Abstract: In this paper it is shown that the generalized Widder's inequality can be established by means of the theory of analytic function and Hölder's inequality and by the help of Hardy's idea. The conditions in which the equality contained in this inequality holds are discussed. In particular, the classical Widder's theorem is attained when p=q= 2.

Key words: Widder's theorem, analytic function,
func t ion;Hölder's inequali ty