zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the iterated order and the fixed points of entire solutions of some complex linear differential equations. (English) Zbl 1120.34070

The iterated p-order σ p (f) of an entire function f is defined as

σ p (f)=lim sup r+ log p T(r,f) logr=lim sup r+ log p+1 M(r,f) logr·

The finiteness degree of the order of an entire function f is defined as i(f)=0 if f is a polynomial or i(f)=min{j:σ j (f)<} if f is a transcendental entire function and σ j (f)< for some j, otherwise i(f)=· The iterated convergence exponent of the sequence of distinct zeros of an entire function f is defined as

λ ¯ p (f)=lim suplog p N ¯(r,1 f) logr,

where N ¯(r,1 f) is the counting function of the distinct zeros of f(z) in Δ r ={z:|z|<1}·

In this paper, the author studies the iterated order and iterated convergence exponent of the sequence of distinct zeros of the entire solutions of the following linear differential equations

f (k) +A k-1 (z)f (k) ++A 1 (z)f ' +A 0 (z)f=0,(1)
f (k) +A k-1 (z)f (k) ++A 1 (z)f ' +A 0 (z)f=F(z),(2)

where A 0 (z),,A k-1 (z) and F(z)¬0 are entire functions. The author proves that if A 0 (z),,A k-1 (z) are entire functions with σ p (A j )σ p (A s )< for some s(0sk-1), then the equation (1) has at least one solution f with i(f)=p+1 and σ p+1 (f)=σ p (A s )· Furthermore, if F(z)¬0 is an entire function with either i(F)=q<p+1 or i(F)=q=p+1 and σ p+1 (F)<σ p (A s )<+, then there exists at least one solution g of the corresponding homogeneous equation (1) of the equation (2) such that all solutions f in the solution subspace {cg+f 0 ,c} satisfying i(f)=p+1 and σ p+1 (f)=λ ¯ p+1 (f)=σ p (A s ), with at most one exception, where f 0 is a solution of (2).

34M10Oscillation, growth of solutions (ODE in the complex domain)
30D35Distribution of values (one complex variable); Nevanlinna theory