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.



Growth comparision between an entire function and a related function
No, over the complex field the result is much weaker. Even this is not true:
$$\max\_{|z|\leq r}\left|\sum\_{n=0}^\infty a\_nb\_nz^n\right|\leq K\max\_{|z|\leq r}\left|\sum\_{n=0}^\infty a\_nz^n\right|,$$
with $K$ independent of $z$. I suppose that the simplest counter-example is $a\_n=e^{i\alpha\_n}/n!$ and $b\_n=e^{-i\alpha\_n}$ where $\alpha\_n=2\pi in^2\alpha$,
and $\alpha=(\sqrt{5}-1)/2$. This can be obtained by combining the results of P. Levy, Sur la croissance des fonctions entières, Bull. Soc. Math. France,
58 (1930) 29–59 et 127–149, and Theorem 4 in my paper with I. Ostrovskii,
On the "pits effect" of Littlewood and Offord, Bull. London Math. Soc., 39 (2007) 929-939.


<http://www.math.purdue.edu/~eremenko/dvi/ostr.pdf>


Combination of these results gives $e^r$ in the LHS and $o(e^r)$ in the RHS.


Alternatively, one can use independent random arguments $\alpha\_n$.
In this case, the LHS is $e^r$ while the RHS is
$$\max\_{|z|\leq r}\left|\sum\_{n=0}^\infty e^{2\pi i\alpha\_n}z^n/n!\right|\approx
r^{-1/4}e^r\sqrt{\log r},$$
P. Erdos and A. Renyi, On random entire functions, Zastosowania Matematyki (Applicationes mathematicae) 10 (1969) 47-55. Here $\approx$ means that the ratio
is between positive constants.


EDIT: For recent results on the subject see
Filevich, On Influence of the Arguments of Coefficients of a Power Series Expansion of an Entire Function on the Growth of the Maximum of Its Modulus,
Siberian math. J., 44 (2003), 3, 529-538.


