A new proof of Frank-Weissenborn inequality
A new proof of the Frank-Weissenborn inequality is given. This proof uses the theory of algebroid functions.
Збережено в:
| Дата: | 2005 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2005
|
| Назва видання: | Журнал математической физики, анализа, геометрии |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/106565 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A new proof of Frank-Weissenborn inequality / A. Gol`dberg // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 1. — С. 71-73. — Бібліогр.: 6 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-106565 |
|---|---|
| record_format |
dspace |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2005, v. 1, No. 1, p. 71�73
A new proof of Frank�Weissenborn inequality
Anatoly Gol'dberg
Department of Mathematics & Statistics, Bar-Ilan University
Ramat-Gan, 52900, Israel
E-mail:misgo@bezeqint.net
Received January 24, 2005
A new proof of the Frank�Weissenborn inequality is given. This proof
uses the theory of algebroid functions.
Let f be a transcendental meromorphic in C function and all the poles of f
be simple. We use the standard notation of the value distribution theory [1]. We
also denote by Q(r; f) any quantity, satisfying Q(r; f) = o(T (r; f)) as r ! 1
possibly outside some system of intervals that have a �nite common length in the
case of a function f of in�nite order.
In [2] the following remarkable inequality was proved:
Lemma 1. Let � > 0: Then
N(r; f) � (1 + �)N(r; 1=f 00) +Q(r; f): (1)
We give a new proof of the inequality (1). This proof uses elements of the
theory of algebroid functions. We prove by the way that (1) holds with � = 0:
Denote
Af (z) :=
�
f 000
f 00
�2
�
3
4
f (4)
f 00
:
Let z0 be a simple pole of f , i.e., f(z) = c(z � z0)
�1 + h(z); where h is an
analytic function at z0: One can suppose, without loss of generality, that c = 1:
We have
f (n)(z) =
(�1)nn!
(z � z0)n+1
+ h(n)(z); n = 1; 2; 3; : : : :
Mathematics Subject Classi�cation 2000: 30D35.
Key words: Frank�Weissenborn inequality, meromorphic functions.
c
Anatoly Gol'dberg, 2005
Anatoly Gol'dberg
Further
f 000(z)
f 00(z)
=
�6(z � z0)
�4 + h000
2(z � z0)�3 + h00
= �
3
z � z0
(1 +O((z � z0)
3))
= �
3
z � z0
+O((z � z0)
2);
f (4)(z)
f 00(z)
=
24(z � z0)
�5 + h(4)
2(z � z0)�3 + h00
=
12
(z � z0)2
(1 +O((z � z0)
3))
=
12
(z � z0)2
+O(z � z0);
Af (z) = O(z � z0):
Hence Af (z0) = 0 and
n(r; 1=Af ) � n(r; f): (2)
Further
n(r;Af ) � n(r; 1=f 00): (3)
Now we will exploit the standart notions of the algebroid functions theory and
some its basic results [3, Ch. 1, �7; Ch. 3, �7]; [4, 5].
Let us consider the algebroid function
Bf (z) :=
q
Af (z):
Since all the poles of Af are of the second order, then all the poles of Bf (z) are
of the �rst order.
Recall ([4, �1]) that Bf (z) can be represented as
Bf (z) = (z � z0)
�=2g((z � z0)
1=2)
in some heigborhood of its zero z0, where g(z) is holomorphic at z = 0 and � 2 N
is the order of z0.
Thus from (2) we have n(r; 1=Bf ) � n(r; f) and hence
N(r; 1=Bf ) � N(r; f): (4)
Inequality (3) implies n(r;Bf ) � n(r; 1=f 00) and hence
N(r;Bf ) � N(r; 1=f 00): (5)
By Logarithmic Derivative Lemma [5, 6] m(r;Bf ) = Q(r; f): By the First Main
Theorem [3, 4]
T (r;Bf ) = m(r;Bf ) +N(r;Bf ) = Q(r; f) +N(r;Bf )
72 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 1
A new proof of Frank�Weissenborn inequality
T (r;Bf ) � Q(r; f) +N(r; 1=Bf );
and thus
N(r;Bf ) � N(r; 1=Bf ) +Q(r; f): (6)
From (4)�(6) we obtain
N(r; f) � N(r; 1=f 00) +Q(r; f);
i.e., (1) with � = 0:
I am grateful to Prof. I.V. Ostrovskii for remarks that were exploited in the
�nal variant of this paper.
References
[1] A.A. Gol'dberg and I.V. Ostrovskii, Distribution of values of meromorphic functions
(Raspredelenie znachenij meromorfnyh funkcij). Nauka, Moscow, 1970. (Russian)
[2] G. Frank and G. Weissenborn, Rational de�cient functions of meromorphic func-
tions. � Bull. London Math. Soc. 18 (1986), 29�33.
[3] V.P. Petrenko, Entire Curves (Tselyje krivyje). Vyshcha Shkola, Kharkov, 1984.
(Russian)
[4] E. Ullrich, �Uber den Ein�uss der Verzweigtheit einer Algebroide auf ihre
Wertverteilung. � J. Reine Angew. Math. 167 (1932), 198�220.
[5] G. Valiron, Sur la deriv�ee des functions algebroides. � Bull. Soc. Math. France 59
(1931), 17�39.
[6] V.D. Mohon'ko, Logarithmic Derivative Lemma. � Teor. Funktsii, Funktsion. Anal.
i ikh Prilozhen. 20 (1974), 112�122. (Russian)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 1 73
|
| spelling |
irk-123456789-1065652016-10-01T03:01:43Z A new proof of Frank-Weissenborn inequality Gol`dberg, A. A new proof of the Frank-Weissenborn inequality is given. This proof uses the theory of algebroid functions. 2005 Article A new proof of Frank-Weissenborn inequality / A. Gol`dberg // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 1. — С. 71-73. — Бібліогр.: 6 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106565 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A new proof of the Frank-Weissenborn inequality is given. This proof uses the theory of algebroid functions. |
| format |
Article |
| author |
Gol`dberg, A. |
| spellingShingle |
Gol`dberg, A. A new proof of Frank-Weissenborn inequality Журнал математической физики, анализа, геометрии |
| author_facet |
Gol`dberg, A. |
| author_sort |
Gol`dberg, A. |
| title |
A new proof of Frank-Weissenborn inequality |
| title_short |
A new proof of Frank-Weissenborn inequality |
| title_full |
A new proof of Frank-Weissenborn inequality |
| title_fullStr |
A new proof of Frank-Weissenborn inequality |
| title_full_unstemmed |
A new proof of Frank-Weissenborn inequality |
| title_sort |
new proof of frank-weissenborn inequality |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106565 |
| citation_txt |
A new proof of Frank-Weissenborn inequality / A. Gol`dberg // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 1. — С. 71-73. — Бібліогр.: 6 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT goldberga anewproofoffrankweissenborninequality AT goldberga newproofoffrankweissenborninequality |
| first_indexed |
2025-07-07T18:38:52Z |
| last_indexed |
2025-07-07T18:38:52Z |
| _version_ |
1837014485301198848 |