A Version of the Lohwater-Pommerenke Theorem for Strongly Normal Functions

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1 Computational Methods and Function Theory Volume ), No. 1, A Version of the Lohwater-Pommerenke Theorem for Strongly Normal Functions Rauno Aulaskari and Hasi Wulan Abstract. A new characterization of strongly normal functions, a version of the well-known Lohwater-Pommerenke Theorem for strongly normal functions, is obtained. The corresponding result for little Bloch functions is also stated. Some applications of this new characterization to algebraic differential equations are given. Keywords. Lohwater-Pommerenke Theorem, normal functions, strongly normal functions MSC. 30D45, 30D50. A meromorphic function f in the unit disc D is said to be normal if 1) sup f # z)1 z 2 ) <, z D where f # z) f z) /1 + fz) 2 ) is the spherical derivative of f. These functions were introduced by Lehto and Virtanen [10]. For subsequent developments, see [1]. Lohwater and Pommerenke [11] gave an alternative characterization of normal functions. Their result can be stated as follows. Theorem LP. Let f be a meromorphic function in D. Then f is not normal if and only if there exist a sequence {z n } D with z n 1 and a sequence {ρ n } of positive numbers satisfying ρ n /1 z n ) 0 such that the sequence {fz n +ρ n ξ)} converges locally uniformly to a non-constant meromorphic function in C. Lohwater and Pommerenke stated this theorem a bit differently, but their result and the theorem as stated here are essentially the same. In somewhat picturesque language, they mentioned that this theorem can be expressed as follows: A function is normal if and only if its Riemann image surface does not contain asymptotically a Riemann surface of parabolic type. A number of important developments have arisen from the Lohwater-Pommerenke Theorem. For instance, answering a question of Pommerenke [13, question 3.2, Received October 10, This research was supported in part by a grant from the Väisälä and Wihuri Funds, Finland. ISSN /$ 2.50 c 2001 Heldermann Verlag

2 100 R. Aulaskari and H. Wulan CMFT p. 357], P. Lappan [9] proved the striking result that a function f meromorphic in D is normal if and only if it satisfies 1) for some five-point subset E of the extended plane, i.e., sup{f # z)1 z 2 ) : z f 1 E)} <. Theorem LP plays a central role in the elegant proof of this result. Shortly thereafter in order to justify a well-known heuristic principle in complex function theory, L. Zalcman [15] proved an analogue of Theorem LP for normal families. Over the years Zalcman s Lemma and its generalizations have been applied to classical complex analysis, complex differential equations, complex dynamics and, to many other areas of analysis; cf. [16]. Applying Theorem LP, Lappan and the first author gave a new characterization of normal functions [2], and obtained a series of integral characterizations of such functions [3]. These results provided the impetus for the introduction of Q p spaces, which have attracted considerable attention because of their generality cf. [8]). Recently, several extended versions of Theorem LP have been given cf. [6], [14] and [7]). One class of meromorphic functions in D which is closely related to the normal functions is the set of strongly normal functions. A meromorphic function f in D is called strongly normal if lim f # z)1 z 2 ) 0 z 1 cf. [5]). Although Theorem LP was proved many years ago, so far as we know, no version of the Lohwater-Pommerenke Theorem for strongly normal functions has appeared in the literature. In this paper, we present such a result. Theorem 1. A function f meromorphic in D is not a strongly normal function if and only if there exist a constant R > 0, a sequence {z n } D with z n 1 and a sequence {ρ n } of positive numbers satisfying ρ n /1 z n ) < 1/2R) such that {fz n + ρ n ξ)} converges locally uniformly to a non-constant meromorphic function in ξ < R. Proof. Assume that f is not a strongly normal function. Then there exists a constant c > 0 and a sequence {z n} D with z n 1 such that 2) f # z n)1 z n 2 ) c, n 1, 2,.... If f is not a normal function, the result follows at once from Theorem LP. So we suppose that f is a normal function and assume that z n > 1/2 for all n N. Let r n 2 z n /1 + z n ) and r n z n /2 z n ), n N. Then 1/3 < r n < z n < r n < 1 and r n 1, r n 1 as n. Also, we have and r n r n 2 z n 1 + z n z n 2 z n 3 z n 1 z n ) 1 + z n )2 z n ) 1 r n 1 z n 1 + z n.

3 1 2001), No. 1 A Version of the Lohwater-Pommerenke Theorem 101 Thus 3) 31 r n ) 3 z n 1 z n ) 1 + z n 3 z n 1 z n ) 1 + z n )2 z n ) r n r n. Choose {z n } D such that 4) M n max 1 z r n z rn 1 z n r n Since r n < z n < r n, by 2), 5) M n r n ) 1/2 z ) 1/2 z n ) 1/2 1 z n r n 1 z n 2 ) 1/2 1 z n 2 z n z n )f # z n) c 4 > 0. ) 1/2 f # z) ) 1/2 f # z n ). ) 1/2 f # z z n) n ) 1/2 f # zn) Hence, r n < z n < r n and z n 1 as n. By 4) and 5), we have f # z n ) as n. Set Then ρ n 0. From 3) we get ρ n 1 1 z ) 1/2 ) 1/2 n M n r n z n 1 f # z n ). 6) 1 z n > 1 r n 1 3 r n r n) 1 3 z n r n), n N. Since 1 z n > r n z n and 1/3 < r n < z n < r n, by 5) and 6), we have 7) ρ n 1 z n 1 1 z ) 1/2 ) 1/2 n 1 z n )M n r n z n r n z n ) 1/2 z n r n) 1/2 M n r n z n ) 1/2 1 z 3 n r n)) 1/2 rn 1/2 z n 1/ : 1 M n r 1/2 n z n 1/2 c 2R. Therefore, the functions g n ξ) fz n + ρ n ξ) are defined for ξ < R. We now show that {g # n ξ)} is uniformly bounded on ξ R < R, R arbitrary. Since

4 102 R. Aulaskari and H. Wulan CMFT f N sup z D f # z)1 z 2 ) < and ξ R < R, we have by 7) g n # ξ) ρ n f # ρ n z n + ρ n ξ) f N 1 z n + ρ n ξ 2 ρ n f N 1 1 z n ρ n R 2R f 1 z n N 1 z n ρ n R 1 1 2R f N 1 ρnr 1 z n 1 R f N. By Marty s criterion, {g n ξ)} is a normal family in ξ < R. Thus there exists a function gξ) meromorphic in ξ < R such that g nk ξ) gξ) locally uniformly in ξ < R. Since g # n k 0) ρ nk f # z nk ) 1, gξ) is not constant. Conversely, suppose that g n ξ) fz n + ρ n ξ) gξ) locally uniformly in ξ < R, where gξ) is a non-constant meromorphic function and z n 1 with ρ n /1 z n ) < 1/2R) for all n N. Choose ξ 0, ξ 0 < R, such that g # ξ 0 ) > 0. The proof will be completed by contradiction. If f is a strongly normal function, then g n # ξ 0 ) ρ n f # z n + ρ n ξ 0 ) ρ n 1 z n + ρ n ξ 0 f # z 2 n + ρ n ξ 0 )1 z n + ρ n ξ 0 2 ) 1 z n 2R1 z n ρ n R) f # z n + ρ n ξ 0 )1 z n + ρ n ξ 0 2 ) 1 R f # z n + ρ n ξ 0 )1 z n + ρ n ξ 0 2 ) 0, which implies that g # ξ 0 ) 0. proof of Theorem 1. This is a contradiction, which completes the As an application, we now use Theorem 1 to describe meromorphic solutions of algebraic differential equations. Let us consider algebraic differential equations of the form ) n dw 8) P j z, w)d j [w], m N, dz where P j z, w) m j i0 a ijz)w i, 1 j m, with coefficients a ij z) analytic in D. Here D j [w] are differential monomials in w of the form ) j1 ) dw d 2 j2 ) w d l jl w D j [w], 1 j m, dz dz 2 dz l with j 1, j 2,..., j l N {0}. We call ν j j 1 + 2j lj l the weight of D j, and the weight of P [w]z) P j z, w)d j [w]

5 1 2001), No. 1 A Version of the Lohwater-Pommerenke Theorem 103 is defined by νp ) max 1 j m {ν j }. Theorem 2. Let f be a meromorphic solution of 8) with n > νp ). If the coefficients a ij z) satisfy the condition lim z 1 1 z )n νp ) max then f is a strongly normal function in D. m j 1 j m i0 a ij z) 0, Proof. Suppose that f is not a strongly normal function. By Theorem 1 there exist a constant R > 0, a sequence of points {z k } in D with z k 1 and a sequence of positive numbers {ρ k } satisfying ρ k /1 z k ) 1/2R) such that the sequence {g k ξ)} {fz k + ρ k ξ)} converges locally uniformly to a nonconstant meromorphic function gξ) in ξ < R. Thus g ξ) is not identically zero. We choose a point ξ 0, ξ 0 < R, such that g ξ 0 ) 0 and gξ 0 ). Replacing z by z k + ρ k ξ 0 in 8) for k sufficiently large, we have 1 ρk g kξ 0 ) ) m n ν P j z k + ρ k ξ 0, g k ξ 0 ))D j [g k ξ 0 )]ρ j k. By the assumption, for each j, 1 j m, g kξ 0 ) n n ν P j z k + ρ k ξ 0, g k ξ 0 )) D j [g k ξ 0 )] ρ j k P j z k + ρ k ξ 0, g k ξ 0 )) 1 z k + ρ k ξ 0 ) n νj ) n νj ) n νj 1 z k ρk D j [g k ξ 0 )] 1 z k + ρ k ξ 0 1 z k mj ) n νj 1 a ij z k + ρ k ξ 0 ) g k ξ 0 ) R) i i0 1 z k + ρ k ξ 0 ) n ν j D j [g k ξ 0 )] 1 z k + ρ k ξ 0 ) n νp ) max a ij z k + ρ k ξ 0 ) 1 j m i0 ) n νj g k ξ 0 ) ) m j D j [g k ξ 0 )] 0 R as k. This is a contradiction. Example. Consider the Riccati differential equation 9) w z + 1 2z)w + z 1)w 2. m j

6 104 R. Aulaskari and H. Wulan CMFT The non-constant meromorphic solutions of 9) are of the form 1 w c z) 1 + z + c exp z, c C, and by Theorem 2 they are strongly normal functions in D. If f is an analytic function in D and sup f z) 1 z 2 ) <, z D then f is said to be a Bloch function. Further, if lim f z) 1 z 2 ) 0, z 1 then f is called a little Bloch function. Corresponding to Theorem LP we have the following result. Theorem 3 [4] or [12]). A function f analytic in D is not a Bloch function if and only if there exist a sequence {z n } D with z n 1 and a sequence {ρ n } of positive numbers satisfying ρ n /1 z n ) 0 such that the sequence {fz n + ρ n ξ) fz n )} converges locally uniformly to a non-constant analytic function in C. Using this theorem and a proof technique similar to that in Theorem 1, we obtain Theorem 4. A function f analytic in D is not a little Bloch function if and only if there exist a constant R > 0, a sequence {z n } D with z n 1 and a sequence {ρ n } of positive numbers satisfying ρ n /1 z n ) < 1/2R) such that {fz n + ρ n ξ) fz n )} converges locally uniformly to a non-constant analytic function in ξ < R. Acknowledgment. We would like to thank Professor Peter Lappan for showing us an error in the early version of this manuscript and the referee for helpful comments. References 1. J. M. Anderson, J. Clunie, and Chr. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math ), R. Aulaskari and P. Lappan, An integral criterion for normal functions, Proc. Amer. Math. Soc ), R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, Complex analysis and its applications, Pitman Research Notes in Mathematics, 305, Longman Scientific & Technical, Harlow, 1994, R. Aulaskari and P. Lappan, A criterion for a rotation automorphic function to be normal, Bull. Inst. Math. Acad. Sinica ), H. Chen and P. Gauthier, On strongly normal functions, Canad. Math. Bull ),

7 1 2001), No. 1 A Version of the Lohwater-Pommerenke Theorem H. Chen and P. Lappan, Products of spherical derivatives and normal functions, J. Austral. Math. Soc. Series A) ), M. Essén and H. Wulan, Carleson type measures and their applications, Complex Variables Theory Appl ), M. Essén and J. Xiao, Q p spaces a survey, Complex function spaces Mekrijärvi, 1999), ed. R. Aulaskari, Univ. Joensuu, Dept. Math. Rep. Ser., No ), P. Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helv ), O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math ), A. J. Lohwater and Chr. Pommerenke, On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math ), D. Minda, Bloch and normal functions on general planar regions, Holomorphic functions and moduli, Vol. I Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, Springer, New York Berlin, 1988, Chr. Pommerenke, Problems in complex function theory, Bull. London Math. Soc ), H. Wulan, On some classes of meromorphic functions, Ann. Acad. Sci. Fenn. Math. Diss ), L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly ), L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc ), Rauno Aulaskari rauno.aulaskari@joensuu.fi Address: Department of Mathematics, University of Joensuu, P. O. Box 111, FIN Joensuu, Finland Hasi Wulan wulan@stu.edu.cn Address: Department of Mathematics, Shantou University, Shantou, Guangdong , P. R. China