New lower bound for the modulus of an analytic function

A lower bound for the minimum of the modulus of an analytic function in terms of the integral norm on a «large» circle is obtained. The proof uses the basic facts of the theory of spaces of analytic functions in the disk and the classical Chebyshev polynomials. From the main theorem a statement for an entire function is derived in which the integral norm replaced by the maximum modulus. A special example of a sequence of entire functions is constructed, showing that this result cannot be much improved. Earlier in the theory of entire functions theorems on lower estimates for the modulus of an entire function on a system of circles expanding to infinity through some degrees of the maximum of the modulus on the same circles were known.