Growth of Functions Analytic in
the Unit Disc which are Generating Functions of Pólya Frequency Sequences

M.T. Alzugaray

For any integer $r \geq 2$ and any positive number $\rho$, an example of the function $f(z)$ is constructed with the following properties: (i) $f(z)$ is analytic in the unit disc, (ii) $f(z)$ is a generating function of the Pólya frequency sequence of the order $r$, (iii) the order of the growth of $f(z)$ in the unit disc is equal to $\rho$.

1. Karlin S., Total Positivity. - Stanford: Stanford University Press, 1968.

2. Baker G.A., Graves-Morris P. Padé Approximants. - London: Addison-Wesley, 1981.

3. Aissen M., Edrei A., Schoenberg I.J., Whitney A. On the Generating Functions of Totally Positive Sequences // Journal d'Analyse Math. 1953. v.2. P.93-109.

4. Schoenberg I.J., On the Zeroes of Generating Functions of Multiple Positive Sequences and Functions // Annals of Math. 1955. P.447-471.

5. Katkova O.M., Ostrovskii I.V. Zero Sets of Entire Generating Functions of Pólya Frequency Sequences of Finite Order // Math.USSR - Izvestija 1990. v.35. P.101-112.

6. Wiman A. Über die Nullstellen der Funktionen $E_{\alpha}(x)$ // Acta Math. 1905. v.29. P.217-234.

7. Katkova O.M. On Indicators of Entire Functions of Finite Order with Multiple Positive Sequences of Coefficients (Russian) // Dep. VINITI 22.02.89. No.1179-B89.

8. Goldberg A.A., Ostrovskii I.V. Distribution of Values of Meromorphic Functions (Russian). – Moscow: Nauka, 1970.

9. Boas R., Entire Functions. – New York: Academic Press, 1954.

10. Levin B.Ja. Distribution of Zeros of Entire Functions. – Providence: Amer.Math.Soc., 1980.

11. Beuermann F. Wachtumsordnung, Koeffizientenwachstum und Nullstellendichte bei Potenzreihen mit endlichem Konvergenzkreis // Math.Z. 1931. Bd.33. S.98-108.

12. MacLane G.R. Asymptotic Values of Holomorphic Functions. – Houston: Rice University, 1963. Department of Mathematics, Kharkov State University, 4 Svobody sq., Kharkov 310077, Ukraine

	Growth of Functions Analytic in the Unit Disc which are Generating Functions of Pólya Frequency Sequences
Author	M.T. Alzugaray Department of Mathematics, Kharkov State University, 4 Svobody sq., Kharkov 310077, Ukraine
Abstract	For any integer $r \geq 2$ and any positive number $\rho$, an example of the function $f(z)$ is constructed with the following properties: (i) $f(z)$ is analytic in the unit disc, (ii) $f(z)$ is a generating function of the Pólya frequency sequence of the order $r$, (iii) the order of the growth of $f(z)$ in the unit disc is equal to $\rho$.
Keywords	analytic function, unit disc, generating function, Pólya frequency sequence
DOI	doi:10.30970/ms.4.1.13-18
Reference	1. Karlin S., Total Positivity. - Stanford: Stanford University Press, 1968. 2. Baker G.A., Graves-Morris P. Padé Approximants. - London: Addison-Wesley, 1981. 3. Aissen M., Edrei A., Schoenberg I.J., Whitney A. On the Generating Functions of Totally Positive Sequences // Journal d'Analyse Math. 1953. v.2. P.93-109. 4. Schoenberg I.J., On the Zeroes of Generating Functions of Multiple Positive Sequences and Functions // Annals of Math. 1955. P.447-471. 5. Katkova O.M., Ostrovskii I.V. Zero Sets of Entire Generating Functions of Pólya Frequency Sequences of Finite Order // Math.USSR - Izvestija 1990. v.35. P.101-112. 6. Wiman A. Über die Nullstellen der Funktionen $E_{\alpha}(x)$ // Acta Math. 1905. v.29. P.217-234. 7. Katkova O.M. On Indicators of Entire Functions of Finite Order with Multiple Positive Sequences of Coefficients (Russian) // Dep. VINITI 22.02.89. No.1179-B89. 8. Goldberg A.A., Ostrovskii I.V. Distribution of Values of Meromorphic Functions (Russian). – Moscow: Nauka, 1970. 9. Boas R., Entire Functions. – New York: Academic Press, 1954. 10. Levin B.Ja. Distribution of Zeros of Entire Functions. – Providence: Amer.Math.Soc., 1980. 11. Beuermann F. Wachtumsordnung, Koeffizientenwachstum und Nullstellendichte bei Potenzreihen mit endlichem Konvergenzkreis // Math.Z. 1931. Bd.33. S.98-108. 12. MacLane G.R. Asymptotic Values of Holomorphic Functions. – Houston: Rice University, 1963. Department of Mathematics, Kharkov State University, 4 Svobody sq., Kharkov 310077, Ukraine
Pages	13-18
Volume	4
Issue	1
Year	1995
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Growth of Functions Analytic in the Unit Disc which are Generating Functions of Pólya Frequency Sequences