Does the ratio of Laplace transforms of powers of a function identify the function?

We study the following question: if f is a nonzero measurable function on [ 0 , ∞ ) and m n are distinct nonnegative integers, does ratio ˆ / of Laplace transforms powers uniquely determine ? By inverse transform, answer yes or zero. Under some assumptions smoothness we show that in general case also affirmative. The question arose from problem economics, specifically auction theory, where cumulative distribution certain random variable. This discussed.