Uniform Estimates of Entire Functions by Logarithmic Sums

We give uniform estimates in the whole complex plane of entire functions of exponential type less than a certain numerical constant (approximately equal to 0.44) having sufficiently small logarithmic sums. In these estimates the entire dependence on the function is through its type and logarithmic sum. This result extends a theorem of Koosis about polynomials and gives a new proof of that theorem. The proof is based on material related to multiplier theorems, first obtained by Beurling and Malliavin.