Let $\mathbf{D}$ be the open unit disc in $\mathbf{C}$ and let $f,g:\mathbf{D}\to \mathbf{C}$ be holomorphic functions such that the real valued function $\vert f\vert^2+\vert g\vert^2$ is bounded from above by some real number $c$ (everywhere on $\mathbf{D}$).
Question. Can we bound $$\left\vert \frac{df}{dz} \right\vert^2(0)+ \left\vert \frac{dg}{dz}\right\vert^2(0)$$ from above in terms of $c$?
Answer. Yes! See answer below.
Question. Let $x\in \mathbf{D}$. Can we bound $$\left\vert \frac{df}{dz} \right\vert^2(x)+ \left\vert \frac{dg}{dz}\right\vert^2(x)$$ from above in terms of $c$ and the norm of $x$?
Differentiating the Cauchy's integral formula is enough: $$ |f'(0)| =\left| \frac{1}{2\pi i} \int_{|z|=r}\frac{f(z)}{z^2}\, dz\right|\le \frac{\sqrt c}{r^2}. $$ Here the radius $r\in(0,1)$ is arbitrary, so $|f'(0)|\le \sqrt c\;$ and $|f'(0)|^2+|g'(0)|^2\le 2c\ $.