30 votes
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....
user avatar
16 votes
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 ...
user avatar
15 votes

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 ...
user avatar
11 votes

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 ...
8 votes

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 ...
7 votes

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....
6 votes

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.
5 votes
Accepted

For estimation on the integral $g(t)=\int_{-t}^{t}\left\vert\sum_{k=1}^Ne^{ikx}\right\rvert^2dx$ for small $t>0$

Using the formula for the sum of the first $n$ terms of a geometric series, we have $$|f(x)|^2=\frac{\sin^2(Nx/2)}{\sin^2(x/2)}$$ and hence for $t\downarrow 0$ $$g(t)=2\int_0^t |f(x)|^2\,dx =2\int_0^t ...
user avatar
4 votes

Is the proof in "On Hilbert’s 8th Problem" published on Brazilian Journal of Probability and Statistics correct?

According to MR it has been retracted by the author, see Braz. J. Probab. Stat. 33 (2019). See also the MathReview for details on the proof.
user avatar
4 votes

Construction of holomorphic function

Such a function does not exist, because the constant value $1$ of $|f(z)|^2$ on the circle $\{z\in\mathbb C\colon|z|^2=1/2\}$ is less than the value $e^{1/4}$ of $|f(z)|^2$ at the center $z=0$ of the ...
user avatar
4 votes

Zeros of hypergeometric functions with complex variables

Much of the literature addresses the case that $a=-n$ is a negative integer, see Real zeros of $2F1$ hypergeometric polynomials (2013) and Zeros of the hypergeometric polynomial $F(-n,b;c;z)$ (2008). ...
user avatar
3 votes

Zeros of hypergeometric functions with complex variables

The hypergeometric function is not a function in the usual meaning of this word: it is multivalued. For real parameters $a,b,c$, the question about real zeros of real branches was investigated by ...
user avatar
3 votes
Accepted

A question about complex Laplacian on compact Hermitian manifolds

For your first question, note that (let $\omega$ be the Hermitian form) $$\int_M\Delta_c(f) \omega^n=\int_M(\mathrm{tr}_\omega \sqrt{-1}\partial \overline{\partial}f)\omega^n=n\int_M \sqrt{-1}\partial ...
user avatar
2 votes
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 $\int|I(0,y)|dy$...
user avatar
2 votes

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-\...
user avatar
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 ...
user avatar
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....
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible