ON HERMITE-HADAMARD-TYPE CHARACTERIZATIONS OF HIGHER-ORDER DIFFERENTIAL INEQUALITIES

Let $ I $ be an open interval of $ \mathbb{R} $ and $ f: I\to \mathbb{R} $. It is well-known that $ f $ is convex in $ I $ if and only if, for all $ x, y\in I $ with $ x<y $, it holds that

$ \frac{1}{y-x}\int_x^y f(z)\, dz\leq \frac{f(x)+f(y)}{2}. $

The above inequality is known in the literature as Hermite-Hadamard inequality. In the first part of this paper, we extend the above result to the set of functions $ f\in C^{2n}(I) $ satisfying the higher-order differential inequality $ (-1)^nf^{(2n)}\leq 0\mbox{ in }I $. In particular, when $ f $ satisfies the above inequality with $ n=2 $, and $ f $ is convex, we obtain an interesting refinement of Hermite-Hadamard inequality. The second part of this paper is devoted to the study of sub-biharmonic functions, i.e., the set of functions $ f\in C^4(\Omega) $, $ \Omega $ is an open subset of $ \mathbb{R}^N $ ($ N\geq 2 $), satisfying $ \Delta^2f\leq 0\mbox{ in }\Omega $. Namely, a characterization of this set of functions is established. In particular, when $ f $ is subharmonic ($ \Delta f\geq 0 $ in $ \Omega $) and $ f $ is sub-biharmonic, an interesting refinement of Hermite-Hadamard inequality in higher dimension is obtained.