For \(f\in C^ n(R)\) and \(0\leq t\leq x\) let \(J_ n(f,t,x)=(-1)^ n f(- x)f^{(n)}(t)+f(x)f^{(n)}(-t)\). The authors establish that the only real-analytic functions satisfying \(J_ n(f,t,x)\geq 0\) for all \(n=0,1,2,\dots\) are the exponential functions \(f(x)=ce^{\lambda x}\), \(c,\lambda\in R\). Further they present a nontrivial class of real- analytic functions satisfying the inequalities \(J_ 0(f,t,x)\geq 0\) and \(\int_ 0^ x(x-t)^{n-1}J_ n(f,t,x)dt\geq 0\) (\(n\geq 1\)).