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Complex Oscillation Theory of Differential Polynomials

Abdallah El FarissiBenharrat Belaïdi — 2011

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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 .

On the growth of solutions of some higher order linear differential equations

Abdallah El FarissiBenharrat Belaidi — 2012

Applications of Mathematics

In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation f ( k ) + A k - 1 f ( k - 1 ) + + A 2 f ' ' + ( D 1 ( z ) + A 1 ( z ) e a z ) f ' + ( D 0 ( z ) + A 0 ( z ) e b z ) f = F ( k 2 ) , where a , b are complex constants that satisfy a b ( a - b ) 0 and A j ( z ) ( j = 0 , 1 , , k - 1 ) , D j ( z ) ( j = 0 , 1 ) , F ( z ) are entire functions with max { ρ ( A j ) ( j = 0 , 1 , , k - 1 ) , ρ ( D j ) ( j = 0 , 1 ) } < 1 . We also investigate the relationship between small functions and the solutions of the above equation.

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