zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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 growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. (English) Zbl 1172.30009

For a meromorphic function f of order σ, the logarithmic derivative f ' /f satisfies the estimate |f ' (z)/f(z)||z| σ-1+ε outside a small exceptional set. This result has many applications, in particular to complex differential equations. In the study of difference equation, a similar role is played by the estimate |f(z+η)/f(z)|exp(|z| σ-1+ε ) which was obtained independently by R. G. Halburd and R. J. Korhonen [J. Math. Anal. Appl. 314, No. 2, 477–487 (2006; Zbl 1085.30026)] and by Y.-M. Chiang and S.-J. Feng [Ramanujan J. 16, No. 1, 105–129 (2008; Zbl 1152.30024)].

In the present paper the authors establish a connection between logarithmic derivatives and differences by showing that

f(z+η) f(z)=expηf ' (z) f(z)+O(r β+ε )

for |z| outside a set of finite logarithmic measure, where β is defined as follows: denoting by λ the maximum of the exponents of convergence of the zeros and poles of f, we have β=max{σ-2,2λ-2} if λ<1 and β=max{σ-2,λ-1} if λ1.

The above result is used to show that

f(z+η)-f(z) f(z)=ηf ' (z) f(z)+Or 2σ-2+ε

outside the exceptional set. Extensions to higher order difference quotients are also included.

Finally the paper contains a difference version of Wiman-Valiron theory which is used to show that entire solutions of first order algebraic difference equations have positive order.

MSC:
30D30General theory of meromorphic functions
30D35Distribution of values (one complex variable); Nevanlinna theory
39A05General theory of difference equations
46E25Rings and algebras of continuous, differentiable or analytic functions