Central gradient, product, product rule

For $(u_i)_{i\in I}$ define the central gradient by$$(D_x^c u)_i = \frac{u_{i+1}-u_{i-1}}{2\Delta x}\textrm{ for all }i\in I.$$Now, for $u=(u_i)_{i\in I}$ and $\bar{u}=(\bar{u}_i)_{i\in I}$, I am wondering what$$(D_x^c D_x^c u)_i\cdot (D_x^c D_x^c\bar{u})_i$$looks like and if there is some kind of product rule for it.

I get$$(D_x^c D_x^c u)_i\cdot (D_x^c D_x^c\bar{u})_i=\frac{1}{16\Delta x^4}(u_{i+2}\bar{u}_{i+2}-u_{i+2}\bar{u}_i+u_{i+2}\bar{u}_{i-2}-u_i\bar{u}_{i+2}+u_i\bar{u}_i-u_i\bar{u}_{i-2}+u_{i-2}\bar{u}_{i-2}-u_{i-2}\bar{u}_i+u_{i-2}\bar{u}_{i-2})$$

Does there hold something like$$(D_x^c D_x^c u)_i\cdot (D_x^c D_x^c\bar{u})_i\leq (D_x^c(D_x^c(u+\bar{u}))_i$$