Analytic functions of bounded index (Q2740464)

Let \(l\) be a positive and continuous function on \([0,\infty)\). An entire function \(f\) is a function of bounded \(l\)-index if there exists \(n\in \mathbb{N}\) such that NEWLINE\[NEWLINE\left|{f^{(n)}(z)\over n!}\right|l^{-n} \bigl(|z|\bigr) \leq\max\left \{\left|{f^{(k)}(z)\over k!}\right |l^{-k} \bigl(|z|\bigr): 0\leq k\leq\mathbb{N}\right\}NEWLINE\]NEWLINE for all \(n\in\mathbb{N}\) and \(z\in \mathbb{C}\). This monograph is devoted to study the functions of bounded \(l\)-index. The book consists of 7 chapters. In Chapter 1-4 the author presents various facts of the behavior and properties of such functions. In Part 5 there are investigated properties of analytic solutions of linear differential equations that coefficienties are functions of bounded \(l\)-index. The question of existence of an entire function of bounded \(l\)-index is considered in the next chapter. And in the last part of this book there are studied the functions of bounded index, i.e. the case \(l\equiv 1\).