Refinements of the Jensen Inequality and Estimates of the Jensen Gap Based on Interval‐Valued Functions - Demir, İzzettin (Mathematical methods in the applied sciences, 2025)
ABSTRACT The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well‐known inequalities such as Hermite–Hadamard, Hölder, Minkowski, arithmetic‐geometric, and Young's inequalities. So, this inequality has become an influential concept in a wide range of scientific fields. Besides, interval analysis provides methods for managing uncertainty in data, making it possible to build mathematical and computer models of various deterministic real‐world phenomena. In this paper, taking into account all of these, we first present several refinements of the Jensen inequality for the left and right convex interval‐valued functions. We also provide examples with corresponding graphs to demonstrate these refinements more clearly. Next, we adopt a novel approach to derive several bounds for the Jensen gap in integral form using the gH‐differentiable interval valued functions as well as various related notions. Moreover, we obtain the proposed bounds by utilizing the renowned Ostrowski inequality. The fundamental benefit of the newly discovered inequalities is that they extend to many known inequalities in the literature, as discussed in this work.