Let $p$ be a prime number, $\mathbb C_p$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valution $v(x)=-\deg(x)$. Let $\sum_{n\ge0}a_nz^n$ be a power series that converge for all $z\in\mathbb C_p$. Schnirelmann's famous result assert that for every $r\in\mathbb R^+$, one has $\sup_{\substack{z\in\mathbb C_p\\\deg(z)\le r}}\limits\deg(f(z))=\sup_{n\in\mathbb N}\limits\{deg(a_n)+nr\}$. From that, we can deduce that for every sequence $(b_n)_n$ of $\mathbb C_p$ with $\deg(b_n)\le 0$, for the power series $$g(z)=\sum_{n\ge0}a_nb_nz^n,$$ one has $$\sup_{\substack{z\in\mathbb C_p\\\deg(z)\le r}}\limits\deg(g(z))\le \sup_{\substack{z\in\mathbb C_p\\\deg(z)\le r}}\limits\deg(f(z))$$

I wonder whether there exists such a result in characteristic $0$ case. Let $f(x)=\sum_{n\ge0}a_nx^n$ be an entire function on $\mathbb C$. Consider a sequence $(b_n)_n$ of complex numbers such that $0<|b_n|\le1$ for all $n\in\mathbb N$. Does one have $$\sup_{\substack{z\in\mathbb C\\|z|\le r}}|\sum_{n\ge0}a_nb_nz^n|\le \sup_{\substack{z\in\mathbb C\\|z|\le r}}|\sum_{n\ge0}a_nz^n|+K$$ where $K$ would be a constant depending on the sequence $(a_n)_n$ only.

Thanks in advance for any answers or hints.