Some integral equalities for special classes of polynomials (Q2901705)

Let \(L,n\in {\mathbb N}\), \(L<\frac{n+1}{2}\), and let \(f(z)=\sum\limits_{k=0}^{L-1}\alpha_kz^k+ \sum\limits_{k=0}^{L-1}\beta_k{\overline{z}}^k\) be a polynomial. Then \(\underset{P_n(z,r)}{\int\int}(\zeta-z)^{n-L}f(\zeta)d\xi d\eta=0\) holds for every regular \(n\)-gon \(p_n(z,r)\) centered at the point \(z\) with inscribed radius \(r\). (\(P_n(z,r)\) is a closed finite region bounded by \(p_n(z,r)\).) Another gives a connection between integrals \(I=\underset{P_n(z,r)}{\int\int}(\zeta-z)^{s}f(\zeta)d\xi d\eta\) and functions of the form \(\sum\limits_{k=0}^h\sum\limits_{l=0}^{m-1}c_{k,l}z^k \overline{z}^l\), where \(n,m,h\in {\mathbb N}\), \(0\leq h< n-s\), \(0\leq s\leq m-1\), and \(c_{k,l}\) are some constants. More precisely, NEWLINE\[NEWLINEI=\sum\limits_{p=s}^{h+s}\frac{nr^{2p+2}\lambda_p}{(2p+2)(p-s)!p!}\left(\frac{\partial}{\partial z}\right)^{p-s}\left(\frac{\partial}{\partial \overline{z}}\right)^{p} f(z)NEWLINE\]NEWLINE holds for some \(\lambda_p\) depending only on \(n\) and \(p\).