#### Glasnik Matematicki, Vol. 33, No.1 (1998), 33-36.

### CERTAIN INTERESTING IMPLICATIONS OF T. J.
RIVLIN'S RESULT ON MAXIMUM MODULUS OF A POLYNOMIAL

### V. K. Jain

Mathematics Department, I.I.T., Kharagpur - 721302, India

**Abstract.** Let *f*(*z*) be an arbitrary
entire function and *M*(*f,r*) =
max_{|z|=r}
|*f*(*z*)|. For a polynomial *p*(*z*) of degree
*n*, having no zeros in |*z*| < 1, Rivlin had obtained

*M*(*p,r*) ≥
((*r*+1)/2)^{n} *M*(*p*,1),
*r* ≤ 1.

Using this and various associated results, we have obtained precise
relationship between
*M*(*p*,ρ),
and *M*(*p*,1) for polynomials *p*(*z*) having
all its zeros on |*z*| = *k*, *k* > 0. Some sort of a
converse of Rivlin's result has also been obtained
**1991 Mathematics Subject Classification.**
30C10, 30A10.

**Key words and phrases.** Maximum modulus, zeros.

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