#### 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*)
≤
*R*^{n}/2 +
(3+2√2)*R*^{n-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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