On the Zeros of a Differential Polynomial and Normal Families

On the Zeros of a Differential Polynomial and Normal Families

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO. 205, 32]42 Ž1997. AY965187 On the Zeros of a Differential Polynomial and Normal Famil...

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

205, 32]42 Ž1997.

AY965187

On the Zeros of a Differential Polynomial and Normal Families* Ye Yasheng Department of Mathematics, East China Uni¨ ersity of Technology, Shanghai, 200093, People’s Republic of China

and

Pang Xuecheng Department of Mathematics, East China Normal Uni¨ ersity, Shanghai, 200062, People’s Republic of China Submitted by J. Mawhin Received December 11, 1992

1. INTRODUCTION Let f Ž z . be a meromorphic function in the complex plane. Throughout this paper we use the familiar notation of value distribution theory Žsee w1x.. For f s f 9 q f n, W. K. Hayman w2x proved that if n G 5 and f is transcendental then f assumes every finite complex number infinitely often. E. Mues w3x then proved that f assumes zero infinitely often in the case n s 4; also, he gave an example to show that for every c / 0, there exists a nonconstant meromorphic function f which satisfies f 9 q f 4 / c. Afterward N. Steinmetz w4x proved that if f 9 q f 4 / c, then f satisfies a Riccati differential equation w9 s 2 p 2 Ž w 2 y p 2 ., p 2 s c / 0. Corresponding to normal family criteria, J. K. Langley w5x and Li w6x proved respectively that if F is a family of meromorphic functions on a domain D and for every f g F, f 9 q f n / c, n G 5, then F is normal on D. Pang Xuecheng w8x and W. Schwick w9x then extended the preceding result. Compare also the discussion in Schiff w10x. * Project supported by the NNSF of China. 32 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

ZEROS OF A DIFFERENTIAL POLYNOMIAL

33

Here we shall improve the above results by proving the following: THEOREM 1. Let f be a meromorphic function in the complex plane, let f s f n q f Ž k . q a1 f Ž ky1. q ??? qak f q a kq1 , where n s k q 3 and f Ž k . q a1 f Ž ky1. q ??? qak f q a kq1 k 0, and let a1 , . . . , a kq1 be k q 1 complex numbers. If N Ž r, 1rf . s SŽ r, f ., then f satisfies the Riccati differential equation w9 s p1w 2 q p 2 w q p 3 , where p1ny2 s Hr2 k!, p 2 s y6 a1rŽ3n y 2. k, p 3 s Hrn, and H is a constant. THEOREM 2. Let f and f be assumed as in Theorem 1. If N Ž r, 1rf . s Ž S r, f . and one of the following conditions is satisfied, Ž1. Ž2.

n is an odd number and a1 s 0, n is an e¨ en number and a1 s a2 s 0,

then f is a constant. Concerning the normality criterion, we have THEOREM 3. Let F be a family of meromorphic functions on a domain D. Suppose that for e¨ ery f g F, f n q f Ž k . q a1Ž z . f Ž ky1. q ??? qa ky1Ž z . f 9 q a k Ž z . f / aŽ z ., where n G k q 3, and a1Ž z ., . . . , a k Ž z ., aŽ z . are k q 1 holomorphic functions on D. Then F is normal.

2. SOME LEMMAS AND PROOFS OF THEOREMS LEMMA 1. Let f be a transcendental meromorphic function and n be a positi¨ e integer. If f n P w f x s Qw f x, and deg Q F n, where P and Q are two differential polynomials in f with coefficient functions satisfying mŽ r, a. s SŽ r, f ., then mŽ r, P . s S Ž r, f .. Lemma 1 is a modified version of Clunie’s theorem w11x. LEMMA 2 w4x. Let f be a meromorphic function in < z < - `, which satisfies mŽ r, f . s SŽ r, f .. Suppose that N1Ž r, f . s N Ž r, f . y N Ž r, f . s S Ž r, f . and in some neighbourhood of the simple poles z ¨ of f f Ž z¨ . s

aŽ z¨ . z y z¨

q b Ž z¨ . q o Ž z y z¨ . ,

where aŽ z . and bŽ z . are small functions of f, i.e., T Ž r, aŽ z .. s oŽ SŽ r, f ... Then f satisfies the Riccati differential equation w9 s A 0 Ž z . w 2 q A1Ž z . w q A 2 Ž z ., where A 0 , A1 , A 2 are small functions of f. Lemma 2 is also true if mŽ r, 1rf . s N1Ž r, 1rf . s SŽ r, f . and for all simple zeros z ¨ of f, z ª z ¨ , f Ž z . s aŽ z ¨ .Ž z y z ¨ . q bŽ z ¨ .Ž z y z ¨ . 2 q O ŽŽ z y z ¨ . 3 .. ŽSee w4, Lemma 2x..

34

YASHENG AND XUECHENG

Proof of Theorem 1. We set w s f 9rf ; then nf ny 1 f 9 q f Ž kq1. q a1 f Ž k . q ??? qa ky1 f 0 q a k f 9 s w f n q f Ž k . q a1 f Ž ky1. q ??? qaky1 f 9 q a k f q a kq1 .

Ž 1.

Rewriting Ž1. in the form Hf ny 1 s w Ž f Ž k . q a1 f Ž ky1. q ??? qa k f q a kq1 . y f Ž kq1. y a1 f Ž k . y ??? ya k f 9,

Ž 2.

where H s nf 9 y w f ,

Ž 3.

we have H k 0. Otherwise by integration, we easily have Ž1 y C . f n q f Ž k . q ??? qa1 f Ž ky1. q ??? qa k f q a kq1 ' 0. But this is impossible according to assumptions n s k q 3 and f Ž k . q a1 f Ž ky1. q ??? qa k f q a kq1 k 0. By Lemma 1, mŽ r, H . s SŽ r, f .. By Ž2. we see that if z 0 is a pole of f, then z 0 cannot be the pole of H because n s k q 3. Only the zeros of w can cause the poles of H, so N Ž r, H . s N Ž r, 1rw . s S Ž r, f .. Hence T Ž r, H . s SŽ r, f .. Again by Ž2., Ž3.,

mŽ r , f . q m r

1

ž / f

s SŽ r , f . .

Ž 4.

From Ž2., it is easy to see that N1. Ž r, f . s T Ž r, f . q S Ž r, f ., where N1.Ž r, f . denotes the N function with respect to the simple poles of f. If z 0 is a simple pole of f and f Ž z . s by1 rŽ z y z 0 . q b 0 q ??? in a neighbourhood of z 0 , then

ws

yn z y z0

q

nb0 by1

q O Ž z y z0 . .

Ž 5.

35

ZEROS OF A DIFFERENTIAL POLYNOMIAL

By computation of Ž2., the left side of Ž2. is ny 1 H0 by1

Ž z y z0 .

ny 1

q

ny 1 ny 2 H1 by1 q H0 Ž n y 1 . b 0 by1

Ž z y z0 .

Ž ny2 .

q terms of higher degree of Ž z y z 0 . and the right side of Ž2. is

Ž y1.

kq 1

q

Ž nk!y Ž k q 1 . ! . by1 kq2 Ž z y z0 .

k k Ž y1. a1 by1 Ž k y 1 . ! Ž n y k . q Ž y1. nk! b 0

Ž z y z0 .

kq 1

q terms of higher degree of Ž z y z 0 . , where H0 s H Ž z 0 . and H1 s H9Ž z 0 .. By comparing their coefficients, ny2 H0 by1 s Ž y1 .

kq 1

2 k!

Ž 6.

and b0 by1

sy

H1

2 3n y 2

ž / H0

y

3a1

Ž 3n y 2 . k

.

Ž 69 .

So by Lemma 2,

w9 s

1 n

w 2 q A1 w q A 2 ,

Ž 7.

where A1 s

4 3n y 2

H9

ž / H

q

6 a1

Ž 3n y 2 . k

and A 2 is a small function. Rewriting Ž3. in the form f9 s

H n

q

1 n

wf

Ž 8.

36

YASHENG AND XUECHENG

and differentiating f and w , we have f0 s f-s

2 n

w2f q

2

6

w fq 3

n3 q

ž

3H n3

A1

wfq

n

6 A1 n2

A2 n

w fq 2

ž

wH

fq

n

5 A2 n2

q

2

AX1

q

H9 n A1

q

n

/

n

w 2 q terms of lower degree of w

wfq

ž

A1 A 2 n

q

AX2 n

/

f

/

.......... f Žk. s

k!

wkf q

nk q

ž

Ck n

ky1

Bk n ky1 A2 q

A1 w ky1 f Dk n

ky2

AX1 q

Ek n ky2

A21 w ky2 f

/

q Ž terms of lower degree of w . f q q terms of lower degree of w

k! 2 nk

Hw kq 1

Ž k G 2. ,

where B1 s C1 s D 1 s E1 s 0 and for k G 2 Bk s Ž k y 1 . Ž Ž k y 1 . !y Bky1 . ;

Ck s Ž k y 1 . Ž k y 1 . !q Ž k y 2 . Cky1 ,

D k s Bky1 q Ž k y 2 . Cky1 ;

Ek s Ž k y 2 . Ž Bky1 q Eky1 . .

If we substitute all f Ž j. in Ž2., then in the case k G 2, Hf ny1 s

ž

2 k! n

kq1

q

ž

w kq1 q

ž

nCk y Ckq1 n

k

nBk y Bkq1 n

A1 q

ky1

A2 q q

3a1 Ž k y 1 . !

nDk y D kq1 n

ky1

nBky 1 y Bk n

ky1

nk AX1 q

q Ž terms of lower degree of w . f q

/

q terms of lower degree of w J Pkq 1 Ž w . f q Pk Ž w . .

n kq 1

wk

nEk y Ekq1 n ky1

a1 A1 q k!

/

A21

4 a2 Ž k y 2 . !

Hw

n ky1 k

/

w ky1

Ž 9.

37

ZEROS OF A DIFFERENTIAL POLYNOMIAL

Differentiation of Ž9. gives H9 f ny 1 q Ž n y 1 . Hf ny 2 f 9 s Ž Pkq 1 Ž w . . 9 f q Pkq1 Ž w . f 9 q Ž Pk Ž w . . 9.

Ž 10 .

If we substitute Ž8. and Ž9. in Ž10., then Ž10. can be written in the form ny1 n

H 2 f ny2 s

žŽ

Pkq 1 Ž w . . 9 q y

ny1

y

n H9 H

Pkq 1 Ž w . n

y

H9

Pkq 1 Ž w . w f q

/

ny1

Pk Ž w . y

Pkq1 Ž w .

H

H n

Pkq1 Ž w . q Ž Pk Ž w . . 9

Pk Ž w . w .

n

Ž 11 .

The coefficient of w kq 1 in H n

Pkq 1 Ž w . q Ž Pk Ž w . . 9 y

H9 H

Pk Ž w . y s

ny1

2 k! H n

Pk Ž w . w

n q

kq 2

kk! H n

kq2

y

Ž n y 1 . k! H n kq2

s 0,

so its degree on w is at most k s n y 3. Let f Ž z . s by1 rŽ z y z 0 . q b 0 q ??? , z ª z 0 ; then in the neighbourny2 Ž hood of z 0 , the left side of Ž11. s ŽŽ n y 1.rn. H02 by1 z y z 0 .yn q2 q terms of higher degree of Ž z y z 0 ., and the right side of Ž11. s uŽ z 0 . n k by1Ž z y z 0 .yn q2 q terms of higher degree of Ž z y z 0 ., where H0 s H Ž z 0 . and uŽ z . is a small function. Hence ny1 n

ny2 H02 by1 s u Ž z 0 . n k by1

and by Ž6., by1 s

Ž n y 1 . H0 Ž y1. n kq1 u Ž z 0 .

kq 1

2 k!

and

ž

b 0 s by1 y

2 3n y 2

H1

ž / H0

y

3a1

Ž 3n y 2 . k

/

.

Again by Lemma 2, f 9 s p1 f 2 q p 2 f q p 3 , where T Ž r, pi . s SŽ r, f ., i s 1 to 3.

Ž 12 .

38

YASHENG AND XUECHENG

It is easy to see that T Ž r, f . s T Ž r, w . q SŽ r, f .. Hence by Ž3. and Ž12.,

w s np1 f q np2 ,

Ž 13 .

and np 3 s H.

Ž 14 .

Differentiating Ž13. gives w 9 s npX1 f q np1 f 9 q npX2 . Combining this with Ž7., Ž12., Ž13., and Ž14., we have pX1

A1 s

y p2 ,

p1

Ž 15 .

and

pX1

A 2 s np1 p 3 q npX2 y np 2

p1

.

Ž 16 .

Rewriting Ž13. in the form f s wrnp1 y p 2rp1 and substituting it in Ž9., then by comparing their coefficients on

H

ny 1

1

ž /

2 k!

s

np1

n

kq 2

,

p1

Ž 17 .

and 2 HCny 1

Ž ny3 .

1

p2

2

ž / ž / np1

sy q

ž ž

p1

nBk y Bkq1 n

nCk y Ckq1 n q

k

A1 q A2 q

nEk y Ekq1 n ky1 q

=

k

1 np1

q

n

k! n kq 1

H.

n

k

nDk y D kq1 n ky1

/

p2 p1 AX1

A21

nBky 1 y Bk ky1

3a1 Ž k y 1 . !

a1 A1 q

4 a2 Ž k y 2 . ! n ky1

/ Ž 18 .

39

ZEROS OF A DIFFERENTIAL POLYNOMIAL

By Ž14. ] Ž17. and the representation of A1 , Ž18. can be written in the form

ž

k! n

kq1

q

s U1

nCk y Ckq1 n kq1 H9

X

ž / H

q U2

/ ž / Hp1

H9

2

q U3

H

H9 H

Ž 19 . q U4 ,

where Ui Ž i s 1 to 4. are constants. It is easy to show that Ck G kk!r4 by induction, so nCk y Ckq1 G 0. Again by Ž17. and Ž19., H must be a constant. This proves the case k G 2. In the case k s 1, in a neighbourhood of the simple zeros z ¨ of f, the right side of Ž3. is

ž

4 b1 q 8 b 2 y b1

ž

2 b 2 q a1 b1 b1 q a2

//

Ž z y z¨ . q terms of higher degree of Ž z y z ¨ . ,

where b1 s f 9 Ž z ¨ .

and

b2 s

H9 Ž z ¨ . Ž b1 q a2 . q a1 b12 6 b1 q 8 H9 Ž z ¨ .

.

Again by Lemma 2, f can be represented as f 9 s p1 f 2 q p 2 f q p 3 . Differentiating f 9 and substituting it in f 0, we have f 0 s 2 p12 f 3 q Ž 3 p1 p 2 q pX1 . f 2 q Ž pX2 q 2 p1 p 3 q p 22 . f q pX3 q p 2 p 3 . Notice that Ž13. is also true in the case k s 1, so Ž2. can be written in the form Hf 3 s 2 p12 f 3 q Ž 5 p1 p 2 y pX1 q 3a1 p1 . f 2 q Ž 3a1 p 2 q 4 a2 p1 q 2 p1 p 3 q 3 p 22 y pX2 . f q Ž 3 p 2 p 3 q 4 a2 p 2 y pX3 y a1 p 3 . . By comparing its coefficients H s 2 p12 5 p1 p 2 y

pX1

Ž 20 .

q 3a1 p1 s 0

Ž 21 .

3a1 p 2 q 4 a2 p1 q 2 p1 p 3 q 3 p 22 y pX2 s 0

Ž 22 .

3 p 2 p 3 q 4 a2 p 2 y

pX3

y a1 p 3 s 0. p1r5 pX1 ,

Ž 23 .

Rewriting Ž21. in the form p 2 s a1r5 q we have T Ž r, p 2 . s SŽ r, p1 .. Also, by Ž20., Ž22., and Ž14., T Ž r, p 3 . s S Ž r, p 2 . and T Ž r, p1 . s SŽ r, p 2 .. Thus all pi are constants and we complete the proof of Theorem 1.

40

YASHENG AND XUECHENG

Proof of Theorem 2. If n is an even number and a1 s a2 s 0, then U4 s 0 and hence H s 0. So f is a constant. If n is an odd number and a1 s 0, then p 2 s 0 and hence f s CtgŽ Az q B .. Substituting f in w gives n

w s Ž Ctg Ž Az q B . . q Ž Ctg Ž Az q B . . q a1 Ž Ctg Ž Az q B . .

Ž ky1 .

Žk.

q ??? qa kq1

s C n tg n Ž Az q B . q Q kq 1 Ž tg Ž Az q B . . , where Q kq 1 is a polynomial of degree k q 1. If N Ž r, 1rw . s SŽ r, f ., then w s C n ŽtgŽ Az q B . q i . p ŽtgŽ Az q b . y q . i , which implies that Ž p y 1. i q Ž q y 1.Žyi . s Ž p y q . i s 0. But this is impossible because n is odd and p q q s n. Hence f must be a constant and the result is proved. COROLLARY 1. Let f be a meromorphic function in C and n and k be two positi¨ e integers If f satisfies one of the following conditions, Ž1. f 4 q f 9 / 0, Ž2. f n q f Ž k . / a, n G k q 3, k G 2 Ž or: n G k q 4, k G 1., a is a finite complex number, then f must be a constant. Remark. The proof of Corollary 1 when n G k q 4 is the same as that given in w2x. In order to prove Theorem 3, we need the following. LEMMA 3 w7x. Let F be a family of meromorphic functions on the unit disc. If F is not normal, then for e¨ ery gi¨ en k Žy1 - k - 1. there exist 1. 2. 3. 4.

a real number r, 0 - r - 1, complex numbers z n , < z n < - r, functions f n g F, n s 1, 2, . . . , and positi¨ e numbers rn

which satisfy lim nª` rn s 0 and lim n ª`ŽŽ r y z n .rrn . s q`, such that rnk f nŽ z n q rn j . ª g Ž j . spherically uniformly on compact subsets of C, where g is a nonconstant meromorphic function on C. Proof of Theorem 3. Without loss of generality, we can assume that D is the unit disc. If F is not normal, then there exist r, z m , rm such that g mŽ j . s rmk rŽ ny1. f mŽ z m q rm j . is convergent to g Ž j . uniformly on compact subsets of C, where g Ž j . is a nonconstant meromorphic function.

41

ZEROS OF A DIFFERENTIAL POLYNOMIAL

If g n Ž j . q g Ž k . Ž j . / 0, then by Corollary 1, g must be a constant and we arrive at a contradiction. Hence there exists j 0 such that g n Ž j 0 . q g Ž k . Ž j 0 . s 0. It is easy to see that g Ž j 0 . / `. So there exists d ) 0 such that g Ž j . is holomorphic in D 2 d s Ž j N < j y j 0 < - 2 d ., and for all sufficiently large m, g mŽ i. Ž j . are holomorphic in Dd s Ž j N < j y j 0 < - d .. Furthermore g mŽ i. Ž j . are convergent to g Ž i. Ž j . uniformly on Dd Ž i s 0 to k .. Now we set ky1

Lm Ž j . s

Ý

a kyi Ž z m q rm j . rmk n rŽ ny1.yi g mŽ i. Ž j .

is0

y rmk n rŽ ny1. a Ž z m q rm j . . Then L mŽ j . tends to zero uniformly on Dd for knrŽ n y 1. y i ) krŽ n y 1. and < a ky i Ž z m q rm j .< F M ŽŽ1 q r .r2, a kyi Ž z .. - ` Ž i s 0 to k y 1.. Therefore g mn Ž j . q g mŽ k . Ž j . q L m Ž j . tends to g n Ž j . q g Ž k . Ž j . uniformly on Dd . But g mn Ž j . q g mŽ k . Ž j . q L m Ž j . s rmk n rŽ ny1.

ž

= f mn Ž z m q rm j . q f mŽ k . Ž z m q rm j . ky1

q

Ý

/

a ky i Ž z m q rm j . f m Ž z m q rm j . y a Ž z m q rm j . / 0,

is0

so by the Hurwitz theorem g n Ž j . q g Ž k . Ž j . ' 0 on Dd . This implies g n q g Ž k . ' 0 on C, but this is also impossible.

REFERENCES 1. W. K. Hayman, ‘‘Meromorphic Function,’’ Oxford Univ. Press, London, 1964. 2. W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. 70 Ž1959., 9]42. 3. E. Mues, Uber ein Problem von Hayman, Math. Z. 164 Ž1979., 239]259. 4. N. Steinmetz, Uber die Nullstellen von Differential polynomes, Math. Z. 176 Ž1981., 255]264. 5. J. K. Langley, On normal families and a result of Drasin, Proc. Roy. Soc. Edinburgh Sect A 98 Ž1984., 385]393.

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YASHENG AND XUECHENG

6. L. Xianjin, Proof of a conjecture of Hayman, Sci. Sinica Ser. A 28 Ž1985., 596]603. 7. P. Xuecheng, A normal criterion of meromorphic function, Sci. Sinica Ser. A Ž1989., 923]928. 8. P. Xuecheng, A normal criterion for a differential polynomial, Kexue Tongbao Ž Chinese. 22 Ž1989., 1690]1693. 9. W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 Ž1989., 241]289. 10. J. L. Schiff, ‘‘Normal Families,’’ Springer-Verlag, New York, 1993. 11. J Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 Ž1962., 17]27.