# Retrospect of Mathematics

Every elementary mathematics branch has a profound background behind it. Some easy tricks are also operational in advanced divisions.

• Geometry Inequality

$\displaystyle\sum{\frac{1}{A}}\ge\frac{9}{2\pi}\sqrt{\frac{R}{2r}}$  (V.Mascioni),

I gave a simple proof to this geometry inequality when I was at high school. The proof is in “A geometry inequality“.

• Summation of series

$\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{3k-2}$ equals what? I showed an easy way to avoid calculus, but with great calculation in “Sum of harmonic series“.

• Reproduce Kernel

Solving a problem with background in Reproduce Kernel from Ahlfors’ Complex Analysis. It is pretty easy, but with interesting properties. See “Feel Reproduce Kernel“.

Using a few tricks to acquire the gradient estimate on the geodesic ball $B(R)$, the process is kind of boring but elementary, however, the method is much more interesting. See “Review Yau” and Cheng_Yau_Gradient_Estimate.

• Cartan Theorem

Without regard of the usual proof, we use the trick of Riemann Mapping Theorem to make the problem trivial. However, Schwarz Lemma does not provide any clues on higher dimensional cases, thus we simply put the proof in $\mathbb{C}$. See “Cartan Theorem“.

• Cartan Theorem(another proof)

This time we use another approach to prove this theorem in $\mathbb{C}^1$ case. The proof is kind of trivial in technique, but this method can be applied for higher dimensional space $\mathbb{C}^n$. See “Cartan Theorem Note 2.

• Koebe Theorem

This theorem illustrates the property around origin point for univalent functions $\mathcal{S}$. See “Koebe Theorem Note 3”.

In Numerical Mathematics, for first order ODE, Liapunov stability is known as zero-stability, which means zero is contained in absolute stable region. I generalized the problem to $y^{(n)}(t)=f(t,y)$, with $f(t,y)$ is uniformly continuous.  I think there will be some similar result if the problem turns out to be $y^{(n)}(t)=f(t,y,y',y'',\dots,y^{(n)})$.(implicit scheme) See “Liapunov stability.

This is one exercise on Gilbarg’s textbook for PDE beginner. And I like this theorem and its generalized form in the reference. The proof won’t be complicated if Poisson Kernel is involved, maybe it is a little bit annoying with the sub-index, however, noting that it is homogeneous is the most important part. See the proof here “Bilinear Harmonic Fucntion“.