Intereting Posts

Proving that the set of limit points of a set is closed
Ideal of polynomials in $k$ vanishing at a point $p$ is $(X_1 – p_1, …,X_n – p_n)$
Combining Two 3D Rotations
If $n_j = p_1\cdot \ldots \cdot p_t – \frac{p_1\cdot \ldots \cdot p_t}{p_j}$, then $\phi(n_j)=\phi(n_k)$ for $1 \leq j,k \leq t$
Countable union of sets of cardinality $c$ has cardinality $c$
Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $
Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.
Modules over commutative rings
Continuity of $\arg (z)$
Closed form for integral of inverse hyperbolic function in terms of ${_4F_3}$
How does Cantor's diagonal argument work?
How best to explain the $\sqrt{2\pi n}$ term in Stirling's?
Limit of Zeta function
Good books on conic section.
Help to understand material implication

I want to prove the following (exercise from Ahlfors’ text):

Prove that a subharmonic function remains subharmonic if the independent variable is subjected to a conformal mapping.

Here is my attempt, please tell me if it’s correct.

- How to integrate $ \int_0^\infty \sin x \cdot x ^{-1/3} dx$ (using Gamma function)
- Formula for calculating residue at a simple pole.
- How to compute the infinite tower of the complex number $i$, that is$ ^{\infty}i$
- problems with singularity $0$ of $\int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz$.
- Entire function having simple zero at the Gaussian integers
- Convergence of Taylor Series

Let $v:\Omega \subset \mathbb C \to \mathbb R$ be a subharmonic function, and let $f: \Omega \to \Omega$ be some conformal map. Suppose that $v \circ f$ is not subharmonic in $\Omega$. Then, there exist a point $z_0 \in \Omega$, a harmonic function $u_0: \Omega_0 \subset \Omega \to \mathbb R$ where $z_0 \in \Omega_0$, such that $v \circ f-u_0$ takes its maximal value in $\overline{\Omega_0}$ at $z_0$ without reducing to a constant. Since $f'(z_0) \neq 0$ we may restrict the domain $\Omega_0$ into some disk $\Delta_0=\Delta_0(z_0,r)$ such that the restricted $f$ results in a biholomorphism.

For $z \in \Delta_0$ we may write $$v \circ f-u_0=v \circ f-u_0 \circ f^{-1} \circ f, $$

and since $\Delta_0$ is simply connected, $u_0$ is the real part of some holomorphic function $f_0$, and thus $u:=u_0 \circ f^{-1}= \Re (f_0 \circ f^{-1})$ is harmonic as well.

Plugging in $f(z)=w$ we see that $$v(w)-u(w) $$ has its maximal value in $\overline{f(\Delta_0)}$ at the (interior) point $w_0=f(z_0)$ without reducing to a constant.

This is in contradiction with the subharmonicity of $v$.

QED (?)

- Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)
- Proving an Entire Function is a Polynomial
- Taylor series expansion of $\log$ about $z=1$ (different branches)
- $f: \Omega \rightarrow \Omega$ holomorphic, $f(0) = 0$, $f'(0) = 1$ implies $f(z) = z$
- Branch cut and $\log(z)$ derivative
- Holomorphic function with zero derivative is constant on an open connected set
- Evaluation of $\int_0^\infty \frac{x^2}{1+x^5} \mathrm{d} x$ by contour integration
- Why are there no discrete zero sets of a polynomial in two complex variables?
- Evaluating real integral using residue calculus: why different results?
- What is wrong with this fake proof $e^i = 1$?

Your proof is basically correct. Once you have the characterization of subharmonic functions in terms of harmonic, it follows quite abstractly that subharmonicity is preserved by whatever transformations preserve harmonicity.

That said, I don’t think that Ahlfors had in mind only conformal automorphisms of $\Omega$. In general $f$ can be a conformal map between different domains, and you should account for that in your proof.

It’s worth pointing out that the composition $v\circ f$ is subharmonic as long as $v$ is subharmonic and $f$ is holomorphic (not necessarily invertible). The proof goes along the same lines, because $h\circ f$ is harmonic whenever $h$ is. Alternatively, one can reduce to smooth $v$ by approximation and compute the Laplacian of composition explicitly.

- Matrix representation of the adjoint of an operator, the same as the complex conjugate of the transpose of that operator?
- roots of $f(z)=z^4+8z^3+3z^2+8z+3=0$ in the right half plane
- Find the ordinary generating function $h(z)$ for a Gambler's Ruin variation.
- How many subsets of size $n+1$ can we have so no two of them have intersection of size $n$
- Interesting GRE problem
- A Book for abstract Algebra
- Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”
- Alternating roots of $f(x) = \exp(x) \sin(x) -1$ and $\exp(x)\cos(x) +1$
- Product of spaces is a manifold with boundary. What can be said about the spaces themselves?
- $f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.
- How to prove that the exponential function is the inverse of the natural logarithm by power series definition alone
- Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_{q^n}$?
- Can we prove directly that $M_t$ is a martingale
- Speed of convergence of Riemann sums
- Is it possible to solve the Zebra Puzzle/Einstein's Riddle using pure math?