Fractional form of Jensen’s Inequality for the Exponential Integral Function with Applications to Modeling Utility

Abstract

Jensen’s inequality is a fundamental result in probability and analysis, that offers simple boundsforthe convex function applied to a random variable. However, the classical form of this process does not include memory effects, which are very important in some physical and financial systems with long-range dependence. In this paper, we introduce a fractional-order generalisation of Jensen’s inequality involving memory effects that can be accounted for by means of fractional calculus. We concentrate on the exponential integral function Ei(x) because of its wide use. For a random variable Xwith mean µsupported on [1, ∞), and for every fractional-order α ∈ (0,1) we show the strict inequality Ei(µ) ≤ E[Ei(X)] − M(α), where the new quantity M(α) is a memory correction defined in terms of Riemann– Liouville fractional integrals of order 1 − αof the function e t /t. The correction term provides a whole family of bounds, controlled by the parameter α ∈ (0, 1], and depending on the specific behavior of X, as α→ 0+ , the bound reduces to Jensen’s gap, while for α → 1 —, the right-hand side approaches a new non-zero and pathdependent bound. The inequality is strict for non-degenerate X. (When M(α) ≥ 0, the bound is two-sided, Ei(µ) ≤ E[Ei(X)] − M(α) ≤ E[Ei(X)].) The proof is based on order-monotonicity properties of fractional integrals. An equivalent formulation of the result in terms of Caputo derivatives is also given. An illustrative interpretation is discussed in the context of economic utility, where the resulting bounds may be viewed as capturing nonlocal averaging effects in convex (risk-seeking) utility evaluations.