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.