scholarly journals Nonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc

2016 ◽  
Vol 20 (1) ◽  
pp. 45
Author(s):  
Benharrat Belaïdi
Keyword(s):  
Differential Equations ◽  
Unit Disc ◽  
Differential Polynomials ◽  
Complex Differential Equations ◽  
Linear Differential Polynomials ◽  
Linear Differential

Pairs of non-homogeneous linear differential polynomials

2006 ◽  
Vol 136 (4) ◽  
pp. 785-794 ◽  
Author(s):  
J. K. Langley
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Linear Differential Polynomials ◽  
Linear Differential ◽  
Image Position ◽  
Homogeneous Linear

Let f be transcendental and meromorphic in the plane and let the non-homogeneous linear differential polynomials F and G be defined by where k,n ∈ N and a, b and the aj, bj are rational functions. Under the assumption that F and G have few zeros, it is shown that either F and G reduce to homogeneous linear differential polynomials in f + c, where c is a rational function that may be computed explicitly, or f has a representation as a rational function in solutions of certain associated linear differential equations, which again may be determined explicitly from the aj, bj and a and b.

Linear Differential Polynomials

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Linear Operators ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Linear Differential Operators ◽  
Differential Field ◽  
Linear Differential Polynomials ◽  
Linear Differential

This chapter introduces the reader to linear differential polynomials. It first considers homogeneous differential polynomials and the corresponding linear operators before proving various basic results on them. In particular, it describes the property of a linear differential operator over a differential field K of defining a surjective map K → K, along with the transformation of a system of linear differential equations in several unknowns to an equivalent system of several linear differential equations in a single unknown. The chapter also discusses second-order linear differential operators, diagonalization of matrices, differential modules, linear differential operators in the presence of a valuation, and compositional conjugation. It concludes with an analysis of the Riccati transform and Johnson's Theorem.

scholarly journals Growth of Logarithmic Derivatives and Their Applications in Complex Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zinelâabidine Latreuch ◽  
Benharrat Belaïdi
Keyword(s):  
Differential Equations ◽  
Entire Functions ◽  
Value Distribution ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Complex Differential Equations ◽  
Linear Differential ◽  
Logarithmic Derivatives

We continue the study of the behavior of the growth of logarithmic derivatives. In fact, we prove some relations between the value distribution of solutions of linear differential equations and growth of their logarithmic derivatives. We also give an estimate of the growth of the quotient of two differential polynomials generated by solutions of the equation f″+A(z)f′+B(z)f=0, where A(z) and B(z) are entire functions.

scholarly journals SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

2010 ◽  
Vol 88 (2) ◽  
pp. 145-167 ◽  
Author(s):  
I. CHYZHYKOV ◽  
J. HEITTOKANGAS ◽  
J. RÄTTYÄ
Keyword(s):  
Differential Equations ◽  
Unit Disc ◽  
Logarithmic Derivative ◽  
Maximum Modulus ◽  
Exceptional Set ◽  
Proximate Order ◽  
Upper Density ◽  
Linear Differential ◽  
Logarithmic Derivatives ◽  
Growth Of Solutions

AbstractNew estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

scholarly journals On the zeros of second order linear differential polynomials

1990 ◽  
Vol 33 (2) ◽  
pp. 265-285 ◽  
Author(s):  
J. K. Langley
Keyword(s):  
Finite Order ◽  
Second Order ◽  
Differential Polynomials ◽  
Order Linear ◽  
Linear Differential Polynomials ◽  
Linear Differential

We determine all functions f(z) meromorphic in the plane such that f′(z)/f(z) has finite order and f(z) and F(z) have only finitely many zeros, where F(z) = f″(z) + Af(z) for some constant A.

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