Suppose $f(z)=a\_0+a\_1z+\cdots+a\_nz^n+\cdots$ is defined in the unit disk and $\|f\|\_{\infty}\leq 1.$ Lets form another series $g$ by interchanging $a\_1$ and $a\_k$ i.e. $g(z)=a\_0+a\_kz+\cdots+a\_1z^k+\cdots$. Is $g$ of norm less than or equal to one?if that is not the case can you provide a counterexample?



Norm of swapped power series in the unit disk
For simplicity, let us assume that the radius of convergence of $f$ is strictly larger than one.
By the maximum principle the absolute value of the analytic function $f:D\to\mathbb C$ obtains its maximum at the boundary $\partial D=S^1$.
Therefore we are interested in the function $g:[0,2\pi]\to\mathbb C$,
$$
h(\phi)
=
\sum\_{n=0}^\infty a\_ne^{in\phi}.
$$
This is a Fourier series.
The norm of this function is given by
$$
|h(\phi)|^2
=
\sum\_{n,m}a\_na\_m^\*e^{i(n-m)\phi}
.
$$
If the coefficients are real, we get
$$
|h(\phi)|^2
=
\sum\_{n,m}a\_na\_m\cos((n-m)\phi)
.
$$
The question is now whether the maximum of the function $\phi\mapsto|h(\phi)|^2$ changes when we change the coefficients.


Let us take a concrete example: $f(z)=z+z^2-z^4$.
Now if $k=4$, the other function is $g(z)=\tilde f(z)=-z+z^2+z^4$.
Now
$$
|h(\phi)|^2=3+2\cos(\phi)-2\cos(2\phi)-2\cos(3\phi)
$$
and the corresponding norm for $\tilde f$ is
$$
|\tilde h(\phi)|^2=3-2\cos(\phi)+2\cos(2\phi)-2\cos(3\phi).
$$
If you plot these functions, you see that
$$
\max\_{\bar D}|f|^2=\max\_{\phi\in[0,2\pi]}|h(\phi)|^2=5
$$
but
$$
\max\_{\bar D}|\tilde f|^2=\max\_{\phi\in[0,2\pi]}|\tilde h(\phi)|^2=6.
$$
(One can of course prove these by hand as well if needed.)
Thus the function $\frac1{\sqrt5}f$ with $k=4$ gives a counterexample.


