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

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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

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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

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