Certain interesting implications of T. J. Rivlin's result on maximum modulus of a polynomial

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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