Growth of Analytic Functions Along a Line.

Let J(z) be regular and of exponential type less than wr in the half plane x _ 0, and let I Xn } be an increasing positive sequence of unit density such that Xn+1 -n _ 5 > 0. It is known that under suitable restrictions and with a suitable measure of growth, the growth of the sequence tf(Xn) determines the growth off(x) for real x > 0. I have established the following results. Let e(x) be a concave increasing differentiable function such that e(x) = o(x), e'(x) _ E(x)/x < 1 and log x = o(E(x)). Let {IXnI satisfy the further condition that Ixn nj < e(n). THEOREM 1. If