Let $z\_1,z\_2,\dots,z\_n\in\Bbb{C}$ be distinct and $w\_1,w\_2,\dots,w\_n\in\Bbb{C}$ be arbitrary. Suppose $f, g$ are two polynomials of degree less than $n$ such that
$$f(z\_j)=w\_j,\qquad g(z\_j)=\bar{w}\_j \qquad\text{for $1\leq j\leq n$}.$$
Define $\Omega(z)=\prod\_{j=1}^n(z-z\_j)$. The following puzzles me.



> 
> **Question 1.** Is this true?
>  $$\sum\_{k=1}^n\frac{\vert f^{\prime}(z\_k)\vert^2}{\vert \Omega^{\prime}(z\_k)\vert^2}\leq
> \sum\_{k=1}^n\frac{\vert g^{\prime}(z\_k)\vert^2}{\vert\Omega^{\prime}(z\_k)\vert^2}.$$
> 
> 
> 


**EDIT.** I'm sorry, one of the conditions was missing. To give credit, I'll leave the question as it stands and ask the correct one below.



> 
> **Question 2.** What if we insist that $f$ and $g$ have equal degrees?
> 
> 
> 



complex polynomials and inequalities
For $n\ge 3$, let's take $z\_j$, for $1\le j\le n$, be the $n$-th roots of unity, and $w\_j:=z\_j+4\bar{z\_j}$, so that the assumption are satisfied by $$f(z):=z+4z^{n-1}$$ $$g(z):=4z+z^{n-1}.$$ However, 
$$ |g'(z\_k)|=|4+(n-1) z\_k^{-2}| \le n+3<\phantom{Z} $$ $$ \phantom{ZZZZ} <4n-5\le|1+4(n-1) z\_k^{-2}|= |f'(z\_k)| $$ for any index $k$ in the sums. 


**Rmk.** For $n=2$, $f$ and $g$ are affine functions, with $|f'|=|g'|={|w\_1-w\_2|\over |z\_1-z\_2|}$, so that the claim is true. 


