On some criteria for completely regular growth of entire functions of exponential type
This paper investigates the relationship between the regularity of growth of an entire function of exponential type (EFET) \(\mathcal{L}(\lambda)\) with only simple zeros \(\lambda_k, \;k=1,2,\ldots.\) For EFET \(\mathcal{L}\) with real zeros the author obtains the growth regularity conditions in terms of the bounds from below for the absolute value of the derivative \(\mathcal{L^{\prime}}\) at points \(\lambda_k.\) These bounds have the form \[ \frac{1}{| {\mathcal{L}}^{\prime}(\lambda _k)| }=O(| \omega(\lambda_k)| ), \;k \to \infty, \] where \(\omega(\lambda)\) is an entire function of zero exponential type with some special properties. Further, the paper deals with the class \(L\) of EFET \(\mathcal{L}\) with only simple zeros \(\lambda _k, \;k=1,2,\ldots,\) satisfying the following condition \[ \limsup_{k \to \infty}\left\{ \frac{1}{| \lambda _k| }\ln \frac{1}{{\mathcal{L}}^{\prime}(| \lambda _k | ) }+h({\mathcal{L}}, \arg \lambda _k)\right\}\leq 0, \] where \(h(\mathcal{L},\theta)\) is the indicator of the function \({\mathcal{L}}, \;\theta \in [0; 2\pi ).\) Leont'ev posed the problem whether or not the function \({\mathcal{L}}\in L\) has a completely regular growth. This problem has not been solved. The author proves the following theorem. \textbf{Theorem.} The indicator diagram of an arbitrary function \({\mathcal{L}}\in L\) with only real zeros is a parallel to the imaginary axis segment . Moreover, if \(\mathcal{L}\) is an even function, then \(\mathcal{L}\) is a function of completely regular growth on the real axis and \( \lim _{k \to \infty}\frac{1}{| \lambda _k| }\ln \frac{1}{{\mathcal{L}}^{\prime}(| \lambda _k| ) }=0.\) Some sufficient conditions for a function \({\mathcal{L}}\in L\) with the positive indicator \(h({\mathcal{L}},\theta)\) to be a function of completely regular growth are obtained. For instance, it is proved that if the inverse to such function \(\mathcal{L}\) admits the representation \( \frac{1}{{\mathcal{L}}(\lambda)} =\sum_{k=1}^{\infty} \frac{a_k}{\lambda-\lambda_k}\) with \(\sum_{k=1}^{\infty}| a_k| < \infty,\) then \(\mathcal{L}\) is a function of completely regular growth. (This fact is an analog of one Krein's theorem).
scientific article; zbMATH DE number 5155503