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We prove that an integral Cauchy-Riemann inequality holds for any pair ofsmooth 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 inequalityto 4-tuples of functions, only valid on the $L^2$-orthogonal complement of asuitable nonzero finite dimensional space of functions. As a consequence weprove that 2-spheres are not $\Omega$-stable surfaces with parallel meancurvature in $\mathbb{R}^7$ for the associative calibration $\Omega$.

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