1. A general inequality satisfied by a convex function

where the x_i are any numbers in the region where f is convex and the a_i are nonnegative numbers whose sum is equal to 1. 2. If a₁, a₂, ..., a_n are positive numbers and s > t > 0, then (a₁^s + a₂^s + . . . + a_n^s)^1/s is less than or equal to (a₁^t + a₂^t + . . . + a_n^t)^1/t.