Turkish Journal of Mathematics
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.
DOI
-
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 31 (1): 89-94. https://doi.org/-
