Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-16T15:12:13.538Z Has data issue: false hasContentIssue false
  • English
  • Français

Three Theorems on the Growth of Entire Transcendental Solutions of Algebraic Differential Equations

Published online by Cambridge University Press:  20 November 2018

Sh. Strelitz
Sh. Strelitz*
Affiliation:
University of Haifa, Haifa, Israel
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

G. Polya [4] has posed the problem as to whether there are entire transcendental functions of order zero satisfying an algebraic differential equation with rational coefficients. G. Polya himself showed that this is impossible for a first order algebraic differential equation. The general problem is now completely solved. G. Valiron demonstrated an example of a third order algebraic differential equation with an entire transcendental solution of order zero (Theorem 1); V. V. Zimogljad (Theorem 2) proved that every entire transcendental solution of a second order algebraic differential equation is of a positive order. It seems to us expedient to bring these results all together. We give here a proof of Theorem 2 different from and in our view simpler than that of V. V. Zimogljad. Theorem 3 refines the results of G. Polya (and of others, see for example [10]) and establishes an exact lower bound for the order of an arbitrary entire transcendental solution satisfying a first order algebraic differential equation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 1110 - 1128
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Blumenthal, O., Sur la mode de croissance des fonctions entières, Bull. Soc. Math. 35 (1907), 213–232.Google Scholar
2. Blumenthal, O., Principes de la théorie des fonctions entières d'ordre infini (Paris, 1910).Google Scholar
3. Hille, E., Analytic function theory, Volume 2 (Chelsea Publ. Comp., New York, 1973).Google Scholar
4. Polya, G., Zur Untersuchung der Grôssenordnung ganzer Funktionen, die einer Differentialgleichung genugen, Acta Math. 42 (1920), 309–316.Google Scholar
5. Strelitz, Sh., On the maximal modulus of an analytic function, Usp. Math. Nauk 4 (1955), 153–160 (Russian).Google Scholar
6. Strelitz, Sh., Asymptotic properties of analytic solutions of differential equations, Vilnius (1972) (Russian).Google Scholar
7. Tannery, J. and Molk, J., Elements de la théorie des fonctions elliptiques, 1, Volumes 1, 2 (Chelsea Publ. Comp., New York, 1972).Google Scholar
8. Valiron, G., Sur les fonctions entières d'ordre nul et les équations différentielle, Bull. Soc. Math. 53 (1925), 34–42.Google Scholar
9. Valiron, G., Sur une jonction entière d'ordre nul qui est solution d'une equation différentielle algébrique, C. R. Acad. Sci., Paris 180 (1925), 571–572.Google Scholar
10. Valiron, G., Fonctions analytiques (Paris, 1954).Google Scholar
11. Zimogljad, V. V., On the growth of entire transcendental solutions of second order algebraic differential equations. Math. Sbor. 85(127), 2(6), (1971), 283–382 (Russian; there is an English translation).Google Scholar