Geodesic
Complex Oscillation Theory of Differential Polynomials
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 50 (2011) no. 1, pp. 43-52 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we investigate the relationship between small functions and differential polynomials $g_{f}(z)=d_{2}f^{\prime \prime }+d_{1}f^{\prime }+d_{0}f$, where $d_{0}(z)$, $d_{1}(z)$, $d_{2}(z)$ are entire functions that are not all equal to zero with $\rho (d_j)1$ $(j=0,1,2) $ generated by solutions of the differential equation $f^{\prime \prime }+A_{1}(z) e^{az}f^{\prime }+A_{0}(z) e^{bz}f=F$, where $a,b$ are complex numbers that satisfy $ab( a-b) \ne 0$ and $A_{j}( z) \lnot \equiv 0$ ($j=0,1$), $F(z) \lnot \equiv 0$ are entire functions such that $\max \left\lbrace \rho (A_j),\, j=0,1,\, \rho (F)\right\rbrace 1.$
In this paper, we investigate the relationship between small functions and differential polynomials $g_{f}(z)=d_{2}f^{\prime \prime }+d_{1}f^{\prime }+d_{0}f$, where $d_{0}(z)$, $d_{1}(z)$, $d_{2}(z)$ are entire functions that are not all equal to zero with $\rho (d_j)1$ $(j=0,1,2) $ generated by solutions of the differential equation $f^{\prime \prime }+A_{1}(z) e^{az}f^{\prime }+A_{0}(z) e^{bz}f=F$, where $a,b$ are complex numbers that satisfy $ab( a-b) \ne 0$ and $A_{j}( z) \lnot \equiv 0$ ($j=0,1$), $F(z) \lnot \equiv 0$ are entire functions such that $\max \left\lbrace \rho (A_j),\, j=0,1,\, \rho (F)\right\rbrace 1.$
Classification : 30D35, 34M10
Keywords: linear differential equations; differential polynomials; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros 
@article{AUPO_2011_50_1_a4,
     author = {El Farissi, Abdallah and Bela{\"\i}di, Benharrat},
     title = {Complex {Oscillation} {Theory} of {Differential} {Polynomials}},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {43--52},
     year = {2011},
     volume = {50},
     number = {1},
     mrnumber = {2920698},
     zbl = {1244.34108},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2011_50_1_a4/}
}
TY  - JOUR
AU  - El Farissi, Abdallah
AU  - Belaïdi, Benharrat
TI  - Complex Oscillation Theory of Differential Polynomials
JO  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY  - 2011
SP  - 43
EP  - 52
VL  - 50
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/AUPO_2011_50_1_a4/
LA  - en
ID  - AUPO_2011_50_1_a4
ER  - 
%0 Journal Article
%A El Farissi, Abdallah
%A Belaïdi, Benharrat
%T Complex Oscillation Theory of Differential Polynomials
%J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
%D 2011
%P 43-52
%V 50
%N 1
%U http://geodesic.mathdoc.fr/item/AUPO_2011_50_1_a4/
%G en
%F AUPO_2011_50_1_a4
El Farissi, Abdallah; Belaïdi, Benharrat. Complex Oscillation Theory of Differential Polynomials. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 50 (2011) no. 1, pp. 43-52. http://geodesic.mathdoc.fr/item/AUPO_2011_50_1_a4/

[1] Amemiya, I., Ozawa, M.: Non-existence of finite order solutions of $w^{\prime \prime }+e^{-z}w^{\prime }+Q(z) w=0$. Hokkaido Math. J. 10 (1981), 1–17 (special issue). | MR

[2] Belaïdi, B.: Growth and oscillation theory of solutions of some linear differential equations. Mat. Vesnik 60, 4 (2008), 233–246. | MR

[3] Belaïdi, B.: Oscillation of fixed points of solutions of some linear differential equations. Acta Math. Univ. Comenian. (N.S.) 77, 2 (2008), 263–269. | MR | Zbl

[4] Belaïdi, B., El Farissi, A.: Relation between differential polynomials and small functions. Kyoto J. Math. Vol. 50, 2 (2010), 453–468. | DOI | MR | Zbl

[5] Chen, Z. X.: Zeros of meromorphic solutions of higher order linear differential equations. Analysis 14 (1994), 425–438. | MR | Zbl

[6] Chen, Z. X.: The fixed points and hyper-order of solutions of second order complex differential equations. (in Chinese), Acta Math. Sci. Ser. A Chin. Ed. 20, 3 (2000), 425–432. | MR | Zbl

[7] Chen, Z. X.: The growth of solutions of $f^{\prime \prime }+e^{-z}f^{\prime }+Q(z)f=0$ where the order $(Q) =1$. Sci. China Ser. A 45, 3 (2002), 290–300. | MR

[8] Chen, Z. X., Shon, K. H.: On the growth and fixed points of solutions of second order differential equations with meromorphic coefficients. Acta Math. Sin. (Engl. Ser.) 21, 4 (2005), 753–764. | DOI | MR | Zbl

[9] Frei, M.: Über die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten. Comment. Math. Helv. 35 (1961), 201–222. | DOI | MR | Zbl

[10] Frei, M.: Über die Subnormalen Lösungen der Differentialgleichung $w^{\prime \prime }+e^{-z}w^{\prime }+( Konst.) w=0$. Comment. Math. Helv. 36 (1961), 1–8. | DOI | MR | Zbl

[11] Gundersen, G. G.: On the question of whether $f^{\prime \prime }+e^{-z}f^{\prime }+B(z) f=0$ can admit a solution $f\lnot \equiv 0$ of finite order. Proc. Roy. Soc. Edinburgh Sect. A 102, 1-2 (1986), 9–17. | MR

[12] Hayman, W. K.: Meromorphic functions. Clarendon Press, Oxford, 1964. | MR | Zbl

[13] Laine, I., Rieppo, J.: Differential polynomials generated by linear differential equations. Complex Var. Theory Appl. 49, 12 (2004), 897–911. | DOI | MR | Zbl

[14] Langley, J. K.: On complex oscillation and a problem of Ozawa. Kodai Math. J. 9, 3 (1986), 430–439. | DOI | MR | Zbl

[15] Levin, B. Ya.: Lectures on entire functions. American Mathematical Society, Providence, RI, 1996 Translations of Mathematical Monographs, Vol. 150. | MR | Zbl

[16] Liu, M. S., Zhang, X. M.: Fixed points of meromorphic solutions of higher order Linear differential equations. Ann. Acad. Sci. Fenn. Math. 31, 1 (2006), 191–211. | MR | Zbl

[17] Nevanlinna, R.: Eindeutige analytische Funktionen. Die Grundlehren der mathematischen Wissenschaften Band 46, Zweite Auflage, Reprint, Springer-Verlag, Berlin–New York, 1974. | MR | Zbl

[18] Ozawa, M.: On a solution of $w^{\prime \prime }+e^{-z}w^{\prime }+( az+b) w=0$. Kodai Math. J. 3, 2 (1980), 295–309. | DOI | MR

[19] Wang, J., Yi, H. X.: Fixed points and hyper order of differential polynomials generated by solutions of differential equation. Complex Var. Theory Appl. 48, 1 (2003), 83–94. | DOI | MR | Zbl

[20] Wang, J., Laine, I.: Growth of solutions of second order linear differential equations. J. Math. Anal. Appl. 342, 1 (2008), 39–51. | DOI | MR | Zbl

[21] Zhang, Q. T., Yang, C. C.: The Fixed Points and Resolution Theory of Meromorphic Functions. Beijing University Press, Beijing, 1988 (in Chinese).