Stability of the Prékopa-Leindler inequality

We prove a stability version of the Prékopa-Leindler inequality. 1 The problem Our main theme is the Prékopa-Leindler inequality, due to A. Prékopa [14] and L. Leindler [13]. Soon after its proof, the inequality was generalized in A. Prékopa [15] and [16], C. Borell [7], and in H.J. Brascamp, E.H. Lieb [8]. Various applications are provided and surveyed in K.M. Ball [1], F. Barthe [5], and R.J. Gardner [12]. The following multiplicative version from [1], is often more useful and is more convenient for our purposes. THEOREM 1.1 (Prékopa-Leindler) If m, f, g are non-negative integrable functions on R satisfying m( r+s 2 ) ≥ √ f(r)g(s) for r, s ∈ R, then