Complex Oscillation Theory of Differential Polynomials

Abdallah El Farissi; Benharrat Belaïdi

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2011)

  • Volume: 50, Issue: 1, page 43-52
  • ISSN: 0231-9721

Abstract

top
In this paper, we investigate the relationship between small functions and differential polynomials g f ( z ) = d 2 f ' ' + d 1 f ' + d 0 f , where d 0 ( z ) , d 1 ( z ) , d 2 ( z ) are entire functions that are not all equal to zero with ρ ( d j ) < 1 ( j = 0 , 1 , 2 ) generated by solutions of the differential equation f ' ' + A 1 ( z ) e a z f ' + A 0 ( z ) e b z f = F , where a , b are complex numbers that satisfy a b ( a - b ) ≠ 0 and A j ( z ) ¬ ≡ 0 ( j = 0 , 1 ), F ( z ) ¬ ≡ 0 are entire functions such that max ρ ( A j ) , j = 0 , 1 , ρ ( F ) < 1 .

How to cite

top

El Farissi, Abdallah, and Belaïdi, Benharrat. "Complex Oscillation Theory of Differential Polynomials." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 50.1 (2011): 43-52. <http://eudml.org/doc/196959>.

@article{ElFarissi2011,
abstract = {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 $\max \left\lbrace \rho (A_j),\, j=0,1,\, \rho (F)\right\rbrace <1.$},
author = {El Farissi, Abdallah, Belaïdi, Benharrat},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {linear differential equations; differential polynomials; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros; linear differential equations; differential polynomials; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros},
language = {eng},
number = {1},
pages = {43-52},
publisher = {Palacký University Olomouc},
title = {Complex Oscillation Theory of Differential Polynomials},
url = {http://eudml.org/doc/196959},
volume = {50},
year = {2011},
}

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
PB - Palacký University Olomouc
VL - 50
IS - 1
SP - 43
EP - 52
AB - 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 $\max \left\lbrace \rho (A_j),\, j=0,1,\, \rho (F)\right\rbrace <1.$
LA - eng
KW - linear differential equations; differential polynomials; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros; linear differential equations; differential polynomials; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros
UR - http://eudml.org/doc/196959
ER -

References

top
  1. Amemiya, I., Ozawa, M., Non-existence of finite order solutions of w ' ' + e - z w ' + Q ( z ) w = 0 , Hokkaido Math. J. 10 (1981), 1–17 (special issue). (1981) MR0662294
  2. Belaïdi, B., Growth and oscillation theory of solutions of some linear differential equations, Mat. Vesnik 60, 4 (2008), 233–246. (2008) MR2465805
  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. (2008) Zbl1174.34528MR2489196
  4. Belaïdi, B., El Farissi, A., 10.1215/0023608X-2009-019, Kyoto J. Math. Vol. 50, 2 (2010), 453–468. (2010) Zbl1203.34148MR2666664DOI10.1215/0023608X-2009-019
  5. Chen, Z. X., Zeros of meromorphic solutions of higher order linear differential equations, Analysis 14 (1994), 425–438. (1994) Zbl0815.34003MR1310623
  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. (2000) Zbl0980.30022MR1792926
  7. Chen, Z. X., The growth of solutions of f ' ' + e - z f ' + Q ( z ) f = 0 where the order ( Q ) = 1 , Sci. China Ser. A 45, 3 (2002), 290–300. (2002) MR1903625
  8. Chen, Z. X., Shon, K. H., 10.1007/s10114-004-0434-z, Acta Math. Sin. (Engl. Ser.) 21, 4 (2005), 753–764. (2005) Zbl1100.34067MR2156950DOI10.1007/s10114-004-0434-z
  9. Frei, M., 10.1007/BF02567016, Comment. Math. Helv. 35 (1961), 201–222. (1961) Zbl0115.06903MR0126008DOI10.1007/BF02567016
  10. Frei, M., 10.1007/BF02566887, Comment. Math. Helv. 36 (1961), 1–8. (1961) Zbl0115.06904MR0151657DOI10.1007/BF02566887
  11. Gundersen, G. G., On the question of whether f ' ' + e - z f ' + B ( z ) f = 0 can admit a solution f ¬ ≡ 0 of finite order, Proc. Roy. Soc. Edinburgh Sect. A 102, 1-2 (1986), 9–17. (1986) MR0837157
  12. Hayman, W. K., Meromorphic functions, Clarendon Press, Oxford, 1964. (1964) Zbl0115.06203MR0164038
  13. Laine, I., Rieppo, J., 10.1080/02781070410001701092, Complex Var. Theory Appl. 49, 12 (2004), 897–911. (2004) Zbl1080.34076MR2101213DOI10.1080/02781070410001701092
  14. Langley, J. K., 10.2996/kmj/1138037272, Kodai Math. J. 9, 3 (1986), 430–439. (1986) Zbl0609.34041MR0856690DOI10.2996/kmj/1138037272
  15. Levin, B. Ya., Lectures on entire functions, American Mathematical Society, Providence, RI, 1996 Translations of Mathematical Monographs, Vol. 150. (1996) Zbl0856.30001MR1400006
  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. (2006) Zbl1094.30036MR2210116
  17. Nevanlinna, R., Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften Band 46, Zweite Auflage, Reprint, Springer-Verlag, Berlin–New York, 1974. (1974) Zbl0278.30002MR0344426
  18. Ozawa, M., 10.2996/kmj/1138036197, Kodai Math. J. 3, 2 (1980), 295–309. (1980) MR0588459DOI10.2996/kmj/1138036197
  19. Wang, J., Yi, H. X., 10.1080/0278107021000037048, Complex Var. Theory Appl. 48, 1 (2003), 83–94. (2003) Zbl1071.30029MR1953763DOI10.1080/0278107021000037048
  20. Wang, J., Laine, I., 10.1016/j.jmaa.2007.11.022, J. Math. Anal. Appl. 342, 1 (2008), 39–51. (2008) Zbl1151.34069MR2440778DOI10.1016/j.jmaa.2007.11.022
  21. Zhang, Q. T., Yang, C. C., The Fixed Points and Resolution Theory of Meromorphic Functions, Beijing University Press, Beijing, 1988 (in Chinese). (1988) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.