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1 NORMAL CRITERIA FOR FAMILIES OF MEROMORPHIC FUNCTIONS Gerd Dethloff a and Tran Van Tan b, and Nuyen Van Thin c a Université de Brest, LMBA, UMR CNRS 6205, 6, avenue Le Goreu - C.S , Brest Cedex 3, France b Department of Mathematics, Hanoi National University of Education, 36 Xuan Thuy Street, Cau Giay, Hanoi, Vietnam c Department of Mathematics, Thai Nuyen University of Education, Luon Noc Quyen Street, Thai Nuyen City, Vietnam Abstract By usin Nevanlinna theory, we prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials enerated by the members of the family. Keywords: Meromorphic function, Normal family, Nevanlinna theory. Mathematics Subject Classification 200: 30D35. Introduction Let D be a domain in the complex plane C and F be a family of meromorphic functions in D. The family F is said to be normal in D, in the sense of Montel, if for any sequence {f v } F, there exists a subsequence {f vi } such that {f vi } converes spherically locally uniformly in D, to a meromorphic function or. In 989, Schwick proved: Correspondin author. addresses: a erd.dethloff@univ-brest.fr, b tranvantanhn@yahoo.com, c thinmath@mail.com The second named author is currently Reular Associate Member of ICTP, Trieste, Italy. This research is funded by Vietnam National Foundation for Science and Technoloy Development (NAFOSTED).

2 Theorem A ([6], Theorem 3.). Let k, n be positive inteers such that n k + 3. Let F be a family of meromorphic functions in a complex domain D such that for every f F, (f n ) (k) (z) for all z D. Then F is normal on D. Theorem B ([6], Theorem 3.2). Let k, n be positive inteers such that n k +. Let F be a family of entire functions in a complex domain D such that for every f F, (f n ) (k) (z) for all z D. Then F is normal on D. The followin normality criterion was established by Pan and Zalcman [7] in 999: Theorem C ([7]). Let n and k be natural numbers and F be a family of holomorphic functions in a domain D all of whose zeros have multiplicity at least k. Assume that f n f (k) is non-vanishin for each f F. Then F is normal in D. The main purpose of this paper is to establish some normality criteria for the case of more eneral differential polynomials. Our main results are as follows: Theorem. Take q (q ) distinct nonzero complex values a,..., a q, and q positive inteers (or + ) l,... l q. Let n be a nonneative inteer, and let n,..., n k, t,..., t k be positive inteers (k ). Let F be a family of meromorphic functions in a complex domain D such that for every f F and for every m {,..., q}, all zeros of f n (f n ) (t ) (f n k ) (t k ) a m have multiplicity at least l m. Assume that a) n j t j for all j k, and l i 2 for all i q, b) q i= l i < qn 2+ k q(n j t j ) n+ k (n j+t j ) Then F is a normal family.. Take q = and l = +, we et the followin corollary of Theorem : Corollary 2. Let a be a nonzero complex value, let n be a nonneative inteer, and n,..., n k, t,..., t k be positive inteers. Let F be a family of meromorphic functions in a complex domain D such that for every f F, f n (f n ) (t ) (f n k ) (t k ) a is nowhere vanishin on D. Assume that a) n j t j for all j k, b) n + k n j 3 + k t j. Then F is normal on D. 2

3 We remark that in the case where n 3, condition a) in the above corollary implies condition b); and in the case where n = 0 and k =, Corollary 2 ives Theorem A. For the case of entire functions, we shall prove the followin result: Theorem 3. Take q (q ) distinct nonzero complex values a,..., a q, and q positive inteers (or + ) l,... l q. Let n be a nonneative inteer, and let n,..., n k, t,..., t k be positive inteers (k ). Let F be a family of holomorphic functions in a complex domain D such that for every f F and for every m {,..., q}, all zeros of f n (f n ) (t ) (f n k ) (t k ) a m have multiplicity at least l m. Assume that a) n j t j for all j k, and l i 2 for all i q, b) q i= l i < qn + k q(n j t j ) n+ k n j Then F is a normal family.. Take q = and l = +, Theorem 3 ives the followin eneralization of Theorem B, except for the case n = k +. So for the latter case, we add a new proof of Theorem B in the Appendix which is slihtly simpler than the oriinal one. Corollary 4. Let a be a nonzero complex value, let n be a nonneative inteer, and n,..., n k, t,..., t k be positive inteers. Let F be a family of holomorphic functions in a complex domain D such that for every f F, f n (f n ) (t ) (f n k ) (t k ) a is nowhere vanishin on D. Assume that a) n j t j for all j k, b) n + k n j 2 + k t j. Then F is normal on D. In the case where n 2, condition a) in the above corollary implies condition b). Remark 5. Our above results remain valid if the monomial f n (f n ) (t ) (f n k ) (t k ) is replaced by the followin polynomial f n (f n ) (t ) (f n k ) (t k) + I c I f n I (f n I ) (t I) (f n ki ) (t ki), where c I is a holomorphic function on D, and n I, n ji, t ji are nonneative inteers satisfyin α I := t ji n I + k n < α := t j ji n + k n. j 3

4 2 Some notations and results of Nevanlinna theory Let ν be a divisor on C. The countin function of ν is defined by N(r, ν) = r n(t) dt (r > ), where n(t) = ν(z). t z t For a meromorphic function f on C with f, denote by ν f the pole divisor of f, and the divisor ν f is defined by ν f (z) := min{ν f (z), }. Set N(r, f) := N(r, ν f ) and N(r, f) := N(r, ν f ). The proximity function of f is defined by m(r, f) = 2π 2π where lo + x = max{lo x, 0} for x 0. The characteristic function of f is defined by 0 lo + f(re iθ ) dθ, T (r, f) := m(r, f) + N(r, f). We state the Lemma on Loarithmic Derivative, the First and Second Main Theorems of Nevanlinna theory. Lemma on Loarithmic Derivative. Let f be a nonconstant meromorphic function on C, and let k be a positive inteer. Then the equality m(r, f (k) ) = o(t (r, f)) f holds for all r [, ) excludin a set of finite Lebesue measure. First Main Theorem. Let f be a meromorphic functions on C and a be a complex number. Then T (r, ) = T (r, f) + O(). f a 4

5 Second Main Theorem. Let f be a nonconstant meromorphic function on C. Let a,..., a q be q distinct values in C. Then (q )T (r, f) N(r, f) + q N(r, ) + o(t (r, f)), f a i i= for all r [, ) excludin a set of finite Lebesue measure. 3 Proof of our results To prove our results, we need the followin lemmas: Lemma 6 (Zalcman s Lemma, see [8]). Let F be a family of meromorphic functions defined in the unit disc. Then if F is not normal at a point z 0, there exist, for each real number α satisfyin < α <, ) a real number r, 0 < r <, 2) points z n, z n < r, z n z 0, 3) positive numbers ρ n, ρ n 0 +, 4) functions f n, f n F such that n (ξ) = f n(z n + ρ n ξ) (ξ) ρ α n spherically uniformly on compact subsets of C, where (ξ) is a non-constant meromorphic function and # (ξ) # (0) =. Moreover, the order of is not reater than 2. Here, as usual, # (z) = is the spherical derivative. (z) + (z) 2 Lemma 7 (see [2]). Let be a entire function and M is a positive constant. If # (ξ) M for all ξ C, then has order at most one. Remark 8. In Lemma 6, if F is a family of holomorphic functions, then by Hurwitz theorem, is a holomorphic function. Therefore, by Lemma 7, the order of is not reater than. We consider a nonconstant meromorphic function in the complex plane C, and its first p derivatives. A differential polynomial P of is defined by P (z) := n p α i (z) ( (j) (z)) S ij, i= j=0 5

6 where S ij ( i n, 0 j p) are nonneative inteers, and α i 0 ( i n) are small (with respect to ) meromorphic functions. Set p p d(p ) := min S ij and θ(p ) := max js ij. i n i n j=0 In 2002, J. Hinchliffe [5] eneralized theorems of Hayman [3] and Chuan [] and obtained the followin result: Proposition 9. Let be a transcendental meromorphic function, let P(z) be a non-constant differential polynomial in with d(p ) 2. Then T (r, ) θ(p ) + d(p ) N(r, ) + d(p ) N(r, ) + o(t (r, )), P for all r [, + ) excludin a set of finite Lebesues measure. In order to prove our results, we now ive the followin eneralization of the above result: Lemma 0. Let a,..., a q be distinct nonzero complex numbers. Let be a nonconstant meromorphic function, let P(z) be a nonconstant differential polynomial in with d(p ) 2. Then T (r, ) qθ(p ) + qd(p ) N(r, ) + qd(p ) j=0 q N(r, ) + o(t (r, )), P a j for all r [, + ) excludin a set of finite Lebesues measure. Moreover, in the case where is a entire function, we have T (r, ) qθ(p ) + N(r, qd(p ) ) + q N(r, ) + o(t (r, )), qd(p ) P a j for all r [, + ) excludin a set of finite Lebesue measure. Proof. For any z such that (z), since p j=0 S ij d(p ) ( i n), we have (z) d(p ) = P (z) P (z) P (z) (z) d(p ) n ( αi (z) i= p (j) (z) S ) ij. (z) j=0 6

7 This implies that for all z C, lo + (z) d(p ) lo+ ( P (z) n ( αi (z) i= p (j) (z) S )) ij. (z) Therefore, by the Lemma on Loarithmic Derivative and by the First Main Theorem, we have d(p )m(r, ) m(r, P ) + o(t (r, )) = T (r, P ) N(r, ) + o(t (r, )) P j=0 = T (r, P ) N(r, ) + o(t (r, )). P On the other hand, by the Second Main Theorem (used with the q+ different values 0, a,..., a q ) we have qt (r, P ) N(r, P ) + N(r, q P ) + N(r, ) + o(t (r, )), P a j Hence, d(p )m(r, ) q ( N(r, P ) + N(r, P ) + Therefore, by the First Main Theorem, we have d(p )T (r, ) = d(p )T (r, ) + O() q N(r, ) ) P a j = d(p )m(r, ) + d(p )N(r, ) + O() q ( N(r, P ) + N(r, P ) + N(r, ) + o(t (r, )). P q N(r, ) ) P a j + d(p )N(r, ) N(r, ) + o(t (r, )). (3.) P We have = d(p ) P (z) n i= ( αi ( p j=0 S ij) d(p ) p j=0 ( (j) )S ij ). 7

8 (note that ( p j=0 S ij) d(p ) 0). Therefore, d(p )ν ν P ν P + max i n {ν α i + + n i= p j=0 ν αi + θ(p )ν, js ij ν } where ν φ is the pole divisor of the meromorphic φ and ν φ := min{ν φ, }. This implies, d(p )ν ν P + q ν P (θ(p ) + q )ν + n ν αi, i= (note that for any z 0, if ν (z 0 ) = 0 then d(p )ν (z 0 ) ν (z 0 ) + ν P q (z 0 ) 0). P Then, d(p )N(r, ) N(r, P ) + q N(r, P ) (θ(p ) + q )N(r, n ) + N(r, α i ) Combinin with (3.), we have i= = (θ(p ) + q )N(r, ) + o(t (r, )). d(p )T (r, ) ( q N(r, P ) + N(r, ) ) + (θ(p ) + q P a j q )N(r, ) + o(t (r, )). On the other hand, by the definition of the differential polynomial P, Pole(P ) n i= Pole(α i ) Pole(). Hence (since N(r, α i ) T (r, α i ) = o(t (r, ) for i =,..., n), we et d(p )T (r, ) ( q N(r, ) + N(r, ) ) + (θ(p ) + q P a j q )N(r, ) + o(t (r, )) ( q T (r, ) + N(r, ) ) + (θ(p ) + q P a j q )N(r, ) + o(t (r, )). (3.2) 8

9 Therefore, T (r, ) qθ(p ) + qd(p ) N(r, ) + qd(p ) q N(r, ) + o(t (r, )). P a j In the case where is an entire function, the first inequality in (3.2) becomes d(p )T (r, ) q q N(r, ) + (θ(p ) + P a j q )N(r, ) + o(t (r, )). This implies that T (r, ) θ(p )q + )N(r, qd(p ) ) + qd(p ) q N(r, ) + o(t (r, )). P a j We have completed the proof of Lemma 0. Proof of Theorem. Without loss the enerality, we may asssume that D is the unit disc. Suppose that F is not normal at z 0 D. By Lemma 6, for α = k t j n+ k n j there exist ) a real number r, 0 < r <, 2) points z v, z v < r, z v z 0, 3) positive numbers ρ v, ρ v 0 +, 4) functions f v, f v F such that v (ξ) = f v(z v + ρ v ξ) ρ α v (ξ) (3.3) spherically uniformly on compact subsets of C, where (ξ) is a non-constant meromorphic function and # (ξ) # (0) =. On the other hand, ( n j v (ξ) ) (t j ) ( f v (z v + ρ v ξ) = ( ) ) n (t j j ) = ρ n jα t j v ρ α v (f n j v ) (tj) (z v + ρ v ξ). 9

10 Therefore, by the definition of α and by (4.), we have fv n (z v + ρ v ξ)(f n v ) (t) (z v + ρ v ξ) (f n k v ) (tk) (z v + ρ v ξ) = v n (ξ)( n v (ξ)) (t)... ( n k v (ξ)) (tk) n (ξ)( n (ξ)) (t)... ( n k (ξ)) (t k) (3.4) spherically uniformly on compact subsets of C. Now, we prove the followin claim: Claim: n (ξ)( n (ξ)) (t )... ( n k (ξ)) (t k ) is non-contstant. Since is non-constant and n j t j (j =,..., k), it easy to see that ( n j (ξ)) (t j) 0, for all j {,..., k}. Hence, n (ξ)( n (ξ)) (t )... ( n k (ξ)) (t k) 0. Suppose that n (ξ)( n (ξ)) (t )... ( n k (ξ)) (t k ) a, a C \ {0}. We first remark that, from conditions a), b), we have that in the case n = 0, there exists i {,..., k} such that n i > t i. Therefore, in both cases (n = 0 and n 0), since a 0, it is easy to see that is entire havin no zero. So, by Lemma 7, (ξ) = e cξ+d, c 0. Then n (ξ)( n (ξ)) (t ) ( n k (ξ)) (t k) = e ncξ+nd (e n cξ+n d ) (t ) (e n kcξ+n k d ) (t k) = (n c) t (n k c) t k e (n+ k n j)cξ+(n+ k n j)d. Then (n c) t (n k c) t k e (n+ k n j)cξ+(n+ k nj)d a, which is impossible. So, n (ξ)( n (ξ)) (t)... ( n k (ξ)) (t k ) is nonconstant, which proves the claim. By the assumption of Theorem and by Hurwitz s theorem, for every m {,..., q}, all zeros of (ξ) n ( n (ξ)) (t ) ( n k (ξ)) (t k ) a m have multiplicity at least l m. For any j {,, k}, we have that ( n j (ξ)) (tj) is nonconstant. Indeed, if ( n j (ξ)) (tj) is constant for some j {,..., k}, then since n j t j, and since is nonconstant, we et that n j = t j and (ξ) = aξ + b, where a, b are constants, a 0. Thus, we can write (ξ) n ( n (ξ)) (t ) ( n k (ξ)) (t k) = c(aξ + b) n+ k (n j t j ), where c is a nonzero constant. This contradicts to the fact that all zeros of (ξ) n ( n (ξ)) (t ) ( n k (ξ)) (t k ) a m have multiplicity at least l m 2 (note that a m 0, and that, by condition b) of Theorem, n+ k (n j t j ) > 0). Thus, ( n j (ξ)) (tj) is nonconstant, for all j {,, k}. 0

11 On the other hand, we can write ( n j ) (t j) = c m0,m,...,m tj m 0 ( ) m... ( (t j) ) mt j, c m0,m,...,m tj are constants, and m 0, m,..., m tj are nonneative inteers such that m m tj = n j, t j jm j = t j. Thus, by an easy computation, we et that d(p ) = n + k n j, θ(p ) = k t j. Now, we apply Lemma 0 for the differential polynomial P = (ξ) n ( n (ξ)) (t ) ( n k (ξ)) (t k). By Lemma 0, we have (note that, by condition b) of Theorem, n + k n j 2) T (r, ) q k t j + qn + q k n j N(r, ) + qn + q k n j q N(r, ) + o(t (r, )). (3.5) P a m m= For any m {,..., q}, we have, by the First Main Theorem, N(r, ) = N(r, P a m n ( n ) (t ) ) ( n k ) (t k) a m N(r, l m n ( n ) (t ) ) ( n k ) (t k) a m l m T (r, n ( n ) (t ) ( n k ) (t k) ) + O() = l m m(r, n ( n ) (t ) ( n k ) (t k) ) + l m N(r, n ( n ) (t ) ( n k ) (t k) ) + O(). (3.6)

12 By the Lemma on Loarithmic Derivative and by the First Main Theorem, m(r, n ( n ) (t ) ( n k ) (t k) ) + N(r, n ( n ) (t ) ( n k ) (t k) ) m(r, n ( n ) (t ) ( n k ) (t k ) n n n k ) + m(r, n n n k ) (n + = (n + (n + + N(r, n ( n ) (t ) ( n k ) (tk) ) n j )m(r, ) + N(r, n ( n ) (t ) ( n k ) (tk) ) + o(t (r, )) n j )m(r, ) + (n + n j )N(r, ) + ( t j )N(r, ) + o(t (r, )) n j )T (r, ) + ( t j )N(r, ) + o(t (r, )). (3.7) Combinin with (3.6), for all m {,..., q} we have N(r, ) (n + P a m l m l m (n + n j )T (r, ) + l m ( n j + t j )N(r, ) + o(t (r, )) t j )T (r, ) + o(t (r, )). (3.8) Therefore, by (3.5) and by the First Main Theorem, we have (qn + q n j )T (r, ) (q (q t j + )N(r, ) + t j + )T (r, ) + (n + This implies that n j + qn + k q(n j t j ) 2 n + k (n T (r, ) j + t j ) q m= q N(r, ) + o(t (r, )) P a m m= t j )( q m= l m T (r, ) + o(t (r, )). l m )T (r, ) + o(t (r, ). Combinin with assumption b) we et that is constant. This is a contradiction. Hence F is a normal family. We have completed the proof of Theorem. 2

13 We can obtain Theorem 3 by an arument similar to the the proof of Theorem : We first remark that althouh condition b) of Theorem 3 is different from condition b) of Theorem, whereever it has been used in the proof of Theorem before equation (3.5), the condition b) of Theorem 3 still allows the same conclusion. And from equation (3.5) on we modify as follows : Since F is a family of holomorphic functions and by Remark 8, is an entire functions. So, similarly to (3.5), by Lemma 0, we have T (r, ) q k t j + qn + q k n N(r, j ) + q(n + k n j) q k t j + qn + q k n T (r, ) + j q(n + k n j) q N(r, ) + o(t (r, )) P a m= m q N(r, ) + o(t (r, )). P a m m= (3.9) Since is a holomorphic function, N(r, ) = 0. Therefore, by (3.6) and (3.7) (which remain unchaned), we have N(r, By (3.9), (3.0), we have ) (n + P a m l m n j )T (r, ) + o(t (r, )). (3.0) qn + k q(n j t j ) n + k n T (r, ) j q m= l m T (r, ) + o(t (r, )). Combinin with assumption b) of Theorem 3, we et that is constant. This is a contradiction. We have completed the proof of Theorem 3. In connection with Remark 5, we note that the proofs of Theorem and Theorem 3 remain valid for the case where the monomial f n (f n ) (t ) (f n k ) (t k ) is replaced by the followin polynomial f n (f n ) (t ) (f n k ) (t k) + I c I f n I (f n I ) (t I) (f n ki ) (t ki), where c I is a holomorphic function on D, and n I, n ji, t ji are nonneative inteers satisfyin α I := t ji n I + k n < α := t j ji n + k n. j 3

14 In fact, since α I < α and by (4.), we et I v(ξ) := f v(z v + ρ v ξ) ρ α I v = ρ α α I v v (ξ) 0, spherically uniformly on compact subsets of C. Therefore, similarly to (3.4) c I (z v + ρ v ξ)f n I v (z v + ρ v ξ)(f n I v ) (ti) (z v + ρ v ξ) (f n ki v ) (tki) (z v + ρ v ξ) n = c I (z v + ρ v ξ) I I v (ξ)( n I v (ξ)) (ti) n... ( Ik I v (ξ)) (tki) 0, spherically uniformly on compact subsets of C. This implies that fv n (z v + ρ v ξ)(f n v ) (t) (z v + ρ v ξ) (f n k v ) (tk) (z v + ρ v ξ) + c I (z v + ρ v ξ)f n I v (z v + ρ v ξ)(f n I v ) (ti) (z v + ρ v ξ) (f n ki v ) (tki) (z v + ρ v ξ) I = v n (ξ)( n v (ξ)) (t)... ( n k v (ξ)) (t k) + n c I (z v + ρ v ξ) I n I v (ξ)( I I v (ξ)) (ti) n... ( Ik I v (ξ)) (t ki) I n (ξ)( n (ξ)) (t)... ( n k (ξ)) (tk). (3.) spherically uniformly on compact subsets of C. We use aain the proofs of Theorem and Theorem 3 for the eneral case above after chanin (3.4) by (3.). 4 Appendix Usin our methods above, we ive a slihtly simpler proof of the case of Theorem B above which did not follow from our Corollary 4: Theorem ([6], Theorem 3.2, case n = k + ). Let k be a positive inteer and a be a nonzero constant. Let F be a family of entire functions in a complex domain D such that for every f F, (f k+ ) (k) (z) a for all z D. Then F is normal on D. In order to prove the above theorem we need the followin lemma: 4

15 Lemma 2 ([4]). Let be a transcendental holomorphic function on the complex plane C, and k be a positive inteer. Then ( k+ ) (k) assumes every nonzero value infinitely often. Proof of Theorem. Without loss the enerality, we may assume that D is the unit disc. Suppose that F is not normal at z 0 D. Then, by Lemma 6, for α = k there exist k+ ) a real number r, 0 < r <, 2) points z v, z v < r, z v z 0, 3) positive numbers ρ v, ρ v 0 +, 4) functions f v, f v F such that v (ξ) = f v(z v + ρ v ξ) ρ α v (ξ) (4.) spherically uniformly on compact subsets of C, where (ξ) is a non-constant holomorphic function and # (ξ) # (0) =. Therefore (f k+ v ) (k) (z v + ρ v ξ) = ( ( f v(z v + ρ v ξ) ρ α v ) k+) (k) = ( v k+ (ξ) ) (k) ( k+ (ξ)) (k) spherically uniformly on compact subsets of C. By Hurwitz s theorem either ( k+ ) (k) a, either ( k+ ) (k) a. On the other hand, it is easy to see that there exists z 0 such that ( k+ ) (k) (z 0 ) = a (the case where is a nonconstant polynomial is trivial and the case where is transcendental follows from Lemma 2). Hence, ( k+ ) (k) a. Therefore has no zero point. Hence, by Lemma 7, (ξ) = e cξ+d, c 0. Then a ( k+ ) (k) (ξ) ((k + )c) k e (k+)(cξ+d), which is impossible. References [] C. T. Chuan, On differential polynomials. In: Analysis of One Complex Variable, World Sci. Publishin, Sinapore, 987, [2] J. Clunie and W. K. Hayman, The spherical derivative of interal and meromorphic functions, Comm. Math. Helv., 40 (966),

16 [3] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 964. [4] W. Hennekemper, Über die Werteverteilun von (f k+ ) (k), Math. Z., 77 (98), [5] J. D. Hinchliffe, On a result of Chuan related to Hayman s Alternative, Comput. Method. Funct. Theory, 2 (2002), [6] W. Schwick, Normal criteria for families of meromorphic functions, J. Anal. Math., 52 (989), [7] X. C. Pan and L. Zalcman, On theorems of Hayman and Clunie, New Zealand J. Math., 28 (999), [8] L. Zalcman, Normal families: New perspective, Bull. Amer. Math. Soc., 35 (998),