Approximation of the Jensen Gap for First Order Differentiable Functions (Contemporary Mathematics, 2025)
This article proposes two upper bounds for the gap of discrete and integral Jensen inequality for the functions f ∈ C[a, b]. First bound is obtained when |f′|q , q > 1 is convex and second is obtained when this function is concave. In the convex case, an example is discussed to authenticate the bound when |f′| is not convex. Accordingly and consequently, bounds for the gap of celebrated Hermite-Hadamard and well-known Hölder inequalities are deduced. Also, some estimates for the quasi-arithmetic and power means, and for some basic divergences are obtained from the main results.
Keywords: convex function; Jensen's inequality; information theory