GROWTH OF SOLUTIONS TO HIGHER ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS IN ANGULAR DOMAINS

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1 Electronic Journal of Differential Equations, Vol 200(200), No 64, pp 7 ISSN: URL: or ftp ejdemathtxstateedu GROWTH OF SOLUTIONS TO HIGHER ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS IN ANGULAR DOMAINS NAN WU Abstract In this article, we discuss the growth of meromorphic solutions to higher order homogeneous differential equations in some angular domains, instead of the whole complex plane Introduction and statement of main results By a transcendental meromorphic function, we mean a function that is meromorphic on the whole complex plane, and is not a rational function; in other words, is an essential singular point We assume the reader is familiar with the Nevanlinna theory of meromorphic functions and basic notation such as: Nevanlinna characteristic T (r, f), integrated counting function N(r, f), and proximity function m(r, f), and the deficiency δ(a, f) of f(z) For the details, see [4, 7] The order λ and the lower order µ are defined as follows: λ(f) = lim sup log T (r, f), µ(f) = lim inf log T (r, f) It is known the growth of meromorphic solutions of differential equations with meromorphic coefficients in the complex plane C attracted a lot research In this article, we discuss the growth of meromorphic solutions of differential equations with transcendental meromorphic coefficients in a proper subset of C Let f(z) be a meromorphic function in an angular region Ω(α, β) = {z : α arg z β} Recall the definition of Ahlfors-Shimizu characteristic in an angle (see [6]) Set Define Ω(r) = Ω(α, β) {z : < z < r} = {z : α < arg z < β, < z < r} S(r, Ω, f) = π Ω(r) ( f (z) ) 2dσ, r T (r, Ω, f) = + f(z) 2 The order and lower order of f on Ω are defined as follows σ α,β (f) = lim sup log T (r, Ω, f), µ α,β (f) = lim inf 2000 Mathematics Subject Classification 30D0, 30D20, 30B0, 34M05 Key words and phrases Meromorphic solutions; Nevanlinna theory; order c 200 Texas State University - San Marcos Submitted February 23, 200 Published November 7, 200 Supported by grant from the NSF of China S(t, Ω, f) dt t log T (r, Ω, f)

2 2 N WU EJDE-200/64 Remark The order σ α,β (f) of a meormorphic function f on an angular region here we give is reasonable, because T (r, C, f) = T (r, f) + O() Nevanlinna theory on the angular domain plays an important role in value distribution of meromrorphic functions Let us recall the following terms [3]: A α,β (r, f) = ω r ( ) π t ω tω r 2ω {log + f(te iα ) + log + f(te iβ ) } dt t, B α,β (r, f) = 2ω πr ω C α,β (r, f) = 2 β α < b n <r log + f(re iθ ) sin ω(θ α)dθ, ( b n ω b n ω r 2ω ) sin ω(θ n α), where ω = π/(β α), and b n = b n e iθn is a pole of f(z) in the angular domain Ω(α, β), appears according to its multiplicity The Nevanlinna s angular characteristic is defined as follows: S α,β (r, f) = A α,β (r, f) + B α,β (r, f) + C α,β (r, f) Some articles define the order and lower order of f on Ω as: σ α,β (f) = lim sup According to the inequality log S α,β (r, f), µ α,β (f) = lim inf 2 T (r, Ω, f) S α,β (r, f) 2ω r ω + ω 3 r log S α,β (r, f) T (t, Ω, f) t ω+ dt + O(), showed by Zheng [3, Theorem 247], if σ α,β (r, f) <, then σ α,β (r, f) < We consider q pairs of real numbers {α j, β j } such that π α < β α 2 < β 2 α q < β q π () and the angular domains X = q {z : α j arg z β j } For a function f meromorphic in the complex plane C, we define the order of f on X as σ X (f) = lim sup log T (r, X, f) It is obvious that σ αj,β j (f) σ X (f) q σ α j,β j (f) j =, 2,, q And σ X (f) = + if and only if there exists at least one j 0 q such that σ αj0,β j0 (f) = + We will establish the following results Theorem 2 Let A 0 (z) be a meromorphic function in C with finite lower order µ < and nonzero order 0 < λ and δ = δ(, A 0 ) > 0 For q pair of real numbers {α j, β j } satisfying () and q (α j+ β j ) < 4 σ arcsin δ/2 (2) where σ > 0 with µ σ λ If A j (z)(j =, 2,, n) are meromorphic functions in C with T (r, A j ) = o(t (r, A 0 )), then every solution f 0 to the equation A n f (n) + A n f (n ) + + A 0 f = 0 has the order σ X (f) = + in X = q {z : α j arg z β j }

3 EJDE-200/64 GROWTH OF SOLUTIONS 3 If we remove the condition µ(a 0 ) < in Theorem 2, we can establish the following result Theorem 3 Let A 0 (z) be a meromorphic function in C with nonzero order 0 < λ and δ(, A 0 ) > 0 Suppose that for q directions arg z = α j ( j q), satisfying π α < α 2 < < α q < π, α q+ = α + 2π, A j (z), j =, 2,, n, are meromorphic functions in C with finite lower order and T (r, A j ) = o(t (r, A 0 )) Then every solution f 0 to the equation A n f (n) + A n f (n ) + + A 0 f = 0 has order σ X (f) = + in X = C\ q {z : arg z = α j} The method in this paper was firstly used by Zheng [4] to investigate the growth of transcendental meromorphic functions with radially distributed values 2 Some auxiliary results To prove the theorems, we give some lemmas [2, 3, 4] The following result is from Lemma 2 Let f(z) be a transcendental meromorphic function with lower order µ < and order 0 < λ, then for any positive number µ σ λ and any set E with finite measure, there exist a sequence {r n }, such that r () r n / E, lim nn n = ; log T (r (2) lim inf n,f) n n σ; (3) T (t, f) < ( + o())( 2t r n ) σ T (r n /2, f), t [r n /n, nr n ]; (4) T (t, f)/t σ εn 2 σ+ T (r n, f)/rn σ εn, t nr n, ε n = [log n] 2 We recall that {r n } is called the Pólya peaks of order σ outside E Given a positive function Λ(r) satisfying lim Λ(r) = 0 For r > 0 and a C, define and D Λ (r, a) = {θ [ π, π) : log + f(re iθ > Λ(r)T (r, f)}, ) a D Λ (r, ) = {θ [ π, π) : log + f(re iθ ) > Λ(r)T (r, f)} The following result is called the spread relation, which was conjectured by Edrei [2] and proved by Baernstein [] Lemma 22 Let f(z) be transcendental and meromorphic in C with the finite lower order µ < and the positive order 0 < λ and has one deficient values a Ĉ = C { } Then for any sequence of Pólya peaks {r n} of order σ > 0, µ σ λ and any positive function Λ(r) 0 as r +, we have lim inf n meas D Λ(r n, a) min{2π, 4 σ arcsin δ(a, f)/2} To make it clearly, we give the definition of R-set on the complex plane C Definition 23 Let B(z n, r n ) = {z : z z n < r n } be an open disk on the complex plane If n= r n <, n=b(z n, r n ) is called an R-set

4 4 N WU EJDE-200/64 Lemma 24 ([8]) Let f be a meromorphic function on the angular region Ω(α, β) with finite order ρ, let Γ = {(n, m ), (n 2, m 2 ),, (n j, m j )} denote a finite set of distinct pair of integers which satisfying n i > m i 0 for i =, 2,, j, and let ε > 0 and δ > 0 be given constants Then there exists K > 0 depending only on f, ε, δ such that f (n) (z) f (m) < K z (n m)(kδ+2ρ++ε) (sin k δ (ϕ α δ)) 2n m, (2) (z) for all (n, m) Γ and all z = re iϕ Ω(α + δ, β δ) except for a R-set, that is, a countable union of discs whose radii have finite sum, where k δ = To prove Theorem 3, we need a result from Edrei [2] π β α 2δ Lemma 25 Let f(z) be a meromorphic function with δ = δ(, f) > 0 Then given ε > 0, we have meas E(r, f) > T ε, r / F, (r, f)[] +ε where E(r, f) = {θ [ π, π) : log + f(re iθ ) > δ T (r, f)} 4 and F is a set of positive real numbers with finite logarithmic measure depending on ε 3 Proof of the Theorems Proof of Theorem 2 We suppose that there exists a nontrival meromorphic solution f such that σ αj,β j (f) < +, j =, 2,, q In view of Lemma 24, there exists a constant M > 0 not depending on z such that f (j) (z) f(z) < z M, j =, 2,, n for all z Ω(α j + ε, β j ε), j =, 2,, q, except for a R-set E For E, we can define a set F = {r > 0 z E, st z = r} thus meas F < (I) λ(a 0 ) > µ(a 0 ) Then λ(a 0 ) > σ µ(a 0 ) By the inequality (2), we can take a real number ε > 0 such that q (α j+ β j + 2ε) + 2ε < 4 σ + 2ε arcsin δ/2, (3) where α q+ = 2π + α, and λ(a 0 ) > σ + 2ε > µ(a 0 ) Applying Lemma 2 to A(z) gives the existence of the Pólya peak {r n } of order σ + 2ε of A(z) such that r n / F, and then from Lemma 22 for sufficiently large n we have meas D(r n, ) > 4 σ + 2ε arcsin δ/2 ε (32) We can assume for all the n, above holds Set K := meas(d(r n, ) q (α j + ε, β j ε))

5 EJDE-200/64 GROWTH OF SOLUTIONS 5 Then from (3) and (32) it follows that K meas(d(r n, )) meas([0, 2π)\ q (α j + ε, β j ε)) = meas(d(r n, )) meas( q (β j ε, α j+ + ε)) q = meas(d(r n, )) (α j+ β j + 2ε) > ε > 0 It is easy to see that there exists a j 0 such that for infinitely many n, we have meas(d(r n, ) (α j0 + ε, β j0 ε)) > K q (33) We can assume for all the n, (33) holds We define a real function by { } T Λ(r) 2 (rn, A j ) = max T (r n, A 0 ), n ; j =, 2,, n, T (r n, A 0 ) for r n r < r n+ Obviously lim Λ(r) = 0 and Set meas D Λ(r n ) = meas{θ : r n e iθ E} = 0 D n = (α j0 + ε, β j0 ε)\d Λ(r n ), E n = D(r n, ) (α j0 + ε, β j0 ε) Thus from the definition of D(r, ) it follows that log + A 0 (r n e iθ ) dθ log + A 0 (r n e iθ ) dθ E n meas(e n )Λ(r n )T (r n, A 0 ) (34) > K q Λ(r n)t (r n, A 0 ) Thus, we have ( = log + A 0 (r n e iθ ) dθ ( log + f (j) (r n e iθ ) ) f(r n e iθ ) + log+ A j (r n e iθ ) dθ ) n ( log + f (j) (r n e iθ ) ) f(r n e iθ ) + log+ A j (r n e iθ ) dθ D Λ (rn) + D n log + A j (r n e iθ ) dθ + O( n ) T (r n, A j ) + O( n ) Λ 2 (r n )T (r n, A 0 ) Therefore, K q Λ(r n) < Λ 2 (r n ) This contradicts that Λ(r) 0 (35)

6 6 N WU EJDE-200/64 (II) λ(a 0 ) = µ(a 0 ) Then λ(a 0 ) = σ = µ(a 0 ) By the same argument as in (I) with all the σ + 2ε replaced by σ, we can derive a contradiction The proof is complete Proof of Theorem 3 Applying Lemma 2 to A 0 (z) confirms the existence of a sequence {r n } of positive numbers such that r n / E and meas E(r n, A 0 ) > where E(r n, A 0 ) is defined as in Lemma 25 Set Then for (36) it follows that T ε, (36) (r n, A 0 )[ n ] +ε ε n = 2q + T ε (r n, A 0 )[ n ] +ε meas(e(r n, A 0 ) q (α j + ε n, α j+ ε n )) meas E(r n, A 0 ) meas( q (α j + ε n, α j+ ε n )) (2q + )ε n 2qε n = ε n > 0 so that there exists a j such that for infinitely many n, we have meas E n > ε n q, (37) where E n = E(r n, A 0 ) (α j + ε n, α j+ ε n ) We can assume that (37) holds for all the n Thus αj+ ε n log + A 0 (r n e iθ ) dθ log + A 0 (r n e iθ ) dθ E n α j+ε n On the other hand, αj+ ε n α j+ε n log + A 0 (r n e iθ ) dθ < Combining (38) and (39) gives so that T ε (r n, A 0 ) ε n T (r n, A 0 ) 4q δ 4q(2q + ) δ meas(e n ) δ 4 T (r n, A 0 ) δε n 4q T (r n, A 0 ) (38) T (r n, A j ) + O( n ) (39) T (r n, A j ) + O( n ), [ n ] +ε n T (r n, A j ) + O(log 2+ε r n ), we have µ(a 0 ) max j q (µ(a j ))/( ε) By the same method as in Theorem 2, we obtain a contradiction, which completes the proof

7 EJDE-200/64 GROWTH OF SOLUTIONS 7 References [] A Baerstein; Proof of Edrei s spead conjecture, Proc London Math Soc, 26 (973), pp [2] A Edrei; Sums of deficiencies of meromorphic functions, J Analyse Math, I 4 (965), pp 79-07; II 9 (967), pp [3] A A Goldberg and I V Ostrovskii; The distribution of values of meromorphic functions (in Russian), Izdat Nauk Moscow 970 [4] W K Hayman; Meromorphic Functions, Oxford, 964 [5] I Laine; Nevanlinna Theory and Complex Differential Equations, W de Gruyter, Berlin, 993 [6] M Tsuji; Potential theory in modern function theory, Maruzen Co LTD Tokyo, 959 [7] L Yang; Value Distribution And New Research, Springer-Verlag, Berlin, 993 [8] S J Wu; Estimates for the logarithmic derivative or a meromorphic function in an angle, and their application Proceeding of international conference on complex analysis at the Nankai Institute of Mathematics, 992, pp [9] S J Wu; On the growth of solution of second order linear differential equation in an angle Complex Variable, 24 (994), pp [0] J F Xu, H X Yi; Solutions of higher order linear differential equations in an angle, Applied Mathematics Letters, Volume 22, Issue 4, April 2009, pp [] J F Xu, H X Yi; On uniqueness of meromorphic functions with shared four values in some angular domains, Bull Malays Math Sci Soc, (2) 3(2008), pp [2] L Yang; Borel directions of meromorphic functions in an angular domain, Science in China, Math Series(I) (979), pp [3] J H Zheng; Value Distribution of Meromorphic Functions, Springer-Verlag, Berlin, 200 [4] J H Zheng; On transcendental meromorphic functions with radially distributed values, Sci in China Ser A Math, 47 3 (2004), pp [5] J H Zheng; On uniqueness of meromorphic functions with shared values in some angular domains, Canad J Math, 47 (2004), pp [6] J H Zheng; On uniqueness of meromorphic functions with shared values in one angular domains, Complex variables, 48 (2003), pp Nan Wu Department of Mathematical Sciences, Tsinghua University, Beijing, 00084, China address: wunan07@gmailcom