Turkish Journal of Mathematics
DOI
-
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, 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, (k \geq 1) and simultaneously thinking of s^{\rm th} derivative (0 \leq s < n) of the polynomial, we have obtained the generalization \begin{displaymath} M(p^{(s)},R) \leq \left\{\begin{array}{l} (1/2){\frac{d^s}{dR^s}(R^n + k^n)}(2/(1+k))^nM(p,1), R \geq k,\ (1/(R^s+k^s))[{\frac{d^s}{dx^s}(1+x^n)}_{x=1}]((R+k)/(1+k))^nM(p,1), 1 \leq R \leq k,\end{array}\right. of Ankeny and Rivlin's result.
Keywords
Polynomial, maximum modulus principle, not vanishing in the interior of unit circle, generalization, s^{th} derivative
First Page
89
Last Page
94
Recommended Citation
JAIN, V. K. (2007) "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: Vol. 31: No. 1, Article 7. Available at: https://journals.tubitak.gov.tr/math/vol31/iss1/7
