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Monomials of Entire Algebroid Functions
Journal of Mathematical Analysis and Applications 229, 32]40 Ž1999. Article ID jmaa.1998.6135, available online at http:rrwww.idealibrary.com on
Monomials of Entire Algebroid Functions Kari Katajamaki* ¨ Department of Mathematics, Uni¨ ersity of Joensuu, P.O. Box 111, 80101 Joensuu, Finland Submitted by H. M. Sri¨ asta¨ a Received May 16, 1997
Hayman has shown that if f is a transcendental entire function and n G 2, then f n f 9 assumes all values except possibly zero infinitely often. We extend his result in three directions by considering an entire algebroid function w, its monomial w n 0 Ž w9. n 1 ??? Ž w Ž k . . n k , and by estimating the growth of the number of a-points of the monomial. Q 1999 Academic Press
1. INTRODUCTION AND RESULTS As Hayman w3, p. 34x noted, the problem of possible Picard values of derivatives of a meromorphic function having no zeros reduces to the problem of whether certain differential polynomials of an entire function necessarily have zeros. In this connection he proved the following theorem: THEOREM A. If f Ž z . is a transcendental entire function and n G 2, then f Ž z . n f 9Ž z . assumes all ¨ alues except possibly zero infinitely often. Clunie w1x showed that Theorem A is also true for n s 1. It is assumed that the reader is familiar with the notation of standard Nevanlinna theory w4x and its algebroid counterpart w9, 11, 12x. A characterization of algebroid functions and the basic results of their value distribution can be found in w6, Sects. 1 and 2x. Sons w10x extended Theorem A to more general differential monomials: THEOREM B.
If f Ž z . is a transcendental entire function and
c Ž z. [ f Ž z.
n0
f 9Ž z .
n1
??? f Ž k . Ž z .
* Supported by the Finnish Academy of Science and Letters. 32 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
nk
,
Ž 1.
33
ALGEBROID FUNCTIONS
where n 0 G 2, n k G 1, and n i G 0 for i / 0, k, then, for a / 0,
d Ž a, c . [ 1 y lim sup
N Ž r , 1r Ž c y a . .
- 1.
T Ž r, c .
rª`
In w8x, Theorem A was generalized as follows: THEOREM C. function and set
Let w Ž z . be a n-¨ alued transcendental entire algebroid n
n
c Ž z . [ w Ž z . 0 w9 Ž z . 1 ,
n 0 , n1 g N.
Then if n 0 G 4n y 2 q 2Ž n y 1.Ž n1 y 1., we ha¨ e for each a g C _ 04 ,
ž
N r,
1
cya
/
G pT Ž r , w . y S Ž r , w . ,
where p [ n 0 y 4n q 3 y 2Ž n y 1.Ž n1 y 1. G 1. In this paper we will consider monomials as in Theorem B and obtain a result analogous to Theorem C. The proof will be different from that of Theorem C, where the transformation uŽ z . [ 1rw Ž z . was applied. Referring to Ž1., denote
g [ n 0 q n1 q ??? qn k , m [ n1 q 2 n 2 q ??? qkn k , s [ n1 q 3n 2 q ??? q Ž 2 k y 1 . n k . We can now state the result: THEOREM. tion and set
Let w Ž z . be a n-¨ alued transcendental entire algebroid funcn
c Ž z . [ w Ž z . 0 w9 Ž z .
n1
??? w Ž k . Ž z .
nk
,
Ž 2.
where k G 2, n k G 1, and n i G 0 for i / 0, k. Then if k
n0 G k q
k
Ý
Ž i y 1. n i q 2 Ž n y 1. k 2 q 2 q
ž
is2
Ý i 2 ni is1
/
\ m,
we ha¨ e for each a g C _ 04 ,
ž
N r,
1
cya
/
G pT Ž r , w . y S Ž r , w . ,
where p [ Ž arŽ a q 2..Ž n 0 y m q 1 y 8Ž n y 1.ra . G nm y 1 G 18Ž n y 1. q 1.
1 3
with a [ g y
KARI KATAJAMAKI ¨
34
COROLLARY. With the same hypotheses, we ha¨ e for each a g C _ 04 , Q Ž a, c . [ 1 y lim sup
N Ž r , 1r Ž c y a . . T Ž r, c .
rª`
p
F1y
g q 2 Ž n y 1. s
.
2. LEMMAS Let w be a n-valued algebroid function. It has expansions of the general form w Ž z . s w Ž z 0 . q ct Ž z y z 0 . w Ž z . s cyt Ž z y z 0 .
tr l
yt r l
q ???
or
q ??? ,
where t , l g N and l F n . We define the counting functions of branch points and multiple points of w by nZ Ž r , w . [
Ý Ž l y 1.
and
< z
nW Ž r , w . [
Ý Ž t y 1. ,
< z
respectively, and denote by NZ Ž r, w . and NW Ž r, w . the corresponding integrated counting functions. We regard c , w9rw, etc., as functions on the Riemann surface of w; see w7, Sect. 2x. By the proof of w13, Theorem 1x, we have the following lemma: LEMMA 1.
Let w be a nonconstant algebroid function. Then for i g N,
T Ž r , w Ž i. . F T Ž r , w . q iN Ž r , w . q Ž 2 i y 1 . NZ Ž r , w . q S Ž r , w . . We also have the following estimate; see w5, Lemma 3x. LEMMA 2. Let w be a nonconstant algebroid function and c be of the form Ž2.. Then T Ž r , c . F g T Ž r , w . q m N Ž r , w . q s NZ Ž r , w . q S Ž r , w . . The next lemma was proved in w7x. LEMMA 3. Let w be an algebroid function and c be a rational function of w, w9, . . . , w Ž k ., k G 0, with meromorphic coefficients. Then NZ Ž r, c . F NZ Ž r, w .. By w2, pp. 213, 215x, we have the following results: LEMMA 4.
Let w be a nonconstant algebroid function. Then
NW Ž r , w . s N r ,
1
ž / w
q N Ž r , w . q NZ Ž r , w . y m r ,
ž
w w9
/
q SŽ r , w.
Ž 3.
35
ALGEBROID FUNCTIONS
and w9
F N r,
1
q N Ž r , w . q NZ Ž r , w . .
ž / ž /
N r,
w
w
Ž 4.
We can now state and prove the final two lemmas. LEMMA 5. With the hypotheses of the theorem, we ha¨ e k
ž
g y m y 2 Ž n y 1.
Ý i 2 ni is1
/
T Ž r , w. F T Ž r , c . q SŽ r , w. .
Ž 5.
Remark. The left-hand side of Ž5. is G kT Ž r, w . by the hypothesis on n0 . Proof of Lemma 5. Set qi [ Ý kjsi n j for i s 0, 1, . . . , k, and write
csw
q0
w9
q1
w0
q2
???
ž /ž / ž w
w9
qk
wŽk. w Ž ky1.
/
.
Using the lemma on the logarithmic derivative with Lemma 1, we get k
q0 T Ž r , w . F T Ž r , c . q
Ý is1
ž
qi N r ,
w Ž i. w Ž iy1.
/
q SŽ r , w. .
By Ž4. and the inequality NZ Ž r, w . F 2Ž n y 1.T Ž r, w . q O Ž1. Žsee w11, Satz II, p. 210x., we have w9
F N r,
1
ž / ž /
N r,
w
w
q NZ Ž r , w . F Ž 1 q 2 Ž n y 1 . . T Ž r , w . q O Ž 1 . .
Again by Ž4., Lemma 1, and the fact that possible poles of the derivatives of w arise only from branch points of w, we obtain
ž
N r,
w Ž i. w Ž iy1.
/
F N r,
ž
1 w
Ž iy1.
/
q N Ž r , w Ž iy1. . q NZ Ž r , w .
F T Ž r , w . q Ž 2 Ž i y 1 . y 1 . NZ Ž r , w . q 2 NZ Ž r , w . q S Ž r , w . s T Ž r , w . q Ž 2 i y 1 . NZ Ž r , w . q S Ž r , w . F Ž 1 q 2 Ž n y 1. Ž 2 i y 1. . T Ž r , w . q S Ž r , w .
KARI KATAJAMAKI ¨
36 for i s 2, 3, . . . , k. Thus k
q0 T Ž r , w . F T Ž r , c . q
Ý qi Ž 1 q 2 Ž n y 1 . Ž 2 i y 1 . . T Ž r , w . q S Ž r , w . is1
and, since q0 s g and
Ý kis1
qi s m ,
k
ž
g y m y 2 Ž n y 1.
Ý qi Ž 2 i y 1 . is1
/
T Ž r , w. F T Ž r , c . q SŽ r , w. .
But k
Ý
k
qi Ž 2 i y 1 . s
is1
Ý
i
ni
is1
k
Ý
Ž 2 j y 1. s
js1
Ý i 2 ni , is1
so that Ž5. holds. LEMMA 6. With the hypotheses of the theorem, we ha¨ e a G 18Ž n y 1. q1. Proof. Since k G 2, we get
a s n 0 q q1 y nm y 1 k
Gkq
k
k
Ý in i y Ý n i q 2 Ž n y 1. is1
is1
k
k
ž
k2 q 2 q
Ý i 2 ni is1
/
q Ý n i y n Ý in i y 1 is1
is1 k
s k y 1 y Ž n y 1.
k
Ý in i q 2 Ž n y 1. is1
ž
k2 q 2 q
Ý i 2 ni is1
/
k
s k y 1 q Ž n y 1. 2 k 2 q 4 q
ž
Ý Ž 2 i y 1. in i is1
/
G 18 Ž n y 1 . q 1,
and the proof is complete.
3. PROOF OF THE THEOREM AND ITS COROLLARY The function c must be nonconstant by Lemma 5. Let a g C _ 04 . The second main theorem Žsee w6, p. 18x. then yields T Ž r , c . F NŽ r , c . q N r ,
1
q N r,
ž / ž c
1
cya
/
q NZ Ž r , c . q S Ž r , c . .
37
ALGEBROID FUNCTIONS
By Lemma 2, we may replace SŽ r, c . by S Ž r, w .. Using also Lemma 3 and noticing that any pole of c must be a branch point of w, we get 1
T Ž r, c . F N r,
q N r,
ž / ž c
1
cya
/
q 2 NZ Ž r , w . q S Ž r , w . .
Ž 6.
Next we estimate the term N Ž r, 1rc .. Let w have an expansion w Ž z . s w Ž z 0 . q ct Ž z y z 0 .
tr l
q ???
Ž 7.
around a point z 0 . Then w9 has an expansion w9 Ž z . s
t l
ct Ž z y z 0 .
Žt y l . r l
q ??? .
Thus we obtain 1
F n r,
1
ž / ž /
n r,
c
w
k
q Ý Ž t y l. q Ý n
q
w/0
is2
ž
1
r,
w Ž i.
/
.
Here q Ý Ž t y l. F Ý Ž t y 1. s n W Ž r , w . y Ý Ž t y 1.
w/0
w/0
s n r,
ws0
1
ž / w
q nW Ž r , w . y n r ,
1
ž / w
.
Integrating logarithmically we get 1
ž /
N r,
c
F 2N r,
1
ž / w
q NW Ž r , w . y N r ,
1
ž / w
k
q
ÝN is2
ž
r,
1 w Ž i.
/
.
The identity Ž3. now gives 1
ž /
N r,
c
F 2N r,
1
ž / w
k
q NZ Ž r , w . q
ÝN is2
ž
r,
1 w Ž i.
/
q S Ž r , w . . Ž 8.
At each zero of w, the function c also has a zero. In fact, if w Ž z 0 . s 0 in Ž7., the expansion of c becomes
c Ž z . s c Ž z y z0 .
Ž gt y m l .r l
q ??? ,
c g C,
KARI KATAJAMAKI ¨
38
where gt y ml G g y mn s a q 1 G 2 by Lemma 6. We thus obtain
an r,
1
ž / w
s
Ý Ž g y mn y 1. ws0
F
Ý Ž gt y ml y 1. F n ws0
1
y n r,
1
ž / ž / r,
c
c
and, integrating logarithmically,
aN r,
1
F N r,
1
y N r,
1
ž / ž / ž / w
c
c
.
Substituting this into Ž8. and recalling Lemma 1, we obtain 1
F
2
1
y N r,
1
ž / ž ž / ž //
N r,
c
N r,
a
c
c
q NZ Ž r , w .
k
q Ý Ž T Ž r , w . q Ž 2 i y 1 . NZ Ž r , w . . q S Ž r , w . . is2
Hence we have
aq2 a
1
ž /
N r,
c
F
2
a
T Ž r , c . q Ž k y 1. T Ž r , w . k
q Ý Ž 2 i y 1 . NZ Ž r , w . q S Ž r , w . , is1
i.e., 1
ž /
N r,
c
F
2
T Ž r, c . aq2 a q Ž Ž k y 1 . T Ž r , w . q k 2 NZ Ž r , w . . q S Ž r , w . . aq2
The inequality Ž6. now gives
a aq2
TŽ r, c . F
a aq2
Ž Ž k y 1 . T Ž r , w . q k 2 NZ Ž r , w . .
q N r,
ž
1
cya
/
q 2 NZ Ž r , w . q S Ž r , w . ,
39
ALGEBROID FUNCTIONS
so that T Ž r , c . F Ž k y 1 . T Ž r , w . q k 2 NZ Ž r , w . q 2 NZ Ž r , w . q
aq2
q
a
ž
N r,
1
cya
/
ž
aq2 a
ž
N r,
1
cya
/
a
NZ Ž r , w .
q SŽ r , w.
F k y 1 q 2 Ž n y 1. Ž k 2 q 2. q q
4
8 Ž n y 1.
a
/
T Ž r, w.
q SŽ r , w. .
Combining this with Ž5., we have k
ž
g y m y 2 Ž n y 1.
Ý i 2 n i y k q 1 y 2 Ž n y 1. Ž k 2 q 2. y
8 Ž n y 1.
a
is1
/
= T Ž r, w. k
s n0 y
ž
is2
q 1y
ž
F
k
Ý Ž i y 1. n i y 2 Ž n y 1.
aq2 a
8 Ž n y 1.
a
ž
N r,
1
cya
ž
k2 q 2 q
Ý i 2 ni is1
y k T Ž r, w.
/ /
/Ž /
T r, w. q SŽ r , w. .
Ž 9.
By Lemma 6 and the hypothesis on n 0 , the left-hand side of Ž9. is ) 59 T Ž r, w ., and we have pT Ž r , w . F N r ,
ž
1
cya
/
q SŽ r , w. ,
where p[
a Ž n0 y m. aq2
q
a y 8 Ž n y 1. aq2
G
1 3
.
This completes the proof. Proof of the Corollary. By Lemma 2, we have T Ž r , c . F Ž g q 2 Ž n y 1. s . T Ž r , w . q S Ž r , w . .
Ž 10 .
KARI KATAJAMAKI ¨
40
Let a g C _ 04 . By Ž10. and the theorem, there exists a set E of finite linear measure such that lim rfE
N Ž r , 1r Ž c y a . . T Ž r, c .
G lim rfE
s
Ž p y o Ž 1. . T Ž r , w . Ž g q 2 Ž n y 1. s q o Ž 1. . T Ž r , w . p
g q 2 Ž n y 1. s
.
The corollary is proved.
REFERENCES 1. J. Clunie, On a result of Hayman, J. London Math. Soc. 42 Ž1967., 389]392. ¨ 2. F. Gackstatter, Uber die logarithmische Ableitung einer algebroiden Funktion, J. Reine Angew. Math. 257 Ž1972., 210]216. 3. W. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 Ž1959., 9]42. 4. W. Hayman, ‘‘Meromorphic Functions,’’ Clarendon Press, Oxford, 1964. 5. Y. He and X. Xiao, Meromorphic and algebroid solutions of higher-order algebraic differential equations, Sci. Sinica Ser. A 26 Ž1983., 1034]1043. 6. K. Katajamaki, ¨ Algebroid solutions of binomial and linear differential equations, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 90 Ž1993., 1]48. 7. K. Katajamaki, ¨ Value distribution of certain differential polynomials of algebroid functions, Arch. Math. Ž Basel . 67 Ž1996., 422]429. 8. K. Katajamaki, ¨ Value distribution of some monomials of entire algebroid functions, Complex Variables Theory Appl. 34 Ž1997., 415]420. ¨ 9. H. Selberg, Uber die Wertverteilung der algebroiden Funktionen, Math. Z. 31 Ž1930., 709]728. 10. L. Sons, Deficiencies of monomials, Math. Z. 111 Ž1969., 53]68. ¨ 11. E. Ullrich, Uber den Einfluß der Verzweigtheit einer Algebroide auf ihre Wertverteilung, J. Reine Angew. Math. 167 Ž1932., 198]220. 12. G. Valiron, Sur la derivee Bull. Soc. Math. France 59 Ž1931., ´ ´ des fonctions algebroıdes, ´ ¨ 17]39. 13. K. Yosida, On algebroid-solutions of ordinary differential equations, Japan. J. Math. 10 Ž1934., 199]208.
Monomials of Entire Algebroid Functions Kari Katajamaki* ¨ Department of Mathematics, Uni¨ ersity of Joensuu, P.O. Box 111, 80101 Joensuu, Finland Submitted by H. M. Sri¨ asta¨ a Received May 16, 1997
Hayman has shown that if f is a transcendental entire function and n G 2, then f n f 9 assumes all values except possibly zero infinitely often. We extend his result in three directions by considering an entire algebroid function w, its monomial w n 0 Ž w9. n 1 ??? Ž w Ž k . . n k , and by estimating the growth of the number of a-points of the monomial. Q 1999 Academic Press
1. INTRODUCTION AND RESULTS As Hayman w3, p. 34x noted, the problem of possible Picard values of derivatives of a meromorphic function having no zeros reduces to the problem of whether certain differential polynomials of an entire function necessarily have zeros. In this connection he proved the following theorem: THEOREM A. If f Ž z . is a transcendental entire function and n G 2, then f Ž z . n f 9Ž z . assumes all ¨ alues except possibly zero infinitely often. Clunie w1x showed that Theorem A is also true for n s 1. It is assumed that the reader is familiar with the notation of standard Nevanlinna theory w4x and its algebroid counterpart w9, 11, 12x. A characterization of algebroid functions and the basic results of their value distribution can be found in w6, Sects. 1 and 2x. Sons w10x extended Theorem A to more general differential monomials: THEOREM B.
If f Ž z . is a transcendental entire function and
c Ž z. [ f Ž z.
n0
f 9Ž z .
n1
??? f Ž k . Ž z .
* Supported by the Finnish Academy of Science and Letters. 32 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
nk
,
Ž 1.
33
ALGEBROID FUNCTIONS
where n 0 G 2, n k G 1, and n i G 0 for i / 0, k, then, for a / 0,
d Ž a, c . [ 1 y lim sup
N Ž r , 1r Ž c y a . .
- 1.
T Ž r, c .
rª`
In w8x, Theorem A was generalized as follows: THEOREM C. function and set
Let w Ž z . be a n-¨ alued transcendental entire algebroid n
n
c Ž z . [ w Ž z . 0 w9 Ž z . 1 ,
n 0 , n1 g N.
Then if n 0 G 4n y 2 q 2Ž n y 1.Ž n1 y 1., we ha¨ e for each a g C _ 04 ,
ž
N r,
1
cya
/
G pT Ž r , w . y S Ž r , w . ,
where p [ n 0 y 4n q 3 y 2Ž n y 1.Ž n1 y 1. G 1. In this paper we will consider monomials as in Theorem B and obtain a result analogous to Theorem C. The proof will be different from that of Theorem C, where the transformation uŽ z . [ 1rw Ž z . was applied. Referring to Ž1., denote
g [ n 0 q n1 q ??? qn k , m [ n1 q 2 n 2 q ??? qkn k , s [ n1 q 3n 2 q ??? q Ž 2 k y 1 . n k . We can now state the result: THEOREM. tion and set
Let w Ž z . be a n-¨ alued transcendental entire algebroid funcn
c Ž z . [ w Ž z . 0 w9 Ž z .
n1
??? w Ž k . Ž z .
nk
,
Ž 2.
where k G 2, n k G 1, and n i G 0 for i / 0, k. Then if k
n0 G k q
k
Ý
Ž i y 1. n i q 2 Ž n y 1. k 2 q 2 q
ž
is2
Ý i 2 ni is1
/
\ m,
we ha¨ e for each a g C _ 04 ,
ž
N r,
1
cya
/
G pT Ž r , w . y S Ž r , w . ,
where p [ Ž arŽ a q 2..Ž n 0 y m q 1 y 8Ž n y 1.ra . G nm y 1 G 18Ž n y 1. q 1.
1 3
with a [ g y
KARI KATAJAMAKI ¨
34
COROLLARY. With the same hypotheses, we ha¨ e for each a g C _ 04 , Q Ž a, c . [ 1 y lim sup
N Ž r , 1r Ž c y a . . T Ž r, c .
rª`
p
F1y
g q 2 Ž n y 1. s
.
2. LEMMAS Let w be a n-valued algebroid function. It has expansions of the general form w Ž z . s w Ž z 0 . q ct Ž z y z 0 . w Ž z . s cyt Ž z y z 0 .
tr l
yt r l
q ???
or
q ??? ,
where t , l g N and l F n . We define the counting functions of branch points and multiple points of w by nZ Ž r , w . [
Ý Ž l y 1.
and
< z
nW Ž r , w . [
Ý Ž t y 1. ,
< z
respectively, and denote by NZ Ž r, w . and NW Ž r, w . the corresponding integrated counting functions. We regard c , w9rw, etc., as functions on the Riemann surface of w; see w7, Sect. 2x. By the proof of w13, Theorem 1x, we have the following lemma: LEMMA 1.
Let w be a nonconstant algebroid function. Then for i g N,
T Ž r , w Ž i. . F T Ž r , w . q iN Ž r , w . q Ž 2 i y 1 . NZ Ž r , w . q S Ž r , w . . We also have the following estimate; see w5, Lemma 3x. LEMMA 2. Let w be a nonconstant algebroid function and c be of the form Ž2.. Then T Ž r , c . F g T Ž r , w . q m N Ž r , w . q s NZ Ž r , w . q S Ž r , w . . The next lemma was proved in w7x. LEMMA 3. Let w be an algebroid function and c be a rational function of w, w9, . . . , w Ž k ., k G 0, with meromorphic coefficients. Then NZ Ž r, c . F NZ Ž r, w .. By w2, pp. 213, 215x, we have the following results: LEMMA 4.
Let w be a nonconstant algebroid function. Then
NW Ž r , w . s N r ,
1
ž / w
q N Ž r , w . q NZ Ž r , w . y m r ,
ž
w w9
/
q SŽ r , w.
Ž 3.
35
ALGEBROID FUNCTIONS
and w9
F N r,
1
q N Ž r , w . q NZ Ž r , w . .
ž / ž /
N r,
w
w
Ž 4.
We can now state and prove the final two lemmas. LEMMA 5. With the hypotheses of the theorem, we ha¨ e k
ž
g y m y 2 Ž n y 1.
Ý i 2 ni is1
/
T Ž r , w. F T Ž r , c . q SŽ r , w. .
Ž 5.
Remark. The left-hand side of Ž5. is G kT Ž r, w . by the hypothesis on n0 . Proof of Lemma 5. Set qi [ Ý kjsi n j for i s 0, 1, . . . , k, and write
csw
q0
w9
q1
w0
q2
???
ž /ž / ž w
w9
qk
wŽk. w Ž ky1.
/
.
Using the lemma on the logarithmic derivative with Lemma 1, we get k
q0 T Ž r , w . F T Ž r , c . q
Ý is1
ž
qi N r ,
w Ž i. w Ž iy1.
/
q SŽ r , w. .
By Ž4. and the inequality NZ Ž r, w . F 2Ž n y 1.T Ž r, w . q O Ž1. Žsee w11, Satz II, p. 210x., we have w9
F N r,
1
ž / ž /
N r,
w
w
q NZ Ž r , w . F Ž 1 q 2 Ž n y 1 . . T Ž r , w . q O Ž 1 . .
Again by Ž4., Lemma 1, and the fact that possible poles of the derivatives of w arise only from branch points of w, we obtain
ž
N r,
w Ž i. w Ž iy1.
/
F N r,
ž
1 w
Ž iy1.
/
q N Ž r , w Ž iy1. . q NZ Ž r , w .
F T Ž r , w . q Ž 2 Ž i y 1 . y 1 . NZ Ž r , w . q 2 NZ Ž r , w . q S Ž r , w . s T Ž r , w . q Ž 2 i y 1 . NZ Ž r , w . q S Ž r , w . F Ž 1 q 2 Ž n y 1. Ž 2 i y 1. . T Ž r , w . q S Ž r , w .
KARI KATAJAMAKI ¨
36 for i s 2, 3, . . . , k. Thus k
q0 T Ž r , w . F T Ž r , c . q
Ý qi Ž 1 q 2 Ž n y 1 . Ž 2 i y 1 . . T Ž r , w . q S Ž r , w . is1
and, since q0 s g and
Ý kis1
qi s m ,
k
ž
g y m y 2 Ž n y 1.
Ý qi Ž 2 i y 1 . is1
/
T Ž r , w. F T Ž r , c . q SŽ r , w. .
But k
Ý
k
qi Ž 2 i y 1 . s
is1
Ý
i
ni
is1
k
Ý
Ž 2 j y 1. s
js1
Ý i 2 ni , is1
so that Ž5. holds. LEMMA 6. With the hypotheses of the theorem, we ha¨ e a G 18Ž n y 1. q1. Proof. Since k G 2, we get
a s n 0 q q1 y nm y 1 k
Gkq
k
k
Ý in i y Ý n i q 2 Ž n y 1. is1
is1
k
k
ž
k2 q 2 q
Ý i 2 ni is1
/
q Ý n i y n Ý in i y 1 is1
is1 k
s k y 1 y Ž n y 1.
k
Ý in i q 2 Ž n y 1. is1
ž
k2 q 2 q
Ý i 2 ni is1
/
k
s k y 1 q Ž n y 1. 2 k 2 q 4 q
ž
Ý Ž 2 i y 1. in i is1
/
G 18 Ž n y 1 . q 1,
and the proof is complete.
3. PROOF OF THE THEOREM AND ITS COROLLARY The function c must be nonconstant by Lemma 5. Let a g C _ 04 . The second main theorem Žsee w6, p. 18x. then yields T Ž r , c . F NŽ r , c . q N r ,
1
q N r,
ž / ž c
1
cya
/
q NZ Ž r , c . q S Ž r , c . .
37
ALGEBROID FUNCTIONS
By Lemma 2, we may replace SŽ r, c . by S Ž r, w .. Using also Lemma 3 and noticing that any pole of c must be a branch point of w, we get 1
T Ž r, c . F N r,
q N r,
ž / ž c
1
cya
/
q 2 NZ Ž r , w . q S Ž r , w . .
Ž 6.
Next we estimate the term N Ž r, 1rc .. Let w have an expansion w Ž z . s w Ž z 0 . q ct Ž z y z 0 .
tr l
q ???
Ž 7.
around a point z 0 . Then w9 has an expansion w9 Ž z . s
t l
ct Ž z y z 0 .
Žt y l . r l
q ??? .
Thus we obtain 1
F n r,
1
ž / ž /
n r,
c
w
k
q Ý Ž t y l. q Ý n
q
w/0
is2
ž
1
r,
w Ž i.
/
.
Here q Ý Ž t y l. F Ý Ž t y 1. s n W Ž r , w . y Ý Ž t y 1.
w/0
w/0
s n r,
ws0
1
ž / w
q nW Ž r , w . y n r ,
1
ž / w
.
Integrating logarithmically we get 1
ž /
N r,
c
F 2N r,
1
ž / w
q NW Ž r , w . y N r ,
1
ž / w
k
q
ÝN is2
ž
r,
1 w Ž i.
/
.
The identity Ž3. now gives 1
ž /
N r,
c
F 2N r,
1
ž / w
k
q NZ Ž r , w . q
ÝN is2
ž
r,
1 w Ž i.
/
q S Ž r , w . . Ž 8.
At each zero of w, the function c also has a zero. In fact, if w Ž z 0 . s 0 in Ž7., the expansion of c becomes
c Ž z . s c Ž z y z0 .
Ž gt y m l .r l
q ??? ,
c g C,
KARI KATAJAMAKI ¨
38
where gt y ml G g y mn s a q 1 G 2 by Lemma 6. We thus obtain
an r,
1
ž / w
s
Ý Ž g y mn y 1. ws0
F
Ý Ž gt y ml y 1. F n ws0
1
y n r,
1
ž / ž / r,
c
c
and, integrating logarithmically,
aN r,
1
F N r,
1
y N r,
1
ž / ž / ž / w
c
c
.
Substituting this into Ž8. and recalling Lemma 1, we obtain 1
F
2
1
y N r,
1
ž / ž ž / ž //
N r,
c
N r,
a
c
c
q NZ Ž r , w .
k
q Ý Ž T Ž r , w . q Ž 2 i y 1 . NZ Ž r , w . . q S Ž r , w . . is2
Hence we have
aq2 a
1
ž /
N r,
c
F
2
a
T Ž r , c . q Ž k y 1. T Ž r , w . k
q Ý Ž 2 i y 1 . NZ Ž r , w . q S Ž r , w . , is1
i.e., 1
ž /
N r,
c
F
2
T Ž r, c . aq2 a q Ž Ž k y 1 . T Ž r , w . q k 2 NZ Ž r , w . . q S Ž r , w . . aq2
The inequality Ž6. now gives
a aq2
TŽ r, c . F
a aq2
Ž Ž k y 1 . T Ž r , w . q k 2 NZ Ž r , w . .
q N r,
ž
1
cya
/
q 2 NZ Ž r , w . q S Ž r , w . ,
39
ALGEBROID FUNCTIONS
so that T Ž r , c . F Ž k y 1 . T Ž r , w . q k 2 NZ Ž r , w . q 2 NZ Ž r , w . q
aq2
q
a
ž
N r,
1
cya
/
ž
aq2 a
ž
N r,
1
cya
/
a
NZ Ž r , w .
q SŽ r , w.
F k y 1 q 2 Ž n y 1. Ž k 2 q 2. q q
4
8 Ž n y 1.
a
/
T Ž r, w.
q SŽ r , w. .
Combining this with Ž5., we have k
ž
g y m y 2 Ž n y 1.
Ý i 2 n i y k q 1 y 2 Ž n y 1. Ž k 2 q 2. y
8 Ž n y 1.
a
is1
/
= T Ž r, w. k
s n0 y
ž
is2
q 1y
ž
F
k
Ý Ž i y 1. n i y 2 Ž n y 1.
aq2 a
8 Ž n y 1.
a
ž
N r,
1
cya
ž
k2 q 2 q
Ý i 2 ni is1
y k T Ž r, w.
/ /
/Ž /
T r, w. q SŽ r , w. .
Ž 9.
By Lemma 6 and the hypothesis on n 0 , the left-hand side of Ž9. is ) 59 T Ž r, w ., and we have pT Ž r , w . F N r ,
ž
1
cya
/
q SŽ r , w. ,
where p[
a Ž n0 y m. aq2
q
a y 8 Ž n y 1. aq2
G
1 3
.
This completes the proof. Proof of the Corollary. By Lemma 2, we have T Ž r , c . F Ž g q 2 Ž n y 1. s . T Ž r , w . q S Ž r , w . .
Ž 10 .
KARI KATAJAMAKI ¨
40
Let a g C _ 04 . By Ž10. and the theorem, there exists a set E of finite linear measure such that lim rfE
N Ž r , 1r Ž c y a . . T Ž r, c .
G lim rfE
s
Ž p y o Ž 1. . T Ž r , w . Ž g q 2 Ž n y 1. s q o Ž 1. . T Ž r , w . p
g q 2 Ž n y 1. s
.
The corollary is proved.
REFERENCES 1. J. Clunie, On a result of Hayman, J. London Math. Soc. 42 Ž1967., 389]392. ¨ 2. F. Gackstatter, Uber die logarithmische Ableitung einer algebroiden Funktion, J. Reine Angew. Math. 257 Ž1972., 210]216. 3. W. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 Ž1959., 9]42. 4. W. Hayman, ‘‘Meromorphic Functions,’’ Clarendon Press, Oxford, 1964. 5. Y. He and X. Xiao, Meromorphic and algebroid solutions of higher-order algebraic differential equations, Sci. Sinica Ser. A 26 Ž1983., 1034]1043. 6. K. Katajamaki, ¨ Algebroid solutions of binomial and linear differential equations, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 90 Ž1993., 1]48. 7. K. Katajamaki, ¨ Value distribution of certain differential polynomials of algebroid functions, Arch. Math. Ž Basel . 67 Ž1996., 422]429. 8. K. Katajamaki, ¨ Value distribution of some monomials of entire algebroid functions, Complex Variables Theory Appl. 34 Ž1997., 415]420. ¨ 9. H. Selberg, Uber die Wertverteilung der algebroiden Funktionen, Math. Z. 31 Ž1930., 709]728. 10. L. Sons, Deficiencies of monomials, Math. Z. 111 Ž1969., 53]68. ¨ 11. E. Ullrich, Uber den Einfluß der Verzweigtheit einer Algebroide auf ihre Wertverteilung, J. Reine Angew. Math. 167 Ž1932., 198]220. 12. G. Valiron, Sur la derivee Bull. Soc. Math. France 59 Ž1931., ´ ´ des fonctions algebroıdes, ´ ¨ 17]39. 13. K. Yosida, On algebroid-solutions of ordinary differential equations, Japan. J. Math. 10 Ž1934., 199]208.