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Here's another elementary proof, again assuming one has checked that the integral converges absolutely despite the singularity at $t=0$. Let $I$ be the average of the $2\pi$-periodic function $\log \l …

The conjecture is easily seen to be true for $n<3$. We give a counterexample for $n=3$. Let $p_i = z^2 - \omega_i$ where the $\omega_i$ are the cube roots of unity. These are linearly dependent becaus …

[expanding on my comment to convert it to an answer] An example is $e^j \varphi$ where $j$ is the $j$-invariant and $\varphi$ is any nonzero form of weight $k$. In general a holomorphic (or meromorphi …

Counterexample to the first question ("is $A$ an algebra of functions?"): Let $M$ be a vertical strip such as {$x + iy : 0 < x < 1$}, and define $f_1,f_2$ as the restriction to $M$ of the entire func …

One expects there to be no such $a_n$ in general, because the "typical" entire functions is surjective (those that aren't are of the special form $z \mapsto c + \exp g(z)$). An explicit example is $ …

No. Either condition implies $|\,f'(z)| = 1$ for all $z$ in the domain of $\,f$, which in turn implies that $\,f'(z)$ is locally constant (for instance using the open mapping theorem) and thus that $ …

Robert Israel's calculations suggest that if $0<a<b$ then $f(z)$ has infinitely many zeros near but not on the imaginary axis, at least on certain rational rays $b=ra$ such as Robert's example with $ …

For the record, one can prove the product formula for the sine without complex analysis (and without the Gamma function), from which (as David Speyer noted) one can recover $\sum_{i \in \bf Z} (x+i)^{ …

In general, once you've proven an inequality like this in ${\bf R}$ it holds automatically in any Euclidean space (including ${\bf C}$) by averaging over projections. ("Inequality like this" = inequa …

That's not true even for power series with real coefficients. Let $$ D(s) = \sqrt{1-2^{-s}} = 1 - \frac12 2^{-s} - \frac18 4^{-s} - \frac1{16} 8^{-s} - \frac5{128} 16^{-s} - \cdots . $$ Then $P(s) = …

It's been noted already in the comments that the problem is still too easy as stated, because one can easily find functions $f(z) = \sum_{n=0}^\infty c_n z^n$ such as $f(z) = \sin(1 / (r-z))$, with in …

At the risk that I'm doing somebody's homework... Yes, $R_{m,n}[a,b]$ is complete with respect to the uniform metric $\left\| \cdot \right\|$. Let $f \in {\cal C}([a,b])$ be a uniform limit of ratio …

No, such an inequality need not hold: one can construct $f$ of exponential type and a sequence $\{a_n\}$ of real numbers such that $$ \frac1{f(a_n)} \int_{a_n - \frac12}^{a_n - \frac12} \left|\phant …

Alexandre Eremenko already showed that for each $n>1$ the function $$ S_n(x) := \frac1{n!} \sum_{k=1}^n {n \choose k} (-1)^{n-k} k^x $$ has infinitely many zeros $x \in {\bf C}$. One can still say mor …

Over ${\bf C}$, An easy counterexample to question 3 is $f(x) = x^2$, $g(x) = cx^2$ where $c$ is a nontrivial cube root of unity. Then $f(f(x)) = g(g(x)) = x^4$ but $f$ and $g$ do not commute. There a …