AN INEQUALITY FOR PROBABILITY DENSITY-FUNCTIONS ARISING FROM A DISTINGUISHABILITY PROBLEM

An integral inequality is established involving a probability density function on the real line and its first two derivatives. This generali zes an earlier result of Sate and Watari. If f denotes the probability density function concerned, the inequality we prove is that integral( -infinity)(+infinity) [f'(x)(2)](gamma alpha/[f(x)](gamma(beta+1)-1) d x less than or equal to (2 alpha-1/beta-1)(gamma alpha) (integral(-inf inity)(+infinity)\f ''(x)\(alpha-1)/[f(x)](beta-alpha) dx)(gamma) unde r the conditions beta > alpha > 1 and 1/(beta + 1) < gamma less than o r equal to 1.