A,B,C and D vertices of a rectangle if we define a function in this region as product of the distance from $z$ to points show max f at the sides

Question

$A,B,C$ and $C$ vertices of a rectangle on the plane we define a function on the region enclosed by the rectangle by $f(z)= |z-A||z-B||z-C||z-D|$ prove that the maximum value of $f$ attained on the boundary. I know I need to use the maximum modulus theorem but how to prove that $f$ is holomorphic? Or is there any other way to prove that? Any hints?

Answer 1

The function $$ f(z)= |z-A||z-B||z-C||z-D| $$ is not holomorphic (any real-valued holomorphic function is necessarily constant). But the function $$ F(z)= (z-A)(z-B)(z-C)(z-D) $$ is holomorphic, and the maximum modulus principle states that $|F(z)|$ attains its maximum on the boundary of the region enclosed by the rectangle.