# Tag Info

Accepted

### Are entire functions “essentially” determined by their maximum modulus function?

This is a classical problem, but only partial results are available: MR3155684 Hayman, W. K.; Tyler, T. F.; White, D. J. The Blumenthal conjecture, in Complex Analysis and Dynamical Systems V, 149–157....
• 77.2k
Accepted

### Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?

This is not true. The optimal estimate from below for transcendental entire functions is $$\limsup_{z\to\infty}\frac{|z||f'(z)|}{\log|z|(1+|f(z)|^2)}=\infty,$$ and this is best possible, J. Clunie and ...
• 77.2k

### Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?

A concrete example is given by $f(z)=\cos{\sqrt{z}}$. Then $$g_f(z) = \frac{|\sin\sqrt{z}|}{2|\sqrt{z}|(1+|\cos^2{\sqrt{z}}|)} .$$ Obviously, this is small for large $|z|$ if $\sin{\sqrt{z}}$ is not ...
• 16.2k

### Ways to prove the fundamental theorem of algebra

Here is a variant of d'Alembert's argument using the minimum of $|p(z)|$. It has the advantage that it proves more generally the Gelfand-Mazur theorem (usually proved by complex analysis): Any Banach ...

### Explicit triples of isomorphic Riemann surfaces

A particular example is that of the Lawson surface $\xi_{g,1}$ of genus $g$. As defined here, it is a compact minimal surface in the 3-sphere, obtained by reflecting the solution of the Plateau ...

### Explicit triples of isomorphic Riemann surfaces

There are indeed very few pairs (except spheres with 3 or 4 singularities, or tori, and what can be obtained from them by finite coverings, where correspondence 2)-3) is completely explicit. See: H. P....

### Explicit triples of isomorphic Riemann surfaces

A classical, wonderful example in which is possible to explicitly see all the three descriptions is the Klein quartic.
Accepted

Accepted

### Estimate for an oscillatory integral of the first kind

Write $s=\eta t$ and note that $I(s,y)$ solves the 1D Schrödinger equation $iI_s-3I_{yy}=0$. Thus it satisfies the sharp estimate $|I(s,y)|\le c_0s^{-1/2}$ where $c_0$ is a multiple of $\int|I(0,y)|dy$...
• 7,498

### The monodromy in the proof of Little Picard via Klein's $J$

The usual proof of Picard's theorem along these lines used another modular function which is called $\lambda$ and which is related to $J$ by J=\frac{4}{27}\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\...
• 77.2k
1 vote

### Measure of preimage of Jordan disk under entire map

What you say is not correct, even for $e^z$. The preimage of the open disk $\{z:|z-1|<1\}$ under $e^z$ does not satisfy your condition. However, if you consider a closed disk which does not contain ...
• 77.2k
1 vote

### Holomorphic vector fields acting on Dolbeault cohomology

Klemyatin proved that this action is trivial if the corresponding ${\Bbb C}$-flow is compatible with some metric (hence can be extended to a compact torus action), https://arxiv.org/abs/1909.04075, (N....
• 7,548

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