Spirallike functions of hyperbolic complex and of dual complex variable (Q2762823)

Starting from a result obtained by \textit{H. Al-Amiri} and \textit{P. T. Mocanu} [J. Math. Anal. Appl. 80, 387-392 (1981; Zbl 0462.30037)], the author studies some function classes of logarithmic spirals, written in hyperbolic and dual complex variables. The hyperbolic and dual complex numbers are two limit cases of the known complex numbers \(z=x+iy\), namely \(z=x+hy\) and \(z=x+dy\), respectively, where \(h^2=1\) and \(D^2=0\). In the first part of the paper he presents some properties of these nonclassical numbers, similar to the classical magnitudes \(\|z\|\), \(\arg z\), \(e^z\), \(\log z\) and others, and some differential relations for the hyperbolic and dual complex numbers, very usefull for the second part of the paper. Here, he begins with the definition of the spirallike functions in the complex plane: \(\Im(e^{i\gamma}\log w(t))=\text{const.}\), where \(\|\gamma\|<\pi/2\), \(w(t)=u(t)+iv(T)\), \(t\in\mathbb{R}\), which he writes in hyperbolic and dual complex variables. For these spirallike functions there are established some differential equivalence relations both in the hyperbolic and in the dual complex cases. A continuation of this paper is under print in the journal ''Mathematica''.