Some inequalities for Laplace transforms

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Some inequalities for characteristic functions, Theory Probab. Appl. 45 (2001) 711–713] about a lower bound of real-valued characteristic functions. In this paper, we investigate the counterparts for Laplace transforms of non-negative random variables. Surprisingly, the resulting inequalities hold true on the right half-line. Besides, we show some more inequalities by applying the convex/concave properties of the remainder in Taylor's expansion for the exponential function.