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

TL;DR

The paper investigates the structural properties of exponential divided differences $\exp[x_0,\dots,x_n]$, establishing log-submodularity (TN$_2$) and a four-point inequality that reveal a latent convexity in these objects. It then derives sharp two-sided bounds for $n!\,\exp[x_0,\dots,x_n]$ at fixed mean $\mu$ and variance $\sigma^2$, along with their large-$n$ asymptotics, by exploiting Hermite–Genocchi representations and Schur-convexity. A suite of closed-form identities is developed, including a convolution identity and repeated-argument summations, via contour-integral definitions and Laplace-transform techniques. Collectively, these results deepen the theoretical understanding of exponential divided differences and supply practical tools for interpolation, operator theory, and numerical exponential integrators, with potential extensions to operator-valued settings and related functions.

Abstract

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.

Theorems & Definitions (40)

• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• proof
• Theorem 1: Log-submodularity
• proof
• Theorem 2: Supermodularity
• proof