivlin, T. J.: On a theorem of S. Bernstein, Pac. J. Maths. 5, 849–852 1955.
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