Inequalities for the exponential function

We study the log-concavity of the function a -> a(delta)J(a)(x), where J(a) (x) = e(x) - (1 + x/a)(a) and use our result to show that J(a)(x) satisfies certain functional inequalities. Among others, we prove that if a, b and lambda, mu are positive real numbers with a &NOTEQUexpressionL; b and lambda + mu = 1, then we have for all x > 0, a(lambda)b(mu)/lambda a + mu b < J(lambda a+mu b)(x)/J(a)(x)(lambda) J(b)(x)(mu). The lower bound is best possible.