NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 71-75

ON THEOREMS OF HAYMAN AND CLUNIE

X u e c h e n g P a n g a n d L a w r e n c e Z a l c m a n '

(Received April 1998)

Abstract. We give a simple proof of the fact that if / is a transcendental entire function, all of whose zeros have multiplicity at least k, then for each natural number n, takes on every nonzero value infinitely often. Correspondingresults on normal families are also proved.

In a celebrated paper [4], Hayman proved that if / is a transcendental entire function, then f ' f n takes on every nonzero value infinitely often for n > 2. The case n — 1 proved refractory and was only settled (affirmatively) several years later, by Clunie [2]. It was another decade and a half before Oshkin proved that a family of analytic functions on a domain all of which satisfy 1 is normal [7].

In this note, we give simple “soft” proofs of slightly more general versions of the theorems of Hayman and Clunie, based on general principles of normal families. We also derive the corresponding analogues for normal families, due to Yang and Chang [12] and Oshkin [7], respectively. What allows us to sidestep all technicalities of Nevanlinna theory is that it turns out to be possible to reduce matters to a consideration of functions of exponential type, where the discussion is elementary.

Our proof is based on the following result.

Lemma 1. Let T be a family of holomorphic functions on the unit disc A all of whose zeros have multiplicity at least k, and let a be a real number satisfying —I < a < k. Then T is not normal on A if and only if there exists

(i) a number r, 0 < r < 1;

(ii) points Zj, \zj\ < r;

(iii) positive numbers pj, pj —> 0; and

(iv) functions fj e Tsuch that p~afj(zj + pjC,) —> g(Q uniformly on compact subsets of C, where g is a nonconstant entire function. The function g may be taken to satisfy the normal­ization g*(z) < g*{ 0) = 1 , z EC, where

- I g Mg*(z) =1 + |0(*)I2

denotes the spherical derivative.

1991 AMS Mathematics Subject Classification: 30D35, 30D45.Research of Pang Xuecheng supported by the NNSF of China Approved No. 19771038 and by the Research Institute for Mathematical Sciences, Bar-Ilan University.

72 XUECHENG PANG AND LAWRENCE ZALCMAN

This is a version of Pang’s generalization [8], [9] of Zalcman’s Lemma [13], [11, p. 101] (where a is taken to be 0), with improvements due to Chen and Gu [1 ]; cf. [14].

The proof of Lemma 1 is based on an auxiliary result for functions on the disc. Let / be meromorphic on A and suppose that each zero of / has multiplicity greater than or equal to k. Fix — 1 < a < k and 0 < r < 1 and define for \z\ < r and0 < p < 1

H f l 2 ) I+V ™F(z,p)

Then we have

Lemma 2 (cf. Lemma 1 of [8]). If sup|z|<r F(z, 1) > 1, then there exist zo, po such that |201 < r, 0 < po < 1 and

sup F(z,po) = F(z0,po) = 1.\z\<r

The proof of Lemma 2 is a simple consequence of the fact that, under the given hypotheses, F(z,p) extends as a continuous function on the compact set { (2, p) : \z\ < r, 0 < p < 1} which vanishes when \z\ = r or p = 0. Once Lemma 2 has been established, the proof of Lemma 1 is identical to that of Lemma 2 in [8].

We wish to emphasize that the proofs of the results in this section are completely elementary and, in particular, make no use of Nevanlinna Theory.

2.We also require the fact, due to Clunie and Hayman [3], that an entire function

with bounded spherical derivative must be of exponential type. Minda [6] has given an extremely simple proof of (a more precise version of) this fact, based on the following result of Hayman, as sharpened by Pommerenke [10].

Theorem 3. Let f be analytic on A and suppose that

sup( 1 - \z\2)f*(z) = a.A

Then, for z G A,(! ~ \z\2)\f'(z)\ < 2 ( log+ \f{z)\ + a) max (|/(z)|, l) (1)

and hence

log+ ,/(2)l - r q f | log+ l/(0)l + ' (2)

Pommerenke’s proof of (1) is based on a simple application of Ahlfors’ version of Schwarz5 Lemma, and (2) follows by integration; cf. [5, pp. 168-169] and [11, pp. 92-93] for the explicit deduction of (2) from (1).

For the reader’s convenience, we reproduce Minda’s argument here.

Theorem 4. Let f be an entire function such that supc f^(z) = A. Then, for all z G C |/(2)| < max(l, |/(0)|) e2Alzl. In particular, f is of exponential type.

ON THEOREMS OF HAYMAN AND CLUNIE 73

Proof. For R > 0 put g(w) = f(Rw). Then(1 — \w\2)g#(w) < g^(w) = Rf^(Rw) < RA.

Applying (2), we obtain for w € A! -4- , / M 1 + M , 4-1 /r\ \ t 2AR\w\log+ |#(w)| < -----— log+ \g(0)\ + ----- —

1 — |w;| 1 — | it; |or, with z = Rw,

‘° g + \ m \ < iog+ i/(o)i +

for \z\ < R. Fixing 2 and letting R —> oo, we obtainlog+ \f(z)\ < log+ |/(0)| + 2A\z\,

which is what we want to prove. □

3.

We now turn to our results.

Theorem 5. Let f be a transcendental entire function all of whose zeros have multiplicity at least k, and let n be a natural number. Then f ^ f n takes on each nonzero value a 6 C infinitely often.

Proof. Suppose first that f has exponential type. If F = f ^ f n takes on the value a 0) only finitely often, then / must have infinitely many zeros. For otherwise, f(z) = Pi(z)eAz and so f^h\z) = P2 {z)eAz, where Pi and P2 are polynomials. Thus F(z) = P(z) eBz, where P is a polynomial, so F takes on the values a and0 only finitely often, in contradiction to Picard’s Theorem. Thus / , and hence F, have infinitely many (common) zeros. Since F = a only finitely often, we have F(z) = a + Q(z) eCz for some polynomial Q.

If max(fc,n) > 2, F' vanishes whenever F does. But F'{z) = [Q'(z)+CQ(z)]eCz, which has only a finite number of zeros, a contradiction. If k = n — 1, F(z) — f { z ) f(z) = a + Q(z) eCz can be integrated to f 2(z) = Q(z) eCz + 2az + D, where Q' + CQ = 2Q. Since f 2 and F have infinitely many common zeros, so does Q f 2 — QF; but this is the (nonzero) polynomial (2az 4- D) Q(z) — aQ(z), again a contradiction. Thus F = a infinitely often.

Now suppose / does not have exponential type. Then /# is unbounded in the plane, so we may find a sequence {u>j} tending to infinity such that f#(wj) —» 00. Write fj(z) = f(z + Wj) and consider the family T = {f j} on the unit disc A. Since f * (0) = f * {w 3) -> 00, T is not normal on A. Taking a subsequence and renumbering if necessary 1, we have by Lemma 1 that there exist Zj € A, pj —> 0+, and a nonconstant entire function g of exponential type such that

pJk/{n+1)f(Wj + z j + pj Q = p j k/in+1)fj(zj +PjC) = 9j(0~*ff(C)uniformly 011 compact subsets of C. It follows that

• PiC) = 9 ? ) ( 0 9 j ( 0 ^ 9 ik)( 0 g n ( 0 -

1 Actually, it is never necessary.

74 XUECHENG PANG AND LAWRENCE ZALCMAN

If g is transcendental, g ^ g n takes on each value a / 0 infinitely often; otherwise, g is a (nonconstant) polynomial. But then so is g ^ g n, since (by Hurwitz’ Theo­rem) each zero of g has order at least k (so that g ^ ^ 0). In either case, there exists Co C such that g k\(o)gn(Co) — a- Hence there exists a sequence {Cj}, Q -> Co, such that (for large j) 9jk)(Cj)9j(Cj) = a, i-e. / (fc)(rj ) / n(rj) = where Tj = Wj+Zj+pjQ. Since Wj —> oo, while Zj, pj, and (j all remain bounded, it is clear that Tj —► oo. Thus there are infinitely many solutions to f^k\z) f n(z) = a. □

Theorem 6. Let T be a family of analytic functions on A all of whose zeros have multiplicity greater than or equal to k. Suppose that there exists n > 1 and a £ C, a / 0 , such that f ^ f n ^ a for each f € T . Then T is a normal family.

Proof. Suppose not. Then, by Lemma 1, there exist Zj £ A, pj —► 0+, and fj G T such that

P j k/(n+1)f j ( Zj + pjC) = 9 j { 0 -> ■ f f ( 0locally uniformly on C, where g is a nonconstant entire function of exponential type. Then

f f ^ i + P i O l f ^ i + P i O = 9 f ) ( C ) s j ’, ) ( C ) - 9 <*, ( C ) 9 n (C)uniformly on compacta. Since ^ a, we must have either yn / a org(k)gn = a. If g is transcendental, the latter alternative obviously cannot occur; and the former is ruled out by Theorem 5. If g is a nonconstant polynomial, then (as we have already seen) so is g ^ g n\ again, neither alternative can obtain. The contradiction establishes the theorem. □

Since this paper was submitted and accepted for publication, we have become aware of Chen Huaihui’s paper, “Yosida functions and Picard values of integral functions and their derivatives,” Bull. Austral. Math. Soc. 54 (1996), 373-381, in which closely related results are obtained by essentially the same method.

References

1 . Chen Huaihui and Gu Yongxing, An improvement of Marty’s criterion and its applications, Sci. China Ser. A 36 (1993), 674-681.

2. J. Clunie, On a result of Hayman, J. London Math. Soc. 42 (1967), 389-392.3. J. Clunie and W.K. Hayman, The spherical derivative of integral and mero-

morphic functions, Comment. Math. Helv. 40 (1966), 117-148.4. W.K. Hayman, Picard values of meromorphic functions and their derivatives,

Ann. Math. (2) 70 (1959), 9-42.5. W.K. Hayman, Meromorphic Functions, Oxford University Press, London,

1964.6. David Minda, Yosida functions, Lectures on Complex Analysis (Xian, 1987),

(Chi-Tai Chuang, ed.), World Scientific Pub. Co., Singapore, 1988, pp. 197-213.7. I.B. Oshkin, On a test of the normality of families of holomorphic functions,

Uspehi Mat. Nauk 37 (2) (1982), 221-222; Russian Math. Surveys 37 (2) (1982), 237-238.

8. Pang Xue-cheng, Bloch’s principle and normal criterion, Sci. China Ser. A 32 (1989), 782-791.

ON THEOREMS OF HAYMAN AND CLUNIE 75

9. Pang Xue-cheng, On normal criterion of meromorphic functions, Sci. China Ser. A 33 (1990), 521-527.

10. Ch. Pommerenke, Estimates for normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I 476 (1970).

11. Joel L. Schiff, Normal Families, Springer, 1993.12. Yang Le and Chang Kuang-hou, Recherches sur la normalite des families de

fonctions analytiques a des valeurs multiples, I. Un nouveau critere et quelques applications, Sci. Sinica, 14 (1965), 1258-1271.

13. Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813-817.

14. Lawrence Zalcman, New light on normal families, in Proceedings of the Ashkelon Workshop on Complex Function Theory (May, 1996), IMCP, vol 11, (L. Zalcman, ed.), Amer. Math. Soc. 1997, 237-245.

xcpang@math.ecnu.edu.cn

Xuecheng Pang Department of Mathematics East China Normal University Shanghai 200062 P.R. CHINA

Lawrence ZalcmanDepartment of Mathematics and Computer ScienceBar-Ilan University 52900 Ramat-Gan ISRAEL

zalcman@macs.biu.ac.il