Approximation of the Jensen Gap for First Order Differentiable Functions

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.