Integral mean estimates for polynomials whose zeros are within a circle

Abdul Aziz and Nisar Ahemad, Srinagar, India

Keywords: maximum modulus, derivative of a polynomial, integral mean estimates AMS Subject classification: 30A06, 30A64

Abstract: Let P (z) be a polynomial of degree n having all its zeros in |z| ≤ K. For K = 1, it is known that for any r > 0,

n \{∫_{0}^{2π} |P (e^{iθ})|^{r} dθ\}^{\frac{1}{r}} ≤ \{∫_{0}^{2π} |1 + e^{iθ}|^{r} dθ\}^{\frac{1}{r}} Max_{|z|=1} |P′(z)|.

In this paper, we shall present several generalizations of this result in this sense that maximum |P′(z)| on |z| = 1 is replaced by a factor involving integral mean of |P′(z)| on |z| = 1, by considering the two cases K ≤ 1 and K > 1.