arXiv subjects
Publications and source records attributed to Abd Raouf Chouikha.
In this paper we are interested in isoperimetric inequalities for plane n-gons inscribed in a circle. More specifically, we will examine a conjecture provided by Paul Levy that we propose to solve in particular cases.
In this paper we are interested in isoperimetric inequalities for plane n-gons in relation with a old conjecture proposed by P. Levy.
We know that the algorithm of Theon of Smyrna (70-135 AD) made it possible to highlight fine frames of $\sqrt2$ by rationals. However, this same algorithm also applies to $\sqrt3$ and makes it possible to find the famous Archimedes inequalities. An interesting question is whether this very simple iterative method exposed by Theon was not known to the Greeks before him, notably by Archimedes or his contemporaries. Knowing that Theon of Smyrna had already compiled previous works. Note that this algorithm had also made it possible to open a breach towards the Pell-Fermat equations and continued fractions.
Infinite products expansions of the Weierstrass elliptic function \ $\wp(z) = \wp(z,1,τ)$\ and $n$-order transformations allow us to provide some modular relations.
In this paper, new sharp bounds for circular functions are proved. We provide some improvements of previous results by using infinite products, power series expansions and a generalisation of the so-called Bernoulli inequality. New proofs, refinements as well as new results are offered.