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

Geometry Inequality(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 seriesequals what? I showed an easy way to avoid calculus, but with great calculation in “Sum of harmonic series“.

Reproduce KernelSolving a problem with background in Reproduce Kernel from Ahlfors’

. It is pretty easy, but with interesting properties. See “Feel Reproduce Kernel“.Complex Analysis

Cheng-Yau gradient estimateUsing a few tricks to acquire the gradient estimate on the geodesic ball , the process is kind of boring but elementary, however, the method is much more interesting. See “Review Yau” and Cheng_Yau_Gradient_Estimate.

Cartan TheoremWithout regard of the usual proof, we use the trick of

Riemann Mapping Theoremto make the problem trivial. However,Schwarz Lemmadoes not provide any clues on higher dimensional cases, thus we simply put the proof in . See “Cartan Theorem“.

Cartan Theorem(another proof)This time we use another approach to prove this theorem in case. The proof is kind of trivial in technique, but this method can be applied for higher dimensional space . See “Cartan Theorem Note 2.”

Koebe TheoremThis theorem illustrates the property around origin point for univalent functions . See “Koebe Theorem Note 3”.

Liapunov StabilityIn 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 , with is uniformly continuous. I think there will be some similar result if the problem turns out to be .(implicit scheme) See “Liapunov stability.”

Bilinear Property of Harmonic FunctionThis 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“.