Regular growth of various characteristics of entire functions of order zero
This article is a continuation of the research of the first author which extended the theory of B. Ya. Levin and A. Pflüger of entire functions of completely regular growth of positive order to entire functions of order zero. In [Math. Notes 63, No. 2, 196--208 (1998; Zbl 0915.30026); translation from Mat. Zametki 63, No. 2, 172--182 (1998)], the first author introduced the notion of strongly regular growth for entire functions \(f\) of order zero. There the author considered the function \(r^{\lambda(r)}\), where \(\lambda(r)\) is the zero proximate order of the counting function \(n(r)\) of the zeros of \(f\) such that \(r^{\lambda(r)}\uparrow+\infty\) as \(r\to+\infty\). It was shown that the existence of the angular density of the zeros of an entire function \(f\) of order zero with respect to the function \(r^{\lambda(r)}\) implies the strongly regular growth of \(f\), and if the zeros of \(f\) are located on a finite system of rays, then also the converse statement is valid. In the general case, the converse statement does not hold. Sufficient conditions under which the strongly regular growth of an entire function of order zero implies the existence of the angular density of its zeros were obtained by the first author [Math. Notes 73, No. 5, 656--661 (2003; Zbl 1126.30021); translation from Mat. Zametki 73, No. 5, 698--703 (2003)]. In the present paper, the authors study the relationship between the strongly regular growth of an entire function \(f\) of zero order, the existence of the angular density of its zeros, the behavior of the Fourier coefficients of \(\log f\), and the regular growth of \(\log|f|\) and the argument of \(f\) in the \(L_p[0; 2\pi]\)-metric, \(p\geq1\). Let \(L\) be the class of continuously differentiable growth functions \(v\) on \(\mathbb{R}_+\) for which \(rv'(r)/v(r)\to0\) as \(r\to+\infty\). Let \(\mathcal{H}_0\) be the class of entire functions of order zero and let \(\mathcal{H}_0(v)\subset\mathcal{H}_0\) be the class of functions of strongly regular growth with respect to \(v(r)\), \(v\in L\). In particular, it is proved that if \(f\in\mathcal{H}_0\), \(v\in L\), and the zeros of \(f\) have an angular \(v\)-density, then \(f\in\mathcal{H}_0(v)\). A formula for the indicator of the function \(f\) is also obtained.
scientific article; zbMATH DE number 6682247
scientific article