Uniform estimates of polynomials by logarithmic sums

It is well known that the size of a polynomial is controlled in the whole complex plane by the logarithmic integral of the polynomial. In the present thesis we show that the logarithmic sum, a discrete analogue of the logarithmic integral, can be used, with suitable precautions, for the same purpose. We obtain uniform estimates of polynomials having sufficiently small logarithmic sums. In these estimates, dependence of the polynomials is expressed entirely through the logarithmic sums of the polynomials.This result was originally established by Paul Koosis in 1966. The proof that we present here is, however, new. We use a suitable version of a so-called multiplier theorem. Theorems of this kind were first published by Beurling and Malliavin in 1962. The particular version that we need is based on least superharmonic majorants and harmonic estimation in slit regions and is due to Koosis.We extend the result to entire functions of exponential type less than a certain numerical constant (approximately one third). This extension is new and is based on the ideas behind the proof for polynomials