A characterization of the exponential function by product
This paper concerns mostly entire functions f(z) of positive integral order p. Conditions are used, implying \(f(z)=e^{P(z)}\), where P(z) is a polynomial of degree p. The main condition is (let \(\omega =\exp (\pi i/p))\) that \(f(z)f(\omega z)...f(\omega^{2p-1}z)\) is bounded as \(z\to \infty\) along some Jordan curve. Especially, for \(p=1\), \(f(z)=Ae^{Bz}\) if \(f(r)f(-r)\) is bounded and - as extra condition - all zeros are negative. Other extra conditions used are, e.g., zero-free sectors, zero being a defective value, only real or only non-real zeros. A skilful use of integral inequalities is made.
scientific article; zbMATH DE number 3886230