Functions of coefficients of univalent funtions

Question

Suppose we have a univalent function on a disk $f(z):=\rho(a_1z+a_2z^2+\cdots)$ ($\rho>0$). Let $n\geq 1$. Let $I=(i_1, i_2, \ldots i_n)$ be a sequence of non-negative integers such that $weight(I):=\sum_k ki_k=n$. Denote $a^I:=a_1^{i_1}\cdots a_n^{i_n}$ and $\overline{a}^I:=\overline{a}_1^{i_1}\cdots \overline{a}_n^{i_n}$. Fix $d\geq 1$. Im trying to think of natural ways of thinking about elements of the form $$\rho^d \sum_{k=1}^Na^{I_k}\overline{a}^{J_k}$$ where $weight(I_k)=weight(J_k)+d$ for all $1\leq k\leq N$. This seems very difficult. Let's simplify the problem. Define $a(z)=a_1z+a_2z^2+\cdots$ and $$\big(\frac{d^m}{dz^m} a(z)\big)^q=\sum_{i} P_{i,m,q}(a_1,a_2,\cdots)z^i\quad (q\geq 0).$$ Is there a good way of thinking of $\rho^d P_{i,m,q} \overline{P_{i',m',q'}}$ ? It's difficult for me to say what I mean by "good". I'm just looking for different ideas.