While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this recent question for the background.

It should be useful to know the best approximation for some specific functions, one of which is the modulus function, $f(x)=|x|$. This function could be of great importance for estimating the distance between two complex variables. Thus it could be utilized as subroutines in solving other problems. My question is what is the best approximation to the modulus function in a given domain such as the unit disk $x\in \mathbb{C},|x|\leq 1$? Some references to this problem would be very welcome.

approximations, so it doesn't matter that $|\cdot|$ itself isn't holomorphic. $\endgroup$4more comments