Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 273–290

ON A PROBLEM IN COMPLEX OSCILLATION THEORY OF PERIODIC SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS AND SOME RELATED PERTURBATION RESULTS

Yik-Man Chiang and Shi-An Gao

The Hong Kong University of Science and Technology, Department of Mathematics Clear Water Bay, Hong Kong, P.R. China; machiang@ust.hk South China Normal University, Department of Mathematics Guangzhou, 510631, P.R. China; gaosha@scnu.edu.cn

Abstract. It was conjectured in a previous work [7] that every non-trivial solution of y′′ + Ay = 0

has infinite exponent of convergence of zeros, where A(z) = B(e^{z}), B(ζ) = ∑_{j=1}^{p} b_{−j}ζ^{−j} +g(ζ), p is an odd positive integer and g(ζ) is an entire function of order not equal to a positive integer. We give an affirmative answer to this conjecture and obtain generalizations of some previous results. In addition, perturbation results for periodic equations are found. Some new properties of periodic equations have been found in order to solve the above problems.

1. Introduction

We use standard notation from Nevanlinna theory in this paper (see [8], [9] and [10]). In addition, we use the notation σ(f) and λ(f) respectively to denote the order of growth and exponent of convergence of the zeros of a meromorphic function f.

The following result was proved in [7].

Theorem A. Let B(ζ) = g(1/ζ)+∑_{j=1}^{p} b_{j}ζ^{j}, where g(ζ) is a transcendental entire function with σ(g) < 1, p is an odd positive integer and b_{p} ≠ 0. Let A(z) = B(e^{z}). Then any non-trivial solution f of

(1.1)  f′′ + A(z)f = 0

must have λ(f) = +∞. In fact, the stronger conclusion

(1.2)  log^{+} N(r, f) ≠ o(r)

holds.

2000 Mathematics Subject Classification: Primary 34M05, 34M10, 30D35.

The research of both authors was partially supported by the University Grants Council of the Hong Kong Special Adminstrative Region, China (HKUST711/96P). The second author was also supported by the Natural Science Fundation of Guangdong Province in China, the National Natural Science Fundation of China and the Education Department of Guangdong Province in China.

We remark that Theorem A remains valid when we define

B(ζ) = g(ζ) + ∑_{j=1}^{p} b_{−j}ζ^{−j}, b_{−p} ≠ 0.

An example is given in [7] showing that Theorem A does not hold when σ(g) is any positive integer. It was conjectured in [7] that the conclusion of Theorem A remains valid when the order σ(g) > 1 but is not a positive integer. We now give an affirmative answer to this problem in this paper. In fact, we show that Theorem A remains valid when σ(g) is not a positive integer or infinity. We also give examples showing that Theorem A is no longer valid when σ(g) is infinity. Thus our results are the best possible. The second author acknowledges useful discussions with Jim Langley while he was visiting the Mathematics Department of St Andrews University in 1990.

The proof of the above conjecture is based on a generalization of the following result.

Theorem B ([7, Lemma 5] or [8, Lemma 3.6]). Let A(z) = B(e^{z}) where B(ζ) is transcendental and analytic in 0 < |ζ| < +∞. Suppose A(z) satisfies

(1.3)  \overline{lim}_{r→+∞} \frac{log T (r, A)}{r} < 1.

Then for any non-trivial solution f(z) of (1.1) with

(1.4)  log^{+} N(r, 1/f) = o(r),

f(z) and f(z + 2πi) must be linearly dependent.

The generalization of Theorem B we obtain is when (1.1) has the coefficient of the form A(z) = B(e^{z}) where

(1.5)  B(ζ) = g_{1}(1/ζ) + g_{2}(ζ),

and g_{1}(ζ) and g_{2}(ζ) are entire functions. Thus the equation (1.1) we consider here is related to the well-known equation of G.H. Hill in mathematical physics concerning lunar theory (see [1, Chapter VII], [11] and [12]). We also prove some related perturbation results for equation (1.1) in the second half of this paper.

The main tools that we use in this paper are Nevanlinna theory in C_{0} (see [3, p. 4], [6, pp. 97–107] and [10, pp. 101–102]) where C_{0} = C\{z : |z| ≤ R_{0}}, Valiron representation for functions analytic in C_{0} [14, p. 15] and the fact that if f_{1} and f_{2} are two non-trivial, linearly independent solutions of (1.1), then the product E(z) = f_{1}(z)f_{2}(z) satisfies the differential equation

(1.6)  4A(z) = (\frac{E′}{E} (z))^{2} − 2\frac{E′′}{E} (z) − \frac{c^{2}}{E(z)^{2}}

where c ≠ 0 is the Wronskian of f_{1} and f_{2} [1, p. 354] (see also [6, p. 81] or [8, p. 5]), and

(1.7)  T (r, E) = N(r, 1/E) + \frac{1}{2}T (r, A) + O(log rT (r, E))  n.e.

In general, we use “n.e.” to denote that an asymptotic relation holds except possibly outside a set of finite linear measure (i.e., finite length). Our argument actually depends on an analogue of (1.7) on the Valiron representation of periodic functions. We also make use of the following properties of periodic entire coefficient in (1.1). Let A(z) = B(e^{z}) where A(z) and B(ζ) are defined in Theorem B. Then B(ζ) = g_{1}(1/ζ) + g_{2}(ζ), where both g_{1} and g_{2} are entire functions in C. Using this decomposition and coupled with the above tools, we show that the oscillation properties of solutions of (1.1) depend largely on the properties of g_{1} or g_{2} . The proof of the conjecture is based on the above tools as well as on new concepts and methods which we shall develop in Sections 2 and 3. This also allows us to obtain new perturbation results for (1.1).

This paper is organized as follows. Some new concepts and preparatory results will be introduced in Sections 2 and 3. The main results are stated in Section 4, and their proofs are given in Section 5. In Section 6, we discuss perturbation type results for periodic equations of the form (1.1). We note that the notations c, c_{1} , c_{2}, …, are used to denote constants that may have different values at different occurrences in this paper.

2. Preliminaries for the proof of the main results

Let A(z) be an entire function. We define

(2.1)  σ_{e}(A) = \overline{lim}_{r→+∞} \frac{log T (r, A)}{r}

to be the e-type order of A(z). We also define

(2.2)  \overline{lim}_{r→+∞} \frac{log^{+} N(r, 1/A)}{r},

denoted by λ_{e}(A), to be the e-type exponent of convergence of the zeros of A(z). We shall have occasions to consider the zeros of A(z) in the right-half plane only.

In that case, we define the upper limit in (2.2) by λ_{eR}(A) when we only count the zeros of A(z) in the right-half plane. Similarly, we define λ_{eL}(A) to be the upper limit in (2.2) when we only count the zeros of A(z) in the left-half plane. We shall derive some new relations for (2.1) and (2.2) when A(z) is in the form B(e^{z}) for some function B(ζ) below.

Let B(ζ) be analytic in 0 < |ζ| < +∞. Hence we have a representation B(ζ) = g_{1}(1/ζ) + g_{2}(ζ), where both g_{1}(ζ) and g_{2}(ζ) are entire functions. Let A(z) = B(e^{z}) = A_{1}(z) + A_{2}(z), where A_{1}(z) = g_{1}(e^{−z}) and A_{2}(z) = g_{2}(e^{z}). Observe that the transformation ζ = e^{z} is a one-one correspondence between the sets {z : − log ϱ ≤ Re z ≤ log ϱ, −π < Im z ≤ π} and {ζ : ϱ^{−1} ≤ |ζ| ≤ ϱ}. By the periodicity of e^{z}, we have

{max \atop ϱ^{−1} ≤ |ζ| ≤ ϱ} |B(ζ)| = max_{− log ϱ≤Re z≤log ϱ −π<Im z≤π} |A(z)| ≤ max |A(z)|_{|z|≤log ϱ+π}

≤ max_{−(log ϱ+π)≤Re z≤log ϱ+π −π<Im z≤π} |A(z)| = {max \atop (e^{π}ϱ)^{−1} ≤ |ζ| ≤ e^{π}ϱ} |B(ζ)|.

We deduce that

(2.3)  σ_{e}(A) = \overline{lim}_{ϱ→+∞} \frac{log log max_{ϱ^{−1}≤|ζ|≤ϱ}|B(ζ)|}{log ϱ}.

From

max_{ϱ^{−1}≤|ζ|≤ϱ}B(ζ) = max{ max_{|ζ|=ϱ^{−1}}|B(ζ)|, max_{|ζ|=ϱ}|B(ζ)|},

and the fact that

(2.4)

\overline{lim}_{ϱ→+∞} \frac{log log max_{|ζ|=ϱ^{−1}}|B(ζ)|}{log ϱ} = σ(g_{1}),

\overline{lim}_{ϱ→+∞} \frac{log log max_{|ζ|=ϱ}|B(ζ)|}{log ϱ} = σ(g_{2}),

we deduce that

\overline{lim}_{ϱ→+∞} \frac{log log max_{ϱ^{−1}≤|ζ|≤ϱ}|B(ζ)|}{log ϱ} = max\{σ(g_{1}), σ(g_{2})\}.

This together with (2.3) yields

(2.5)  σ_{e}(A) = max{σ(g_{1}), σ(g_{2})}.

On a problem in complex oscillation theory 277

In particular,

(2.6) σ_{e}(A_{1}) = σ(g_{1}), σ_{e}(A_{2}) = σ(g_{2}).

Let us now turn to the discussion of zeros. Let n(D, 1/F) be the number of zeros of F (z) in the set D. Then we deduce

n(ϱ^{−1} ≤ |ζ| ≤ ϱ, 1/B(ζ)) = n({− log ϱ ≤ Re z ≤ log ϱ −π < Im z ≤ π}, 1/A(z))
≤ n(|z| ≤ log ϱ + π, 1/A(z))

≤ 2(\frac{(log ϱ + π) − π}{2π} + 1) (2.7)

× n({−(log ϱ + π) ≤ Re z ≤ log ϱ + π −π < Im z ≤ π }, 1/A(z))

= (\frac{log ϱ}{π} + 2)n((e^{π}ϱ)^{−1} ≤ |ζ| ≤ e^{π}ϱ, 1/B(ζ)).

Thus

(2.8)  λ_{e}(A) = \overline{lim}_{ϱ→+∞} \frac{log n(ϱ^{−1} ≤ |ζ| ≤ ϱ, 1/B(ζ))}{log ϱ}.

From n(ϱ^{−1} ≤ |ζ| ≤ ϱ, 1/B(ζ)) = n(1 < |ζ| ≤ ϱ, 1/B(ζ)) + n(ϱ^{−1} ≤ |ζ| ≤ 1, 1/B(ζ)), we deduce

\overline{lim}_{ϱ→+∞} \frac{log n(ϱ^{−1} ≤ |ζ| ≤ ϱ, 1/B(ζ))}{log ϱ}

= max\{ \overline{lim}_{ϱ→+∞} \frac{log n(1 < |ζ| ≤ ϱ, 1/B(ζ))}{log ϱ}, \overline{lim}_{ϱ→+∞} \frac{log n(ϱ^{−1} ≤ |ζ| ≤ 1, 1/B(ζ))}{log ϱ}\}.

As in (2.8), we have

(2.9)

λ_{eR}(A) = \overline{lim}_{ϱ→+∞} \frac{log n(1 < |ζ| ≤ ϱ, 1/B(ζ))}{log ϱ},
λ_{eL}(A) = \overline{lim}_{ϱ→+∞} \frac{log n(ϱ^{−1} ≤ |ζ| ≤ 1, 1/B(ζ))}{log ϱ}.

Thus

(2.10)  λ_{e}(A) = max{λ_{eR}(A), λ_{eL}(A)}.

If f is analytic in C_{0} , then [14, p. 15] implies that

(2.11)  f(z) = z^{n}Θ(z)u(z),

where n is an integer, Θ(z) is analytic and non-vanishing on C_{0} ∪ {∞}, and u(z) is an entire function with

(2.12)  u(z) = π(z)e^{h(z)}.

The function π(z) is a Weierstrass product formed from the zeros of f in C_{0} , and h(z) is an entire function. We remark that the assumption that f has an essential singularity at infinity made in [14, p. 15] in order for f to have (2.11) is redundant (see [5, Section 2]).

Letting R_{0} = 1, we may regard B(ζ) to be analytic in C^{∗}, where C^{∗} := C\{z : |z| ≤ 1}. By (2.11) we have a similar representation

(2.13)  B(ζ) = ζ^{n}R(ζ)b(ζ),

where n is an integer, R(ζ) is analytic and non-vanishing on C^{∗} ∪ {∞}, and b(ζ) is an entire function. From (2.13) we easily deduce

(2.14)  \overline{lim}_{ϱ→+∞} \frac{log log max_{|ζ|=ϱ} |B(ζ)|}{log ϱ} = \overline{lim}_{ϱ→+∞} \frac{log log M(ϱ, b(ζ))}{log ϱ} = σ(b(ζ)).

From (2.4) we obtain

(2.15)  σ(g_{2}) = σ(b).

Let us now consider B(ζ) as an analytic function in 0 < |ζ| < 1. If t = 1/ζ, then the function B^{∗}(t) = B(1/t) is analytic in C^{∗}. Thus we have a similar Valiron representation in C^{∗} ∪{∞} as (2.13) with b(ζ) replaced by another entire function, denoted by b^{∗}(t). We deduce from (2.4) again that

(2.16)  σ(g_{1}) = σ(b^{∗}).

We further deduce, from (2.5), (2.15) and (2.16), that

(2.17)  σ_{e}(A) = max{σ(b(ζ), σ(b^{∗}(t))}.

We also deduce, from (2.6), (2.15) and (2.16), that

(2.18)  σ_{e}(A_{1}) = σ(b^{∗}(t)),  σ_{e}(A_{2}) = σ(b(ζ)).

Since the zeros of B(ζ) and b(ζ) coincide in 1 < |ζ| < +∞, we deduce from (2.9) that

(2.19)  λ_{eR}(A) = λ(b(ζ)).

Similarly

(2.20)  λ_{eL}(A) = λ(b^{∗}(t)).

It follows from (2.10) that

(2.21)  λ_{e}(A) = max{λ(b(ζ)), λ(b^{∗}(t))}.

3. Nevanlinna characteristic functions in |z| > R_{0}

Suppose w(z) is meromorphic in C_{0} := {z : R_{0} < |z| < +∞}. By a similar argument as in Valiron [14, p. 15], w(z) has a representation

(3.1)  w(z) = z^{n}Θ(z)f(z),

where n is an integer, Θ(z) is analytic and non-vanishing on C_{0} ∪ {∞}, f is a meromorphic function in C. In fact we may write

(3.2)  f(z) = \frac{u(z)}{v(z)}e^{g(z)},

where u(z) and v(z) are Weierstrass products formed respectively from the zeros and poles of w in C_{0} , and g(z) is an entire function. Thus we can apply the Nevanlinna theory to (3.1) in the region C_{0} (see [6, pp. 97–107]). Let T (r, w) denote the usual Nevanlinna characteristic function in C and T_{1}(r, w) denote the Nevanlinna characteristic function (see [3, p. 4]) for w(z) in C_{0} , which is defined by

T_{1}(r, w) = m_{1}(r, w) + N_{1}(r, w),

where

(3.3)  m_{1}(r, w) = \frac{1}{2π} ∫_{0}^{2π} log^{+} |w(re^{iφ})| dφ,

and N_{1}(r, w) is the counting function for the poles of w in C_{0} .

We deduce, from (3.1), that

(3.4)  m_{1}(r, w) = m(r, f) + O(log r),

and

(3.5)  N_{1}(r, w) = N(r, f).

Thus

(3.6)  T_{1}(r, w) = T (r, f) + O(log r).

But T (r, f) = T (r, 1/f) + O(1), so

(3.7)

T_{1}(r, 1/w) = T (r, 1/f) + O(log r)
= T (r, f) + O(log r)
= T_{1}(r, w) + O(log r).

That is,

(3.8)  T_{1}(r, 1/w) = T_{1}(r, w) + O(log r).

Note that (3.8) is similar to a special case of the first fundamental theorem of the usual Nevanlinna characteristic function.

As in [3, p. 4], we define the order of w in C_{0} by

(3.9)  σ_{1}(w) = \overline{lim}_{r→+∞} \frac{log T_{1}(r, w)}{log r}.

4. Main results

Theorem 1. Let A(z) = B(e^{z}), where B(ζ) = g_{1}(1/ζ) + g_{2}(ζ), g_{1} and g_{2} are entire functions with g_{2} transcendental and σ(g_{2}) not equal to a positive integer or infinity, and g_{1} arbitrary.

(i) Suppose σ(g_{2}) > 1. (a) If f is a non-trivial solution of (1.1) with λ_{e}(f) < σ(g_{2}), then f(z) and f(z + 2πi) are linearly dependent. (b) If f_{1} and f_{2} are any two linearly independent solutions of (1.1), then λ_{e}(f_{1}f_{2}) ≥ σ(g_{2}).

(ii) Suppose σ(g_{2}) < 1. (a) If f is a non-trivial solution of (1.1) with λ_{e}(f) < 1 then f(z) and f(z + 2πi) are linearly dependent. (b) If f_{1} and f_{2} of (1.1) are any two linearly independent solutions, then λ_{e}(f_{1}f_{2}) ≥ 1.

We remark that the conclusion of Theorem 1 remains valid if we assume σ(g_{1}) is not equal to an integer or infinity, and g_{2} arbitrary and still assume B(ζ) = g_{1}(1/ζ) + g_{2}(ζ). In the case when g_{1} is transcendental with its order not equal to an integer or infinity and g_{2} is arbitrary, we need only consider B^{∗}(η) = B(1/η) = g_{1}(η) + g_{2}(1/η) in 0 < |η| < +∞, η = 1/ζ.

From Part (ii)(a) of Theorem 1, and (2.5), we immediately generalize Theorem B on the growth restriction on A(z), and on λ_{e}(f). Part (i) and (ii)(b) of Theorem 1 are new results.

We easily deduce the following result from Theorem 1.

Corollary 1. Under the assumption of A(z) = B(e^{z}) in Theorem 1, any two linearly independent solutions f_{1} and f_{2} of (1.1) must have λ_{e}(f_{1}f_{2}) ≥ 1, and hence λ(f_{1}f_{2}) = +∞.

Theorem 1 also leads to an affirmative answer to the conjecture in [7] mentioned in Section 1.

Theorem 2. Let g(ζ) be a transcendental entire function and its order be not a positive integer or infinity. Let A(z) = B(e^{z}), where B(ζ) = g(1/ζ)+∑_{j=1}^{p} b_{j}ζ^{j} and p is an odd positive integer, then λ(f) = +∞ for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.

We remark that the above conclusion remains valid if

B(ζ) = g(ζ) + ∑_{j=1}^{p} b_{−j}ζ^{−j}.

The following examples show that the above results are the best possible.

Example 1. Let E(z) = e^{pz/2} exp(−\frac{1}{2}e^{mz}), where m is a positive integer and p is an odd positive integer. We further let

(4.1)  f_{j}(z) = E(z)^{1/2} exp(∫_{0}^{z} \frac{(−1)^{j}}{E(t)} dt),

j = 1, 2. Then f_{1} and f_{2} are two linearly independent zero-free entire solutions of (1.1). In fact, the Wronskian of f_{1} and f_{2} is W (f_{1}, f_{2}) = 2. The coefficient A(z) can be calculated from (1.5):

−4A(z) = \frac{2^{2}}{E^{2}} − (\frac{E′}{E} )^{2} + 2\frac{E′′}{E} = \frac{4}{E^{2}} + 2(\frac{E′}{E} )′ + (\frac{E′}{E} )^{2}

= \frac{4}{e^{pz}} exp(e^{mz}) + \frac{p^{2}}{4} − (m^{2} + \frac{pm}{2})e^{mz} + \frac{m^{2}}{4} e^{2mz}

= \frac{4}{ζ^{p}} e^{ζ^{m}} + \frac{p^{2}}{4} − (m^{2} + \frac{pm}{2})ζ^{m} + \frac{m^{2}}{4} ζ^{2m}

= ∑_{j=1}^{p} b_{−j}ζ^{−j} + g(ζ) = −4B(ζ),

where ζ = e^{z} and b_{−j} are constants, b_{−p} ≠ 0, and B(ζ) has a pole of odd order at ζ = 0. Moreover, σ(g) = m.

This example shows that Theorem 2 is the best possible in the sense that the assumption that σ(g) is not a positive integer cannot be dropped. The case when σ(g) = +∞ will be discussed in the next example. Since the solutions f_{1} and f_{2} are zero-free, the example also shows that both (i)(b) and (ii)(b) of Theorem 1 under the assumption that σ(g) is not a positive integer are the best possible. The same situation applies to Corollary 1. Moreover, since f_{1}(z + 2πi) = cf_{2}(z), the example shows that (i)(a) and (ii)(a) of Theorem 1 are the best possible.

Example 2. Let E(z) = e^{pz/2} exp(−\frac{1}{2}e^{e^{z}}), where p is an odd positive integer. If f_{1} and f_{2} are given by (4.1), then W (f_{1}, f_{2}) = 2. Thus, they are linearly independent, zero-free solutions of (1.1) with A(z) given by

−4A(z) = \frac{4}{e^{pz}} exp(e^{e^{z}}) + \frac{p^{2}}{4} − (1 + \frac{p}{2})e^{e^{z}}e^{z} − e^{e^{z}}e^{2z} + \frac{1}{4}e^{2e^{z}}e^{2z}

= \frac{4}{ζ^{p}} exp(e^{ζ}) + \frac{p^{2}}{4} − (1 + \frac{p}{2})e^{ζ}ζ − e^{ζ}ζ^{2} + \frac{1}{4}e^{2ζ}ζ^{2}

= ∑_{j=1}^{p} b_{−j}ζ^{−j} + g(ζ) = −4B(ζ),

where ζ = e^{z}, b_{−j} are again constants, b_{−p} ≠ 0, and B(ζ) has a pole of odd order at ζ = 0. Moreover, σ(g) = +∞.

This example shows that Theorem 2, (i)(a), (ii)(a), (i)(b), (ii)(b) of Theorem 1 and Corollary 1 are the best possible, in the sense that none of the above holds if the order of g(ζ) is replaced by infinity.

5. Proofs of Theorems 1 and 2

Proof of Theorem 1. (i)(a) Let us assume that f(z) and f(z + 2πi) are linearly independent. Since λ_{e}(f) < σ(g_{2}) < +∞, [5, Theorem 1] implies that f(z) and f(z +4πi) must be linearly dependent. Let E(z) = f(z)f(z +2πi), then E(z + 2πi) = c_{1}E(z) for some non-zero constant c_{1} . Clearly E′/E and E′′/E are both periodic functions with period 2πi, while A(z) is periodic by definition. Hence (1.6) shows that E(z)^{2} is also periodic with period 2πi. Thus we can find an analytic function Φ(ζ) in 0 < |ζ| < +∞, so that E(z)^{2} = Φ(e^{z}). Substituting this expression into (1.6) yields

(5.1)  −4B(ζ) = \frac{c^{2}}{Φ} + ζ \frac{Φ′}{Φ} − \frac{3}{4}ζ^{2}(\frac{Φ′}{Φ} )^{2} + ζ^{2} \frac{Φ′′}{Φ} .

Since both B(ζ) and Φ(ζ) are analytic in C^{∗} := {z : 1 < |ζ| < +∞}, the Valiron theory gives their representations as

(5.2)  B(ζ) = ζ^{n}R(ζ) b(ζ),  Φ(ζ) = ζ^{n_{1}}K_{1}(ζ) ϕ(ζ),

where n, n_{1} are some integers, R(ζ) and K_{1}(ζ) are functions that are analytic and non-vanishing on C^{∗} ∪ {∞} and b(ζ) and ϕ(ζ) are entire functions. We deduce from (5.1) that

(5.3)  m_{1}(ϱ, 1/Φ) = m_{1}(ϱ, B) + S_{1}(ϱ, Φ),

where

S_{1}(ϱ, Φ) = 3m_{1}(ϱ, \frac{Φ′}{Φ} ) + m_{1}(ϱ, \frac{Φ′′}{Φ} ) + O(log ϱ).

It is not difficult to show that

(5.4)  S_{1}(ϱ, Φ) = S(ϱ, ϕ) = o(T (ϱ, ϕ))  n.e.

Thus, (5.3) implies

(5.5)  T_{1}(ϱ, 1/Φ) = N_{1}(ϱ, 1/Φ) + T_{1}(ϱ, B) + S_{1}(ϱ, Φ).

Applying (3.6), (3.7), (5.4) to (5.5) and using the fact that N_{1}(ϱ, 1/Φ) = N(ϱ, 1/ϕ), we deduce

(5.6)  T (ϱ, ϕ) = N(ϱ, 1/ϕ) + T (ϱ, b) + S(ϱ, ϕ).

Notice that (5.6) satisfied by ϕ is an analogous formula to (1.7) satisfied by E. It is easy to see that λ_{e}(f) = λ_{e}(E) = λ_{e}(E^{2}). Since λ_{e}(f) < σ(g_{2}), so λ_{eR}(E^{2}) ≤ λ_{e}(E^{2}) < σ(g_{2}). As in (2.19), λ(ϕ) = λ_{eR}(E^{2}). But σ(g_{2}) = σ(b(ζ)) by (2.15). Hence λ(ϕ) < σ(b). It follows from (5.6) and Fact (A) of [2, Section 2] (see also [8, Lemma 1.5] or [10, Lemma 1.1.1]) that σ(ϕ) = σ(b). Thus λ(ϕ) < σ(ϕ). So σ(ϕ) is either a positive integer or infinity. So is σ(b). But σ(g_{2}) = σ(b(ζ)) which is a contradiction. Hence f(z) and f(z + 2πi) must be linearly dependent.

(i)(b) Let us assume that λ_{e}(f_{1}f_{2}) < σ(g_{2}). Hence we have λ_{e}(f_{1}) < σ(g_{2}) and λ_{e}(f_{2}) < σ(g_{2}). From (i)(a), f_{j}(z) and f_{j}(z + 2πi) are linearly dependent, for j = 1, 2. Let E(z) = f_{1}(z)f_{2}(z). Then, as in (i)(a), E(z + 2πi) = c_{2}E(z) for some non-zero constant c_{2} , and E(z)^{2} is a periodic function with period 2πi. Following the argument in (i)(a) yields σ(g_{2}) to be a positive integer or infinity, which is a contradiction. Hence λ_{e}(f_{1}f_{2}) ≥ σ(g_{2}).

(ii)(a) Let us assume that σ(g_{2}) < 1 and that f is a non-trivial solution of (1.1) with λ_{e}(f) < 1. Suppose, as in the proof of (i)(a), that f(z) and f(z + 2πi) are linearly independent, and we obtain (5.6) by following the same argument there.

As in (i)(a), we deduce from λ_{e}(f) = λ_{e}(E) = λ_{e}(E^{2}) and (2.19) that λ(ϕ) = λ_{eR}(E^{2}) ≤ λ_{e}(E^{2}) = λ_{e}(f) < 1. We further deduce from the hypothesis on g_{2} and (2.15) that σ(b) = σ(g_{2}) < 1. Using this result, (5.6) and Fact (A) in [2, Section 2] (see also [8, Lemma 1.5] or [10, Lemma 1.1.1]), we deduce σ(ϕ) < 1.

The remaining proof now closely parallels certain parts of [7, Lemma 5]. We offer the full details here for the sake of completeness.

We turn to the representation (5.2). Since K_{1}(ζ) is analytic at ∞, we deduce K_{1}^{(k)}(ζ)/K_{1}(ζ) = o(1), as |ζ| → +∞. It follows from (5.2) and a standard estimate on the logarithmic derivative (see [8, Section 3.6] or [10, Proposition 5.12]) that there exists a positive constant M such that

|ζ \frac{Φ′}{Φ} − \frac{3}{4}ζ^{2}(\frac{Φ′}{Φ} )^{2} + ζ^{2} \frac{Φ′′}{Φ} | ≤ |ζ|^{M},

for ζ ∉ V , where V is an R-set (see [8, Section 3.6] or [10, p. 84]).

In addition to ζ ∉ V , let us further assume that |ϕ(ζ)| > 1. Then we easily see from (5.1) that there exists a positive integer N so that

(5.7)  |B(ζ)| ≤ |ζ|^{N}.

On the other hand, we have the expansion

B(ζ) = ∑_{k=−∞}^{+∞} b_{k}ζ^{k}, 0 < |ζ| < +∞,

where g_{2}(ζ) = ∑_{k=0}^{+∞} b_{k}ζ^{k} is a transcendental entire function with σ(g_{2}) < 1. We now rewrite B(ζ) = h_{1}(ζ) + h_{2}(ζ), where

h_{1}(ζ) = ∑_{k≤N} b_{k}ζ^{k}, h_{2}(ζ) = ∑_{k>N} b_{k}ζ^{k}.

Clearly |h_{1}(ζ)| = O(|ζ|^{N}). Let u_{1}(ζ) = ϕ(ζ), and by the earlier choice that ζ ∉ V and |u_{1}(ζ)| = |ϕ(ζ)| > 1 we deduce from (5.7) that |h_{2}(ζ)| = O(|ζ|^{N}) holds. We can therefore find a positive constant K such that |ζ^{−N}h_{2}(ζ)/K| ≤ 1, where ζ ∉ V and |u_{1}(ζ)| > 1. Let u_{2}(ζ) = ζ^{−N}h_{2}(ζ)/K, then it is easy to see that u_{2}(ζ) is an entire transcendental function with σ(u_{2}) < 1. It follows from above that if ζ ∉ V, |u_{1}(ζ)| > 1, then |u_{2}(ζ)| ≤ 1.

In addition, we note that since g_{2}(ζ) is transcendental, B(ζ) = g_{1}(1/ζ)+g_{2}(ζ) and hence b(ζ) is also transcendental by (5.2). Therefore formula (5.6) implies that u_{1}(ζ) = ϕ(ζ) must also be transcendental.

We define D_{j}^{∗} = {ζ : |u_{j}(ζ)| > 1}, j = 1, 2. Clearly both D_{1}^{∗} and D_{2}^{∗} are open sets. We denote the boundary of D_{j}^{∗} by ∂D_{j}^{∗}, j = 1, 2, and so we have |u_{j}(ζ)| = 1 for ζ belongs to ∂D_{j}^{∗}, j = 1, 2. But both u_{1}(ζ) and u_{2}(ζ) are transcendental, so each D_{j}^{∗} must contain an unbounded component D_{j} for j = 1, 2. Denote the boundary of D_{j} by ∂D_{j}, j = 1, 2. Let E_{j}(ϱ) = {θ : ϱe^{iθ} ∈ D_{j}}, j = 1, 2, and E(ϱ) = {θ : ϱe^{iθ} ∈ V }. Clearly E_{1}(ϱ)∩E_{2}(ϱ) ⊂ E(ϱ). We also let θ_{j}(ϱ), j = 1, 2, and θ(ϱ) respectively, to be the angular measures of E_{j}(ϱ), j = 1, 2, and E(ϱ).

We note that since V is an R-set, so given ε > 0, there exists ϱ_{0} > 0 such that θ(ϱ) < ε for ϱ > ϱ_{0} . We also note that we can choose ϱ > ϱ_{0} so that the circle |ζ| = ϱ intersects D_{j}, j = 1, 2.

We now define

(5.8)  θ_{j}^{∗}(ϱ) = \{ \begin{matrix}θ_{j}(ϱ), if E_{j}(ϱ) ≠ [0, 2π], \\ + ∞, if E_{j}(ϱ) = [0, 2π]. \end{matrix}

Then the Beurling–Tsuji inequality [13, Theorem III 68, p. 117] gives

(5.9)  π ∫_{ϱ_{0}}^{ϱ/2} \frac{dt}{tθ_{j}^{∗}(t)} < log log M(ϱ, u_{j}) + O(1),  j = 1, 2,

where M(ϱ, u_{j}), j = 1, 2, denotes the usual maximum modulus of u_{j} on |ζ| = ϱ. It was shown in the remark in [7, pp. 153–154] (see also [8, pp. 96–97]) that even when E_{j}(ϱ) = [0, 2π], j = 1, 2, the inequality (5.9) is replaced by

(5.10) π ∫_{ϱ_{0}}^{ϱ/2} \frac{dt}{tθ_{j}(t)} < log log M(ϱ, u_{j}) + \frac{ε}{2π} K_{j} log ϱ + O(1),  j = 1, 2,

if K_{i} > σ(u_{j}), i ≠ j and ϱ is sufficiently large. Since σ(u_{j}) < 1, j = 1, 2, we deduce from (5.10) that there exists a constant 0 < β < 1 so that

π ∫_{ϱ_{0}}^{ϱ/2} \frac{dt}{tθ_{j}(t)} < (1 − β) log ϱ,

for ϱ > ϱ_{1} > ϱ_{0} , j = 1, 2. Summing the above inequalities for j = 1, 2 yields

π ∫_{ϱ_{0}}^{ϱ/2} \frac{θ_{1}(t) + θ_{2}(t)}{θ_{1}(t)θ_{2}(t)} \frac{dt}{t} < 2(1 − β) log ϱ.

But 2\sqrt{ab} ≤ a + b, (a, b ≥ 0). Hence

4π ∫_{ϱ_{0}}^{ϱ/2} \frac{dt}{t(θ_{1}(t) + θ_{2}(t))} < 2(1 − β) log ϱ.

Notice that θ_{1}(ϱ) + θ_{2}(ϱ) ≤ 2π + ε for ϱ > ϱ_{1} . This gives

(5.11)  \frac{4π}{2π + ε} log \frac{ϱ}{2ϱ_{0}} < 2(1 − β) log ϱ.

Since ε > 0 is arbitrary, we obtain a contradiction after dividing both sides by log ϱ. Thus we conclude that f(z) and f(z + 2πi) must be linearly dependent.

(ii)(b) Suppose f_{1} and f_{2} are linearly independent and λ_{e}(f_{1}f_{2}) < 1. Then λ_{e}(f_{1}) < 1 and λ_{e}(f_{2}) < 1. We deduce from the conclusion of (ii)(a) that f_{j}(z) and f_{j}(z + 2πi) are linearly dependent, j = 1, 2. Let E(z) = f_{1}(z)f_{2}(z), then we can find a non-zero constant c_{3} such that E(z + 2πi) = c_{3}E(z). Repeating the same argument in (ii)(a) by using the fact that E(z)^{2} is also periodic, we obtain a contradiction as (5.11). Hence λ_{e}(f_{1}f_{2}) ≥ 1.

This completes the proof of Theorem 1.

Proof of Theorem 2. Suppose there exists a non-trivial solution f that satisfies (1.4). We deduce λ_{e}(f) = 0. Theorem 1(i)(a) and (ii)(a) imply that f(z) and f(z +2πi) are linearly dependent. However, [7, Lemma 6] or [8, Theorem 3.7] implies that f(z) and f(z + 2πi) are linearly independent. This is impossible. Hence (1.2) holds for each non-trivial solution f. This completes the proof of Theorem 2.

6. Perturbation results

Suppose (1.1) admits a non-trivial solution that has a finite e-type exponent of convergence of zeros. If ∏(z) is a periodic entire function with period 2πi and σ_{e}(Π) < σ_{e}(A), what can we say about the e-type exponent of convergence of zeros of any two linearly independent solutions of the equation

(6.1)  f′′ + (A(z) + ∏(z))f = 0?

A perturbation result for (6.1) where the coefficients are not necessarily periodic is given in [4, Theorem 3.1]. We answer the above perturbation problem based on the method used in [4] coupled with the special properties of periodic coefficients established in Section 2.

Theorem 3. Let B(ζ) = g_{1}(1/ζ) + g_{2}(ζ), C(ζ) = g_{3}(ζ), where g_{1} , g_{2} and g_{3} are entire functions of finite order such that σ(g_{2}) is a positive integer, and σ(g_{3}) < σ(g_{2}). Suppose A(z) = B(e^{z}), ∏(z) = C(e^{z}) and furthermore that (1.1) admits a non-trivial solution f with λ_{e}(f) < σ(g_{2}) and that f(z) and f(z + 2πi) are linearly independent. Then

(i) for any non-trivial solution h of equation (6.1) with λ_{e}(h) < σ(g_{2}), h(z) and h(z + 2πi) are linearly dependent, and

(ii) for any two linearly independent solutions h_{1} and h_{2} of (6.1), we must have λ_{e}(h_{1}h_{2}) ≥ σ(g_{2}).

Let us consider Example 1 in Section 4 where f_{1}(z + 2πi) = cf_{2}(z) for some non-zero constant c. This implies that f_{1}(z), f_{1}(z +2πi) are linearly independent solutions of (1.1) with the A(z) given in Example 1. Clearly λ_{e}(f) = 0 < σ(g). Suppose h is a non-trivial solution of (6.1) with λ_{e}(h) < σ(g) = m, then Theorem 3(i) shows that h(z) and h(z + 2πi) are linearly dependent. Part (ii) implies λ_{e}(h_{1}h_{2}) ≥ σ(g) = m for any two non-trivial linearly independent solutions h_{1} and h_{2} .

Theorem 2 investigates the oscillation properties of any non-trivial solution to (1.1) with B(ζ) = ∑_{j=1}^{p} b_{−j}ζ^{−j} + g(ζ), where p is an odd positive integer and σ(g) is not a positive integer or infinity. We now investigate perturbation problem for (6.1) precisely when σ(g) is a positive integer.

Theorem 4. Let g(ζ) be a transcendental entire function of an integer order σ(g) and C(ζ) ≢ 0 be an entire function with σ(C) < σ(g). Let A(z) = B(e^{z}), where B(ζ) = ∑_{j=1}^{p} b_{−j}ζ^{−j}+g(ζ), p is an odd positive integer and ∏(z) = C(e^{z}). Suppose (1.1) admits a non-trivial solution f with λ(f) < +∞. Then any nontrivial solution h of (6.1) must have λ(h) = +∞. In fact, the stronger conclusion (1.2) holds.

Example 1 shows that A(z) satifies the hypothesis of Theorem 4, and that (1.1) possesses two zero-free solutions. Thus any non-trivial solution h to (6.1) must have λ(h) = +∞.

Proof of Theorem 3. (i) Let f be a non-trivial solution of (1.1) with λ_{e}(f) < σ(g_{2}), and f(z) and f(z + 2πi) be linearly independent. Let E(z) = f(z)f(z + 2πi), then (1.5) implies that E(z)^{2} is a periodic function with period 2πi since both E′/E and E′′/E are periodic functions with period 2πi. Thus, we can find an analytic function Φ(ζ) in 0 < |ζ| < +∞ so that E(z)^{2} = Φ(e^{z}). Substituting this representation into (1.5) with ζ = e^{z} yields (5.1). We recall that B(ζ) and Φ(ζ) have the Valiron representations in C^{∗} given in (5.2).

Although the proof now closely parallels that of [4, Theorem 3.1], the modifications needed to apply the argument there are sufficiently intricate that it warrants the inclusion of the details here. Let us now suppose (6.1) possesses a non-trivial solution h(z) such that σ_{e}(h) < σ(g_{2}) but h(z) and h(z + 2πi) are linearly independent. Let F (z) = h(z)h(z + 2πi). By a similar argument that we have applied to E(z) above, we conclude that there exists an analytic function Ψ(ζ) in 0 < |ζ| < ∞ such that F (z)^{2} = Ψ(e^{z}). Similary, Ψ has a Valiron representation

(6.2)  Ψ(ζ) = ζ^{n_{2}}K_{2}(ζ)ψ(ζ)

in C^{∗}, where n_{2} is an integer, K_{2}(ζ) is analytic and non-vanishing on C^{∗} ∪ {∞}, and ψ(ζ) is an entire function in C.

We now substitute F (z)^{2} = Ψ(e^{z}), with ζ = e^{z}, into (1.5) with A(z) replaced by A(z) + ∏(z). This yields

(6.3)  −4(B(ζ) + C(ζ)) = \frac{c_{1}^{2}}{Ψ} + ζ \frac{Ψ′}{Ψ} − \frac{3}{4}ζ^{2}(\frac{Ψ′}{Ψ} )^{2} + ζ^{2} \frac{Ψ′′}{Ψ} ,

where c_{1} ≠ 0 is the Wronskian of h(z) and h(z + 2πi).

Since, as in (2.19), we have λ(ϕ) = λ_{eR}(E^{2}) = λ_{eR}(f) < σ(g_{2}) = σ(b(ζ)), so (5.6) implies that σ(ϕ) = σ(b(ζ)) = σ(g_{2}), and hence σ(ϕ) is an integer with λ(ϕ) < σ(ϕ). Thus we may write ϕ(ζ) = π_{1}e^{P} , where P (ζ) = αz^{σ}, α is a constant, σ = σ(g_{2}) and σ(π_{1}) < σ.

In a similar fashion, we have λ(ψ) = λ_{eR}(F^{2}) = λ_{eR}(h) < σ(g_{2}) = σ(b(ζ)). We then apply a similar argument to (6.3) to obtain

(6.4)  T (ϱ, ψ) = N(ϱ, 1/ψ) + T (ϱ, d) + S(ϱ, ψ),

as to (5.1) for (5.6), where d(ζ) is an entire function appearing in the Valiron representation of

B(ζ) + C(ζ) = ζ^{n_{3}}R_{d}(ζ)d(ζ),

where the functions R_{d}(ζ) and d(ζ) play the same roles of the corresponding functions in (5.2), and it is easy to check that σ(d) = σ(b). But then σ(ψ) = σ(d(ζ)) = σ(g_{2}) > λ(ψ). Thus we may write ψ(ζ) = π_{2}e^{Q}, where Q = βz^{σ}, β is a constant, σ = σ(g_{2}), and σ(π_{2}) < σ.

Let Φ(ζ) = R_{1}e^{P} and Ψ(ζ) = R_{2}e^{Q}, where R_{1}(ζ) = ζ^{n_{1}}K_{1}(ζ)π_{1}(ζ), R_{2}(ζ) = ζ^{n_{2}}K_{2}(ζ)π_{2}(ζ), σ_{1}(R_{1}), σ_{1}(R_{2}) < σ = σ(g_{2}), and substitute them into (5.1) and (6.3) respectively yielding

(6.5) \begin{aligned}−4B(ζ) & = \frac{c^{2}}{R_{1}e^{P}} + ζ(\frac{R′_{1}}{R_{1}} + P′) − \frac{3}{4}((ζ \frac{R′_{1}}{R_{1}} )^{2} + 2ζ \frac{R′_{1}}{R_{1}} P′ + ζ^{2}P′^{2}) \\ & + ζ^{2}(\frac{R′′}{R_{1}} + 2\frac{R′_{1}}{R_{1}} P′ + P′^{2} + P′′) \end{aligned}

and

(6.6)  \begin{aligned}−4(B(ζ) + C(ζ)) & = \frac{c_{1}^{2}}{R_{2}e^{Q}} + ζ(\frac{R′_{2}}{R_{2}} + Q′) \\ & − \frac{3}{4}((ζ \frac{R′_{2}}{R_{2}} )^{2} + 2ζ \frac{R′_{2}}{R_{2}} Q′ + ζ^{2}Q′^{2}) \\ & + ζ^{2}(\frac{R′′_{2}}{R_{2}} + 2\frac{R′_{2}}{R_{2}} Q′ + Q′^{2} + Q′′). \end{aligned}

Substracting (6.6) from (6.5) yields

(6.7)  4C(ζ) = \frac{c^{2}}{R_{1}e^{P}} − \frac{c_{1}^{2}}{R_{2}e^{Q}} + H(ζ),

where H(ζ) is meromorphic in C^{∗}. In fact H(ζ) is a differential polynomial in R′_{1}/R_{1} , R′_{2}/R_{2} , P′, Q′ and their derivatives. We can deduce from the definitions of R_{1} , R_{2} , P, Q that σ_{1}(H) < σ(g_{2}). Rewriting (6.7) as

(6.8)  e^{−P} + H_{1}e^{−Q} = H_{2},

where H_{1} and H_{2} are meromorphic functions in C^{∗} with max{σ_{1}(H_{1}), σ_{1}(H_{2})} < σ(g_{2}) and H_{1} = −c_{1}^{2}/c^{2}R_{1}/R_{2} . Differentiating (6.8) and using the resulting equation to eliminate e^{−P} from (6.8) yields

(6.9)  H_{3}e^{−Q} = H_{4},

where H_{3} = H′_{1} + (P′ − Q′)H_{1} and H_{4} is a meromorphic function in C^{∗} with σ_{1}(H_{3}), σ_{1}(H_{4}) < σ(g_{2}) = σ. Thus H_{3} ≡ 0, i.e., H_{1} = c_{2}e^{Q−P} , where c_{2} is a nonzero constant, from a simple order consideration in (6.9). This is a contradiction to σ_{1}(H_{1}) < σ(g_{2}) unless P ≡ Q = αz^{σ}. Thus R_{1} = c_{3}R_{2} , where c_{3} is a constant, and R′_{1}/R_{1} = R′_{2}/R_{2} . Substituting R′_{1}/R_{1} = R′_{2}/R_{2} , P = Q into (6.5) and (6.6) and subtracting the resulting equations yields (6.7) with H(ζ) ≡ 0. Substituting R_{1} = c_{3}R_{2} into this new equation and considering its order yields immediately C(ζ) ≡ 0 in C^{∗}, and hence in C. This is a contradiction. Hence h(z) and h(z + 2πi) must be linearly dependent.

(ii) Suppose (6.1) possesses two non-trivial solutions h_{1} and h_{2} that are linearly independent and λ_{e}(h_{1}h_{2}) < σ(g_{2}). Part (i) implies that h_{j}(z), and h_{j}(z + 2πi) are linearly dependent, for j = 1, 2. Let E(z) = f(z)f(z + 2πi) and F (z) = h_{1}(z)h_{2}(z). Then F (z + 2πi) = c_{4}F (z), for some non-zero constant c_{4} . Applying a similar argument to E(z) and F (z) as in (i) yields C(ζ) ≡ 0. This is a contradiction. Hence λ_{e}(h_{1}h_{2}) ≥ σ(g_{2}). This completes the proof of Theorem 3.

290 Yik-Man Chiang and Shi-An Gao

Proof of Theorem 4. Let g, C and A be defined in Theorem 4. Suppose (1.1) possesses a non-trivial solution f with λ(f) < +∞. Hence λ_{e}(f) = 0 < σ(g). Thus [7, Lemma 6] implies that f(z) and f(z + 2πi) are linearly independent solutions of (1.1). Thus f(z) and f(z + 2πi) satisfy the hypotheses of Theorem 3. Suppose that (6.1) admits a non-trivial solution h with λ_{e}(h) = 0 < σ(g). Then [7, Lemma 6] again implies that h(z) and h(z + 2πi) are linearly independent. Theorem 3(ii) implies that λ_{e}(h(z)h(z + 2πi)) ≥ σ(g) > 0. But λ_{e}(h) = λ_{e}(h(z)h(z + 2πi)) > 0. Hence (1.2) holds. This completes the proof.

Acknowledgement. The authors would like to thank the referee for his detailed comments that improved the readability of the paper.

References

[1] Arscott, F.M.: Periodic Differential Equations. - Internat. Ser. Monogr. in Pure and Applied Math. 66, Pergamon Press, 1964.

[2] Bank, S.B., and I. Laine: On the oscillation theory of f′′ + Af = 0 where A is entire. - Trans. Amer. Math. Soc. 273, 1982, 351–363.

[3] Bank, S.B., and I. Laine: Representations of solutions of periodic second order linear differential equations. - J. Reine Angew. Math. 344, 1983, 1–21.

[4] Bank, S.B., I. Laine, and J. Langley: Oscillation results for solutions of linear differential equations in the complex domain. - Resultate Math. 16, 1989, 3–15.

[5] Bank, S.B., and J. Langley: Oscillation theorems for higher order linear differential equations with entire periodic coefficients. - Comment. Math. Univ. St. Paul. 41, 1992, 65–85.

[6] Bieberbach, L.: Theorie der gewöhnlichen Differentialgleichungen. - Springer-Verlag, Berlin–Heidelberg–New York, 1965.

[7] Gao, Shi-An: A further result on the complex oscillation theory of periodic second order linear differential equations. - Proc. Edinburgh Math. Soc. 33, 1990, 143–158.

[8] Gao, Shi-An, Zong-Xuan Chen, and Te-Wei Chen: Oscillation Theory of Linear Differential Equations. - Huazhong University of Science and Technology Press, 1998 (Chinese).

[9] Hayman, W.K.: Meromorphic Functions. - Clarendon Press, Oxford, 1975 (with appendix).

[10] Laine, I.: Nevanlinna Theory and Complex Differential Equations. - Walter de Gruyter, Berlin, 1993.

[11] Magnus, W., and S. Winkler: Hill’s Equation. - Interscience Tracts in Pure and Applied Mathematics 20, Interscience Publishers/John Wiley, New York–London–Sydney, 1966.

[12] Slavyanov, S.Yu., and W. Lay: Special Functions – A Unified Theory Based on Singularities. - Oxford University Press, 2000.

[13] Tsuji, M.: Potential Theory in Modern Function Theory, 2nd ed. - Chelsea, New York, 1975.

[14] Valiron, G.: Lectures on the General Theory of Integral Functions. - Chelsea, New York, 1949.

Received 1 March 2001