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 $\Omega$-stable surfaces with parallel mean curvature in $\mathbb{R}^7$ for the associative calibration $\Omega$.