A functional inequality for real-analytic functions

Summary Forf ( Cⁿ() and 0 t x letJ_n(f, t, x) = (–1)ⁿf(–x)f⁽ⁿ⁾(t) +f(x)f⁽ⁿ⁾(–t). We prove that the only real-analytic functions satisfyingJ_n(f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ₀(f, x, x) 0 and₀^x(x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).