Cauchy-Riemann inequalities on 2-spheres of ${\mathbb{R}}^7$

We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere ${\mathbb{S}}^2$, and equality holds iff $f$ and $h$ are related ${\lambda }_1$-eigenfunctions. We extend such inequality to 4-tuples of functions, only valid on the $L^2$-orthogonal complement of a suitable nonzero finite dimensional space of functions. As a consequence we prove that 2-spheres are not $\mathrm{\Omega }$-stable surfaces with parallel mean curvature in ${\mathbb{R}}^7$ for the associative calibration $\mathrm{\Omega }$.