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An example is easy. Denote $z=a^{-(s-1)}$ and take the logarithm. Then $$ \log f(z)=\sum_n a_n a^{-n}z^n+\sum_n a_n [\log(1+a^{-n}z^n)-a^{-n}z^n]\,. $$ Assume that $a_na^{-n}=\frac 1n+O(b^{-n})$ wit …

Looks like we are closing the question anyway, so I'll just provide a counterexample quickly before the final vote is cast. If you think a bit of what is asked and what the natural freedoms and scal …

It is easy to design a function like $z/(1-az)$ with small $a$ that maps the circle to a nice domain whose closure does not contain $-1/{a_2}$. Now take this domain and grow a blob that contains $-1/{ …

It is true for $k=1,2$ but not for $k\ge 3$. Write $f=g^2$ near $0$ and let $g(z)=\sum_{m\ge 0}a_m z^m$. Then $|f|(z)=\overline{g(z)}g(z)$, so we want (up to $k!$ on both sides) that $$ \left|\sum_{ …

Let's start with the simple reduction. Notice that $f(\Delta)$ and $g(\Delta)$ are connected open sets containing small disks near the origin, so if one of them is unbounded, $f(\Delta)g(\Delta)=\math …

There is, indeed, a grain of truth in your conjecture. One possible formalization of it is as follows. Suppose that there is a sequence of analytic in the unit disk $\mathbb D$ functions $f_n$ such th …

It took me some time to find a solution that satisfies both requirements: a) If should be based on the power series expansion b) It should use no tools heavier than contour integration. So, let $g( …

It can get arbitrarily ugly. Indeed, approximate $1/z$ by a polynomial $p$ in the domain $K\subset\mathbb D$ whose complement is connected but goes from $0$ to the boundary along a long winding narrow …

The answer is "Quite often yes, but the error terms are seldom as good as in the meromorphic case". The reason the asymptotics for the meromorphic functions works is that we know the exact coefficient …

First of all, notice that if $D$ is a compact and $\Gamma$ is a compact Jordan curve that starts on the boundary of $D$ but otherwise lies in the infinite component of the open complement of $D$, then …

Now, since you call erfc(1) "a closed form expression", I should confess I do not understand the rules of this game. What's the big difference between $\int_1^\infty e^{-x^2/2} dx$ and the original in …

In general it is impossible. We can take the square root, map the circle conformally to the half-plane, and arrive at the following problem: given any nonnegative $f\in L^2(\mu)$ with finite logarithm …

OK, here are full details. Hope that I don't say any nonsense. The first observation is that $x\mapsto \frac{x+1}{x-1}$ maps the unit disk $|x|\le 1$ to the left half-plane $z\le 0$. You can see it by …

Take your favorite $C^\infty$ function $F(x)$ on the line that is not analytic and write $F(x)=\sum_{k\ge 1}P_k(x)$ where $P_k$ are polynomials of degree $k$ such that for all $m\ge 0, d>0$, we have $ …

OK, here is (I hope) a proof. I prefer to consider the integral $\int_{\mathbb D}\left|\frac 1{z-a}+\frac 1{z-\bar a}\right|\,d\mu(z)$ where $a=e^{i\theta}, \theta\in(0,\pi/2)$. This is just a one-par …