Lower Bounds for the L p ‐Norm in Terms of the Mellin Transform

Given a measurable function f on (0, ∞) with Mellin transform F ( s ), let | f | p denote the L p ‐norm of f with respect to the measure dx/x . We prove that under certain assumptions, for instance if f is real and non‐negative and F (α) converges for α in an open interval and F (α) ≠ 0, then| f | p ⩾ c p| α |1 ‐ 1 / p| F ( α ) exp ( ‐ αF ' F ( α ) )| , where c p ⩾ (2 e ) −1 . We derive similar inequalities for complex‐valued f , for the L p ‐norm of the derivative of f , and for the supremum of real‐valued f and of its derivative. The lower bounds are eminently applicable when f is a convolution product.