Let \Omega \subset \mathbb{R}^{n} be a bounded open set and take p' = \frac{p}{(p-1)}. Let a: \Omega \times \mathbb{R} \times \mathbb{R} be a Caratheodory function satisfying |a(x,s,\xi)| \leq k(x) + \beta(|s|^{p-1} + |\xi|^{p-1}) for almost every x \in \Omega, for every (s, \xi) \in \mathbb{R} \times \mathbb{R}^{n} and for some k \in L^{p^{'}}(\Omega), \beta \geq 0, p > 1.

If we further assume that x_{0} is a Lebesgue point of a and let s \in \mathbb{R}, \xi, \eta \in \mathbb{R}^{n} ( \xi \neq \eta) be such that \langle a(x_{0},s,\xi)-a(x_{0},s,\eta); \xi-\eta \rangle < 0. Then how does it follow that since x_{0} is a Lebesgue point of a gives us

\lim\limits_{R \rightarrow 0}\{\frac{1}{\text{meas}H_{R}(x_{0})}}\int_{H_{R}(  x_{0})}\langle a(x,s,\xi)-a(x,s,\eta); \xi-\eta \rangle dx \}< 0, where H_{R}(x_{0}) is Hypercube of \mathbb{R}^{n}?

Let me know if any further info is required. Thanks.