Existence of an entire transcendental function of bounded \(l\)-index
Let \(L\) be a class of positive continuous functions on \([0, +\infty)\) and let \(l\in L\). An entire function \(f\) is called a function of bounded \(l\)-index if \[ {|f^{(n)}(z)|\over n! l^n(|z|)}\leq \max_{0\leq k\leq N} {|f^{(k)}(z)|\over k! l^k(|z|)} \] for some \(N\) and all \(n\geq 0\), \(z\in \mathbb{C}\). The authors prove that for every \(l\in L\) such that \(rl(r)\to \infty\) \((r\to \infty)\), there exists an entire function of bounded \(l\)-index. Together with earlier investigations this gives the following assertion: Let a function \(l\in L\) be non-increasing. Then an entire transcendental function of bounded \(l\)-index exists if and only if \(rl(r)\to \infty\) \((r\to \infty)\).
scientific article; zbMATH DE number 846465