Glasnik Matematicki, Vol. 38, No.2 (2003), 269-272.
A GENERALIZATION OF A RESULT ON MAXIMUM
MODULUS OF POLYNOMIALS
V. K. Jain
Mathematics Department, Indian Institute of Technology,
Kharagpur - 721302, India
e-mail: vkj@maths.iitkgp.ernet.in
Abstract. For an arbitrary entire function
f(z), let
M(f,d) = max|z|=d
|f(z)|.
It is known that if the geometric mean of the moduli of the zeros
of a polynomial p(z) of degree n is at least 1,
and M(p,1) = 1, then for R > 1,
M(p,R)
≤
R/2 + 1/2 if n = 1,
M(p,R)
≤
Rn/2 +
(3+2√2)Rn-2/2 if n
≥ 2.
We have obtained a generalization of this result, by assuming the
geometric mean of the moduli of the zeros of the polynomial to be
at least k, (k > 0).
2000 Mathematics Subject Classification.
30C10, 30A10.
Key words and phrases. Polynomials, zeros, geometric mean, maximum modulus.
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