BANK-LAINE FUNCTIONS WITH SPARSE ZEROS

Transcription

1 BANK-LAINE FUNCTIONS WITH SPARSE ZEROS J.K. LANGLEY Abstract. A Bank-Laine function is an entire function E satisfying E (z) = ± at every zero of E. We construct a Bank-Laine function of finite order with arbitrarily sparse zero-sequence. On the other hand, we show that a real sequence of at most order, convergence class, cannot be the zero-sequence of a Bank-Laine function of finite order.. Introduction A Bank-Laine function is an entire function E such that E (z) = ± at every zero z of E. These arise from differential equations in the following way [, 2]. Let A be an entire function, and let f, f 2 be linearly independent solutions of () w + A(z)w = 0, normalized so that the Wronskian W = W(f, f 2 ) = f f 2 f f 2 satisfies W =. Then E = f f 2 satisfies (2) 4A = (E /E) 2 2E /E /E 2. Further, E is a Bank-Laine function while, conversely, if E is any Bank-Laine function then [3] the function A defined by (2) is entire, and E is the product of linearly independent normalized solutions of (). Extensive work in recent years has concerned the exponent of convergence λ(f j ) of the zeros of solutions f j, in connection with the order of growth ρ(a) of the coefficient A, these defined by (3) λ(f j ) = limsup r It has been conjectured that (4) log + N(r, /f j ), ρ(a) = limsup log r r log + T(r, A). log r A transcendental, ρ(a) <, max{λ(f ), λ(f 2 )} < implies that ρ(a) is a positive integer, and this has been proved in [] under the stronger assumption max{λ(f ), λ(f 2 )} < ρ(a) <. Further, (4) implies that ρ(a) > /2 [6, 7] and that E has finite order []. We refer the reader to [5, 0, 2, 5] for further results. It was observed by Shen [8] that if (a n ) is a complex sequence tending to infinity without repetition, then there exists a Bank-Laine function F with zero-sequence (a n ), the construction based on the Mittag-Leffler theorem. A natural question arising from both this observation and the conjecture above is the following: for which sequences (a n ) with finite exponent of convergence does there exist a Bank- Laine function E of finite order with zero-sequence (a n )? In [6] the answer was 99 Mathematics Subject Classification. 30D35.

2 2 J.K. LANGLEY shown to be negative for certain special sequences, such as a n = n 2. The following theorem shows that the answer is negative for a large class of sequences. Theorem.. Let L be a straight line in the complex plane and let (a n ) be a sequence of pairwise distinct complex numbers, all lying on L, such that a n as n and (5) a n <. a n 0 Then there is no Bank-Laine function of finite order with zero-sequence (a n ). Obvious examples such as E(z) = sinz show that the hypothesis (5) is not redundant in Theorem.. We shall see in Theorem.3 below that the hypothesis that all a n lie on a line cannot be deleted either. One obvious way to make Bank-Laine functions of finite order is to choose A to be a polynomial in (): if A is not identically zero and has degree n then ρ(e) = (n+2)/2 []. However, there are very few examples in the literature of Bank- Laine functions of finite order associated via (2) with transcendental coefficient functions A. The simplest [, 4, 8] are of the following form: given any polynomial P having only simple zeros, there exists a non-constant polynomial Q such that Pe Q is a Bank-Laine function. A second class arises from equations having periodic coefficients [2, 4], leading to Bank-Laine functions of form E(z) = P(e αz )exp(βz), with P a polynomial and α, β constants. In view of the conjecture above and nonexistence results such as Theorem., it seems worth looking for further examples. Theorem.2 ([4]). There exists a Bank-Laine function F(z) of finite order, with infinitely many zeros and with transcendental associated coefficient function A, but having no representation of the form F(z) = P(e αz )exp(q(z)), with P, Q polynomials and α constant. It is relatively straightforward to show that the examples F of Theorem.2 cannot have a representation F(z) = P (z)p 2 (e αz )e Q(z), with P, P 2, Q polynomials and α a non-zero constant. For if P 2 (β) = 0 and e αz = β then P (z) 2 e 2Q(z) = (αβ) 2 P 2 (β) 2 and Q(z) + log P (z) would be a polynomial, by Lemma 5 of [3]. However, the use of quasiconformal modifications in the proof of Theorem.2 makes it difficult to determine precisely the form of the examples F, although it is clear from the distortion theorems used there that the exponent of convergence of the zeros of F will always be positive. A natural question is then whether there exist Bank- Laine functions of finite order with zeros which are infinite in number but have zero exponent of convergence, and we give a strongly affirmative answer to this question. Theorem.3. Let (c n ) be a positive sequence tending to +. Then there exists a Bank-Laine function E(z) = e z ( z/α n ), n= with α n > c n for each n. Further, ρ(e) = and λ(e) = 0 and E is the product f f 2 of normalized linearly independent solutions of an equation (), with A transcendental, and f has no zeros.

3 BANK-LAINE FUNCTIONS WITH SPARSE ZEROS 3 Thus there exist Bank-Laine functions of finite order with arbitrarily sparse zerosequences. The proof of Theorem.3 is lengthy but elementary, and it will be seen in the proof that the α n lie close to, but not on, the imaginary axis. 2. Proof of Theorem. We assume that (a n ) is as in the statement of Theorem., and that there exists a Bank-Laine function E of finite order, with zero-sequence (a n ). There is no loss of generality in assuming that L is the real axis and all the a n are non-zero, and that infinitely many a n are positive. By (5) and [9, Chapter ] we may write (6) E(z) = e P(z)+iQ(z) ( z/a n ) = e P(z)+iQ(z) W(z), n= in which P and Q are polynomials, real on the real axis. Since the a n are real and E is a Bank-Laine function, (6) implies that e 2iQ(an) is real and positive and hence e iq(an) = ± for each n. Thus E(z)e iq(z) is a Bank-Laine function and there is no loss of generality in assuming that Q(z) 0. Now E is the product f f 2 of normalized linearly independent solutions of an equation (), with A an entire function of finite order, and A and E are related by (2). By (2) and [9, Theorem., p.27], we have (7) T(r, A) = O(T(r, E)), T(r, W) = o(r), r. Lemma 2.. Let ε > 0 and let z = re iθ with r > 0 and ±θ (ε, π ε). Then (8) log W(z) = o(r), W (z)/w(z) + W (z)/w(z) = o(), r. Lemma 2. is an immediate consequence of the Poisson-Jensen formula [9, p.] and its differentiated form [9, p.22], as well as of the fact that for z as in Lemma 2. the distance from z to the nearest zero of E is at least cr, in which the positive constant c depends only on ε. Lemma 2.2. P is not constant. Proof. Suppose that P(z) is constant. Let y be real, with y large. Then (9) 2 log W(iy) = log( + y 2 /a 2 n ) = log M(y2, G), G(z) = ( + z/a 2 n ), n= and so W(iy) is large, since G is a transcendental entire function in (9). Thus A(iy) = o(), using (2) and (8). A standard application of the Phragmén-Lindelöf principle now shows that either A(z) 0, which is obviously impossible, or A has at least order, mean type. However, (7) gives T(r, A) = o(r), and this is a contradiction. n= Thus P is a non-constant real polynomial. Now if P(x) is negative for large positive x, we have W (x)e P(x) 0 as x +, using (7), which contradicts our earlier assumption that E has infinitely many zeros on the positive real axis. There must therefore exist positive constants c j such that (0) argp(z) < π/2 c, z > c 2, argz < c 3.

4 4 J.K. LANGLEY Let δ be a small positive constant. Then (2), (8) and (0) give () A(z) = 4 P (z) 2 ( + o()), for z > c 2, δ < argz < c 3. We now apply the Phragmén-Lindelöf principle to the function A(z)P (z) 2, which has finite order, and deduce that () holds for large z with argz < c 3. The contradiction required to prove Theorem. arises at once upon applying the following lemma. Lemma 2.3. Let c be a positive constant. Then there exists a positive constant δ such that the following is true. Suppose that A(z) is analytic and satisfies () as z in the region S given by z r 0, arg z δ, in which P is a polynomial of positive degree N satisfying arg P(z) < π/2 2c as z in S. Let f be a non-trivial solution of () in S. Then ff has finitely many zeros in S. Proof. This is a standard application of Green s transform as in [, pp.286-8]. Let ε be small and positive, and assume that ff has infinitely many zeros in S. We may write P(z) = bz N ( + o()), arg P (z) = (N )argz + α + o(), α = arg b, as z. Thus, without loss of generality, we have (2) α π/2 c, 2c π + 2α 2π 2c. Also, as z in S, provided δ was chosen small enough, (3) π + 2α ε arga(z) π + 2α + ε. Suppose now that z 0 and z are zeros of ff in S with z 0 and z /z 0 large. Following [, pp.286-8], write z = z 0 + re is, z = z 0 + Re is, F(r) = f(z 0 + re is ), H(r) = F(r)F (r) with r, R > 0 and s real. Then and hence (4) H (r) = F (r) 2 + F(r)F (r) = F (r) 2 e 2is A(z) f(z) 2 I = R 0 F (r) 2 dr = R 0 e 2is A(z 0 + re is ) f(z 0 + re is ) 2 dr. If z is large enough then without loss of generality s < 4δ and hence, using (3), π + 2α ε 8δ arg I π + 2α + ε + 8δ. On the other hand we obviously have I > 0, by (4). Provided ε and δ were chosen small enough we thus have c + 2kπ < π + 2α < c + 2kπ for some integer k, which contradicts (2). From Lemma 2.3 we deduce the following result. Theorem 2.. Let E = We P be a Bank-Laine function, with P a polynomial of positive degree N and W an entire function of order ρ(w) < N. Let θ < θ 2 and c > 0 and suppose that Re(P(z)) > c z N as z in the sector S given by θ argz θ 2. Then E has finitely many zeros in S.

5 BANK-LAINE FUNCTIONS WITH SPARSE ZEROS 5 Thus zeros of E can only accumulate near the rays on which Re(P(z)) = o( z N ). A example illustrating this result is E(z) = (/π)sin(πz)exp(2πiz 2 ). Proof. Obviously we have Re(P(z)) > (c/2) z N as z in a slightly larger sector S. Now suppose that θ θ θ 2 and that E has infinitely many zeros in every sector arg z θ < δ, δ > 0. We may assume that θ = 0. Now if Re(P(z)) < (c/2) z N as z in S then E and E are small in S and the result is obvious. Suppose now that Re(P(z)) > (c/2) z N for large z in S. By (2) there exists an entire function A of finite order such that E is the product of linearly independent solutions of (). Further, by standard estimates [8, 9] there is a set H 0 of measure 0 such that for all real θ not in H 0 we have, for z = re iθ, r > 0, log W(z) = o(r N ), W (z)/w(z) = o(r N ), W (z)/w(z) = o(r 2N 2 ). Then we have () for large z in S with arg z H 0 and hence, by the Phragmén- Lindelöf principle, for all large z in S. Applying Lemma 2.3 gives a contradiction, if δ is small enough. 3. Proof of Theorem.3 Let λ be a large positive constant. There is no loss of generality in assuming that (5) c > λ 2, c j+ /c j > λ 2, j =, 2,.... Choose A, A 2,... inductively, so that A > λc and e A ( /A ) =, while (6) and (7) A j > λc j, A j+ /A j > λ 2, e Aj ( /A j ) µ<j ( A j /A µ ) = for each j. To see that such A j exist, we need only note that the left hand side of (7) is a meromorphic function of A j with finitely many zeros and poles. Let (8) D j = {A j + α + iβ : π α π, π β π }. Provided λ was chosen large enough we then have, by (6), (9) We also have (20) a j > c j, a µ /a j > λ µ j, a j D j, a µ D µ, µ > j. a µ a j ( /λ)max{ a j, a µ }, a j D j, a µ D µ, j µ. For positive integer n and j n and a j lying in an open neighbourhood of D j, define (2) F j,n (a,..., a n ) = e aj G j,n (a,..., a n ) = e aj ( /a j ) ( a j /a µ ). For the proof of Theorem.3 we need a number of lemmas. Lemma 3.. Suppose that δ > 0 and that a j, b j D j and a j b j δ for j =,..., n. Then, for j =,..., n, log G (22) j,n(a,..., a n ) G j,n (b,..., b n ) 6δ λ( /λ) 2.

6 6 J.K. LANGLEY Proof. By (2) we may write (23) G j,n (a,..., a n ) = Now, using (20), µ n a µ (a µ a j ). a µ a j b µ b j 2δ ( /λ)max{ b µ, b j }. Using (9) and the fact that log( + z) 2 z for z /2, this gives log a µ a j b µ b j 4δ n (24) ( /λ) b µ 4δ λ( /λ) 2. Similarly µ n log b µ a µ 2 µ n µ= On combination with (24) this proves Lemma 3.. δ a µ 2δ λ( /λ). Lemma 3.2. Let n be a positive integer and let a j D j for j n. Then the Jacobian matrix ( ) Fj,n J = a k is non-singular. Proof. It suffices to show that the Jacobian matrix ( ) gj (25) H =, g j = log F j,n, a k is non-singular, since the mapping φ(w,..., w n ) = (e w,..., e wn ) has non-singular Jacobian matrix. Now, by (2), g j = + a j a j a j a µ and so, using (9) and (20), we have (26) g j a j a j + ( /λ) Further, for k j, using (2), g j a k = a j a k (a k a j ) which gives, using (9) and (20) again, (27) g j a k ( /λ) a k ( /λ)λ k. Using (26) and (27) we may now write (28) H = I n + C, C = (c j,k ), a µ λ( /λ) 2.

7 BANK-LAINE FUNCTIONS WITH SPARSE ZEROS 7 in which I n is the n by n identity matrix and the entries c j,k of C satisfy (29) c j,j λ( /λ) 2, c j,k ( /λ)λ k, j k. Let d be a column vector with entries d,..., d n and let d r have greatest modulus, say σ. Then by (29), each entry of Cd has modulus at most ( ) σ λ( /λ) 2 + n 2σ ( /λ) λ k λ( /λ) 2 < σ provided λ was chosen large enough. Thus Hd cannot be the zero vector. k= Lemma 3.3. Suppose that a µ D µ for µ n and that a j D j for some j with j n. Then (30) F j,n (a,..., a n ) 4. Proof. By (7) and (2) we have and so (3) F j,n (A,..., A n ) = j<µ n log F j,n (A,..., A n ) 2 j<µ n ( A j /A µ ) A j A µ 2 λ, using (6). In particular, F j,n (A,..., A n ) is close to, provided λ was chosen large enough. Also, (32) F j,n (a,..., a n ) F j,n (A,..., A n ) = eaj Aj X j = e aj Aj G j,n(a,..., a n ) G j,n (A,..., A n ). Now if Re(w) = π then e w e π /2 while if Re(w) = π then e w e π /2. If Im(w) = ±π then e w is real and negative and e w. Thus for a j D j we have e aj Aj /2. But X j is close to, by Lemma 3., provided λ was chosen large enough, and Lemma 3.3 now follows. The next lemma is the key step in proving Theorem.3. Lemma 3.4. For each positive integer n there exist a,,..., a n,n with a j,n D j and F j,n (a,n,..., a n,n ) =, j n. Proof. We set a, = A and the result is trivially true for n =. Assume now that b j = a j,n have been chosen so that (33) Now for j n, by (2), b j D j, F j,n (b,..., b n ) =, j n. F j,n+ (b,..., b n, A n+ ) = e bj ( /b j )( b j /A n+ ) = F j,n (b,..., b n )( b j /A n+ ) ( b j /b µ )

8 8 J.K. LANGLEY and so (34) F j,n+ (b,..., b n, A n+ ) = using (9) and (33). Also, by (7), b j A n+ λj n, F n+,n+ (b,..., b n, A n+ ) = F n+,n+(b,..., b n, A n+ ) F n+,n+ (A,..., A n, A n+ ) = G n+,n+(b,..., b n, A n+ ) G n+,n+ (A,..., A n, A n+ ) and applying Lemma 3. gives (35) For a j D j, j n +, set (36) F n+,n+ (b,..., b n, A n+ ) 24π λ( /λ) 2. n+ h(a,..., a n+ ) = F j,n+ (a,..., a n+ ) 2. j= Then by (34) and (35), provided λ was chosen large enough, (37) (24π) 2 n h(b,..., b n, A n+ ) λ 2 ( /λ) 4 + λ 2(j n ) < 6. However, if a µ D µ for µ n + and at least one a j lies on D j, then by Lemma 3.3 we have h(a,..., a n+ ) /6. Choose d j D j such that j= h(a,..., a n+ ) h(d,..., d n+ ), a j D j. Then d j is an interior point of D j for each j and, at (d,..., d n+ ), n+ ( 0 = Fj,n+ )( ) F j,n+, k n +, a k j= so that by Lemma 3.2 we have F j,n+ (d,..., d n+ ) = for j n +. To complete the proof of Theorem.3, set E n (z) = e z q n (z), q n (z) = µ n Then E n has one zero a j,n in each D j, for j n, and ( z/a µ,n ). E n (a j,n) = F j,n (a,n,..., a n,n ) =, by Lemma 3.4. Let r be large and positive, with A N r < A N+. Then for positive integer m and z r we have, using (9), q m (z) ( + r) N+ ( + r/ a j,m ) N+2 j m ( + r) d log r ( + λ p ) p= exp(2d(log r) 2 ),

9 BANK-LAINE FUNCTIONS WITH SPARSE ZEROS 9 using d to denote a positive constant independent of r and m. It follows that a subsequence q nk converges locally uniformly in the plane to an entire function q of order 0, and q(0) =. Set E(z) = e z q(z). By the usual diagonalization process we may assume that lim k a j,n k = α j D j for each j. Thus E(α j ) = 0 and E (α j ) = for each j. Further, if E(α) = 0 then by Hurwitz theorem each q nk, for k large, has a zero near α. Thus the α j are the only zeros of E and E has precisely one zero in each D j. It remains only to observe that the coefficient function A associated with E has order at most, by (2), and is transcendental, since m(r, /E) O(log r), while f has no zeros since E (α j ) = and W(f, f 2 ) =. Theorem.3 is proved. A natural question to ask is whether examples such as that above could be constructed more elegantly using techniques of interpolation theory [7]. However Theorem. makes it clear that one cannot arbitrarily specify the zero-sequence of a Bank-Laine function of finite order, and it seems necessary to allow the location of the zeros to vary as in Lemma 3.4 above. References [] S. Bank and I. Laine, On the oscillation theory of f + Af = 0 where A is entire, Trans. Amer. Math. Soc. 273 (982), [2] S. Bank and I. Laine, Representations of solutions of periodic second order linear differential equations, J. reine angew. Math. 344 (983), -2. [3], On the zeros of meromorphic solutions of second-order linear differential equations, Comment. Math. Helv. 58 (983), [4] S. Bank, I. Laine and J. K. Langley, On the frequency of zeros of solutions of second order linear differential equations, Results. Math. 0 (986), [5] S. Bank and J. K. Langley, On the oscillation of solutions of certain linear differential equations in the complex domain, Proc. Edin. Math. Soc. 30 (987), [6] S. M. ElZaidi, On Bank-Laine sequences, Complex Variables 38 (999), [7] J. B. Garnett, Bounded analytic functions, Academic Press, New York 98. [8] G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (988), [9] W. K. Hayman, Meromorphic functions, Oxford at the Clarendon Press, 964. [0] S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of certain linear differential equations, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 7 (992), [] E. Hille, Ordinary differential equations in the complex domain, Wiley, New York, 976. [2] I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Math. 5, Walter de Gruyter, Berlin/New York 993. [3] J. K. Langley, On second order linear differential polynomials, Results. Math. 26 (994), [4], Quasiconformal modifications and Bank-Laine functions, Arch. Math. 7 (998), [5] J. Miles and J. Rossi, Linear combinations of logarithmic derivatives of entire functions with applications to differential equations, Pacific J. Math. 74 (996), [6] J. Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (986), [7] L. C. Shen, Solution to a problem of S. Bank regarding the exponent of convergence of the solutions of a differential equation f + Af = 0, Kexue Tongbao 30 (985), [8], Construction of a differential equation y +Ay = 0 with solutions having prescribed zeros, Proc. Amer. Math. Soc. 95 (985), School of Mathematical Sciences, University of Nottingham, NG7 2RD UK address: jkl@maths.nott.ac.uk