arXiv subjects

Faruk F. Abi-Khuzam

Publications and source records attributed to Faruk F. Abi-Khuzam.

3 recordsLinked to original sources

arXiv2026

A Jacobian Group Structure on a Hyperbolic Pencil of circles and its Applications

Using Jacobian Elliptic functions, we introduce a novel parametrization of a hyperbolic pencil of coaxal circles which reveals a remarkable group structure on the pencil. The geometric properties of the group elements lead to a new proof of of the general Poncelet theorems, which in turn leads to a proof of the so called closure theorem. In particular we prove: if $T$ and $% D $ are members of the pencil, then an interscribed $n$-gon to $T$ and $D$ exists, if and only if $D$, the inside circle, is an element of order $n$ in the group.

math.CV↗

arXiv2022

Dimension Free Growth Results for Vector-Valued Functions of Several Complex Variables

Let f be an entire function of finite order less than 1. The maximum modulus M(r) of f and the counting function of the zeros N(r) are connected by a best possible growth inequality known as Valiron's Theorem: For functions subharmonic in d-dimensional Euclidean space, Hayman obtained a corresponding result with a best possible constant involving the dimension d. For the special case of an entire function on d-dimensional complex space, we obtain a corresponding dimension-free, best possible inequality.

math.CV↗

arXiv2019

Decay estimates connected with Toeplitzness of composition operators

In a higher dimensional version of an earlier conjecture of Nazarov and Shapiro, the truth of which would imply that any composition operator on the second Hardy space is weakly asymptotically Toeplitz, Shayya proved that the arithmetic means of the main diagonal entries of the matrix of coefficients connected with the conjecture, decay like 1/log N. But the proof does not extend to other diagonal entries. We prove that the arithmetic means of the entries along each diagonal decay like 1/(log N loglog N) uniformly.

math.FA↗