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Let p > 3 be a prime. Gauss first introduced the polynomial S p (x) = ∏ c (x − ζ p c) , where 0 < c < p and c varies over all quadratic residues modulo p and ζ p = e 2 π i / p . Later Dirichlet investigated this polynomial and used this to solve problems involving the Pell equations. Recently, Sun studied some trigonometric identities involving this polynomial. In this paper, we generalize their results. As applications of our result, we extend Chowla's result on the congruence concerning the fundamental unit of ℚ (p) and give an equivalent form of the extended Ankeny et al. conjecture. [ABSTRACT FROM AUTHOR]

For a polynomial p z of degree n , it follows from the maximum modulus theorem that max z = R ≥ 1 p z ≤ R n max z = 1 p z . It was shown by Ankeny and Rivlin that if p z ≠ 0 for z < 1 , then max z = R ≥ 1 p z ≤ R n + 1 / 2 max z = 1 p z . In 1998, Govil and Mohapatra extended the above two inequalities to rational functions, and in this paper, we study the refinements of these results of Govil and Mohapatra. [ABSTRACT FROM AUTHOR]

Let p(z) = ∑ν=0n aνzν be a polynomial of degree n, M(p, R) := max|z|=R≥0 |p(z)|, and M(p, 1) := ∥p∥. Then according to a well-known result of Ankeny and Rivlin, we have for R ≥ 1, M(p, R) ≤ (...) ∥p∥. This inequality has been sharpened among others by Govil, who proved that for R ≥ 1, .... In this paper, we sharpen the above inequality of Govil, which in turn sharpens inequality of Ankeny and Rivlin. We present our result in terms of the LerchPhi function Φ(z, s, a), implemented in Wolfram's MATHEMATICA as LerchPhi [z, s, a], which can be evaluated to arbitrary numerical precision, and is suitable for both symbolic and numerical manipulations. Also, we present an example and by using MATLAB show that for some polynomials the improvement in bound can be considerably significant. [ABSTRACT FROM AUTHOR]

Let ... be a polynomial of degree n, ... Then by well-known result due to Ankeny and Rivlin [1], we have ... In this paper, we sharpen and generalizes the above inequality by using a result due to Govil [5]. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of finding some upper bound estimates for the maximum modulus of the derivative and higher order derivatives of a complex polynomial on a disk under the assumption that the polynomial has no zeros in another disk. The estimates obtained strengthen the well-known inequality of Ankeny and Rivlin about polynomials. [ABSTRACT FROM AUTHOR]

Let p (z) be a polynomial of degree n having no zeros in | z | < 1 . In this paper, we generalize and extend a well-known result proven by Ankeny and Rivlin for the s t h derivative of the polynomial. Furthermore, another well-known result proven by Rivlin is also improved, generalized and extended for the s t h derivative of p (z) . Our results also give a number of interesting consequences as special cases. [ABSTRACT FROM AUTHOR]