On the Polya Type of an Entire Function

Malyutin, K. G.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 1.

Abstract:
Let \(f\) be an entire function and let \(M(r,f)=\max\nolimits_{|z|=r}|f(z)|\) be maximum of the modulus of the function \(f\) in disk \(|z|\leq r\). The article considers the density functions of the maximum modulus of the function \(f\), which are calculated using the formulas  \( M(\alpha)=\varlimsup\nolimits_{r\to\infty}\frac{M(r+\alpha r,f)-M(r,f)}{r^{\rho(r)}},\quad \underline M(\alpha)=\varliminf\nolimits_{r\to\infty}\frac{M(r+\alpha r,f)-M(r,f)}{r^{\rho(r)}},\quad\alpha\geq 0, \) where \(\rho(r)\) is proximate order in the sense of Valiron, \(\lim\nolimits_{r\to+\infty}\rho(r)=\varrho\geq0\). It is proved, that \(M(\alpha)\) and \(\underline M(\alpha)\) are \(\varrho\)-semi-additive functions. The definition of the type \(\sigma_p(f)\) and the minimum type \(\underline\sigma_p(f)\) in the sense of Polia of the function \(f\) is introduced by the formulas \(\sigma_p(f)=\lim\nolimits_{\alpha\to+0}\frac{M(\alpha)}{\alpha},\quad \underline\sigma_p(f)=\lim\nolimits_{\alpha\to+0}\frac{\underline M(\alpha)}{\alpha}. \) These quantities give more information about the behavior of the function than its type and lower type in the classical sense. This definition is an extension of the concepts of maximum and minimum density of a sequence of positive numbers introduced by Polya, who proved their existence if the growth of the counting function of a sequence of numbers has normal type with respect to \(r\). The existence of the quantities \(\sigma_p(f)\) and \(\underline\sigma_p(f)\) is proved if the growth \(\ln|f|\) has type not higher than normal type with respect to \(r^{\rho(r)}\) in the classical sense, i. e. \(\ln M(r,f)\leq Kr^{\rho(r)}\) for some \(K>0\). Some properties of functions \(M(\alpha)\) and \(\underline M(\alpha)\) are considered.

Keywords: entire function, density function, semi-additive function, Polya theorem, maximum type, minimum type

+ References