Inequalities, identities, and bounds for divided differences of the exponential function

Let $\exp[x_0,x_1,\dots,x_n]$ denote the divided difference of the exponential function. (i) We prove that exponential divided differences are log-submodular. (ii) We establish the four-point inequality $ \exp[a,a,b,c]\,\exp[d,d,b,c]+\exp[b,b,a,d]\,\exp[c,c,a,d]-\exp[a,b,c,d]^2 \ge 0 $ for all $ a,b,c,d \in \mathbb{R} $. (iii) We obtain sharp two-sided bounds for $\exp[x_0,\dots,x_n]$ at fixed mean and variance; as a consequence, we derive their large-input asymptotics. (iv) We present closed-form identities for divided differences of the exponential function, including a convolution identity and summation formulas for repeated arguments.