Estimates of entire functions of exponential type less than π in terms of logarithmic sums over real Duffin and Schaeffer sequences

We give uniform estimates of entire functions of exponential type less than p having sufficiently small logarithmic sums over real sequences {lambda(n)} satisfying \lambda(n)-n\ less than or equal to L and lambda(n+1)-lambda(n)greater than or equal todelta for fixed positive constants L and delta. We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.