on the zeros of a certain wronskian - CiteSeerX

ON THE ZEROS OF A CERTAIN WRONSKIAN NORBERT STEINMETZ 1. Notations L e t / b e a meromorphic function in the complex plane. We will use the following standard notations of Nevanlinna theory, m(r,f),N(r,f), T(r,f),S(r,f), ... (see Hayman [2]). If g is another meromorphic function (which always means a meromorphic function in the plane), we will say that g is small compared with / if T(r,g) = S(r,f). Given q meromorphic functions yx, ..-,yq,

\,-,yq) =

their Wronskian will be denoted by L

•••

yQ

[

•••

y' satisfies a growth condition \(z)\ = O(|z|m) as z -*• oo. Let S be a sector |argz — 0\ < S, \z\ > R, and let 0 be holomorphic in S. Then 0 is said to have the asymptotic representation 00

k—m

if

UOO- E atz-fc/p z n/p ^Oasz->ooin \

fc—m

/

for every n = —m, — m+ 1, .... It is well known that an asymptotic representation may be differentiated term by term in any smaller subsector. The class of functions having an asymptotic representation (2) in S for fixed p will be denoted by (S)p.

Received 1 December 1987. 1980 Mathematics Subject Classification 30D35, 34A20. Bull London Math. Soc. 20 (1988) 525-531

526

NORBERT STEINMETZ

2. Introduction An essential step in the proof of the Second Main Theorem for Small Functions (see [1, 5]) is an estimate of the quantity m(r, \/L(f)), where L is generated by the meromorphic functions yv ..., yQ which are assumed to be small compared with the meromorphic function/. Recently, Frank and Weissenborn [1] proved the inequality (in an equivalent form)

mk j^J ^ m(r, L(f)) + (2 + e)N(r, f) + S(r, f)

(3)

for any e > 0. (Here S(r,f) depends on £ > 0, but the associated exceptional set is independent of e.) They also showed by an example that the condition

T(r,yv) = S(r,f)

(4)

is essential. EXAMPLE 1. yi=\,y2

= e*\y3 = e-2i\

L(y)=y'" + 4y',f= tanz, L(f) = 6(1 +tan 2 z) 2 . Here we have max 7(r, >>„) ~

2r

T(r,f),

71

while 1 \ 8r 4r m a n dm ' 77 77A ~ ~ ^ W)) + 2N(rf) ~ We remark t h a t / = — i(y2— yi)/(yz+ yO »s a rational function of yt,y2

(and y3).

3. Results The main result of this paper will be Theorem 1, which in particular shows that the example of Section 2 describes the typical exceptional case. THEOREM 1. Let the linear differential operator y^ + a(/_1(z)/^

+ ...+a0(z)y

(5)

have rational coefficients a0, ..., aQ_x and let f be a nonrational meromorphic function in rational function of the plane. Then, either inequality (3) holds for every e>0orfisa a (local) fundamental set yv ..., yq of the differential equation L(y) = 0. REMARK. If the coefficients of L are polynomials, then the solutions of L(y) = 0 are entire functions and the representation/= R(yv ..., yQ) holds true throughout the plane. In the contrary case the solutions of L(y) = 0 are in general not single-valued (and may have finite essential singularities), and the representation/= R(y^ ..., yq) is valid locally (in any simply connected domain containing no poles of any coefficient) and then globally by analytic continuation. For the proof of Theorem 1 we need the following result on the solutions of linear differential equations with rational coefficients, which seems to have its own interest.

ON THE ZEROS OF A CERTAIN WRONSKIAN

527

THEOREM 2. Let K and L be linear differential operators whose coefficients are rational near infinity. Then there exists a third linear differential operator M with coefficients of the same type such that the {local) solutions of M(w) = 0 are exactly the function elements w = £ u} v}, where uj and v} are local solutions of K(u) = 0 and L(v) = 0, respectively.

REMARK. The set K of solutions of K(u) = 0, say, has to be regarded as a set of function elements (u, D): u is meromorphic in the domain D and satisfies K(u) = 0 in D. It is thus asserted in Theorem 2 that the set M of solution elements is the linear hull of the set K L . We do not know whether in Theorem 2 the term 'rational near infinity' may be replaced (both times!) by 'rational'. Of course, if K and L are of Fuchsian type, then the coefficients of M are rational functions. In general there is possibly a finite number of essential singularities.

4. Linear differential equations with rational coefficients Let N be a linear differential operator of order n and let N be the set of all solution elements (w, D): D is a domain and w is meromorphic in D and satisfies there N(w) = 0. The common domain of regularity of the coefficients of N is denoted by Q. It is well known that N has the following properties: (a) Completeness. Every element (w, D) may be continued analytically along any arc in Q, and any continuation belongs to N. (b) Linearity. Locally, N is a linear space of dimension n, that is, for a fixed simply connected domain D c Q the set of all elements (w, D) of N forms a ndimensional linear space over C. Conversely, if a set N of function elements has properties (a) and (b), then there is a linear differential operator N of order n such that N is exactly the set of solutions of N(w) = 0. However, the coefficients of N may have (infinitely many) poles in ft. However, if the coefficients of N are rational near infinity, we have a third important property. (c) Asymptotic representation. Given any direction argz = 0, there exists a sector , S : | a r g z - 0 | < S, \z\ > R and a local base (w15 5), ..., (wn, S) such that

^zhQfalogz),

(6)

where P} is a polynomial, p is a positive integer, Xj is a complex constant and Qj is a polynomial in log z over (S)p (for this notation see Section 1). REMARK. The existence of an asymptotic representation is usually proved in the theory of matrix differential equations Y' = A{z) Y (see, for example, Wasow [7, Theorem 19.1, p. I l l ] ; A(z) is a n x n matrix whose entries are rational near infinity, say). It is, however, not easy to derive the desired results for a single «th order

equation from the general theory. We thus prefer to use the much older result (c) by Sternberg [6], which has been kindly pointed out to me by Giinther Frank.

528

NORBERT STEINMETZ

We will prove that conditions (a)-(c) are characteristic for sets of solutions of linear differential equations whose coefficients are rational near infinity. THEOREM 3. Let N be a set of function elements (w, D), D always contained in \z\ > R, such that (a)-(c) hold. Then there exists a linear differential operator N with coefficients rational near infinity such that N is exactly the solution set of N(w) = 0.

Proof. Let (wl5 D), ..., (wn, D) be a local base of N. Then W(wv ..., wn) =£ 0, and it is clear that N has to be defined (locally) by N(W) =

W

^ - ^ n ^ ) WiW

=

„ +

flni(z)

„ + _ _ +

flo(z) M;

(7)

Wj

Since N does not depend on the special choice of the base and since w1? ..., wn can be continued analytically along any arc in \z\ > R, the coefficients afS,...,an_l are meromorphic in \z\ > R. We have to show that the point at infinity is not an essential singularity of any of the coefficients. To this aim we consider an arbitrary direction argz = 6 and the corresponding sector Sand the distinguished base wv ...,wn belonging to S(condition (c)). Diminishing S if necessary we may assume that the derivatives h;ave similar representations p llP)x ]» « (6J (J = 1, ..., n;k = 1, 2, ...). From this representation it is easily seen that the coefficients aj are rational functions in log z over (,S)p, a^(z) = R^z, logz), and, in particular, that \a.j(z)\ = O(\zn (8) holds as z -*• oo for some m = m(S) in 5":|z| > R, |argz — #| < \S. Thus, an easy compactness argument shows that (8) holds without restriction of argz for some m > 0. This proves Theorem 3. 5. Proof of Theorem 2 Let M be the linear hull of K • L, where K and L are the solution sets of K(u) = 0 and L{v) = 0, respectively. We will show that M has the properties (a)-(c) of Section 4 and thus, by Theorem 3, M is the set of solutions of some differential equation M(w) = 0, where M has coefficients rational near infinity. The completeness and linearity of M are obvious. If argz = 6 is an arbitrary direction, let SK and Sh be the corresponding sectors and ux, ...,uk and vv ...,vt be the distinguished fundamental sets of K(u) = 0 and L(v) = 0, respectively. We set S = SK n SL and choose a base wv ..., wm of M consisting of products uKv}. Then it is clear that the functions wv ...,wm have an asymptotic representation, and (c) is also satisfied with p = lcm {pK, p,), say.

6. Proof of Theorem 1 Let 5 ^ 2 be an arbitrary integer and let Mg be the linear hull of L • L • • • L (s times), where L is the set of solutions of L(y) = 0.

ON THE ZEROS OF A CERTAIN WRONSKIAN

529

We choose arbitrary local bases w1} ..., un of Ms and Uv ..., Uk of M g+1 and consider W(U1,...,Uk,uJ,...,unf) U)

W(u1,...,uk)w(u1,...,uny

K)

Clearly, Q is independent of the choice of bases and can be continued analytically along every arc in \z\ > R, except perhaps for poles. Here R > 0 is chosen in such a way that the disk \z\ < R contains all poles of the coefficients of the operator L. We remark that the differential polynomial Q has also been used in [1] and [5], but without the normalization by the two Wronskians in the denominator. This normalization has two advantages. First, it makes O a meromorphic function in \z\ > R, of course independent of the special bases. The second advantage is that the coefficients of Q are rational near infinity (this will be shown in Section 7). Even if the coefficients of L are constants, the coefficients of Q would be exponentials without this normalization, and the following proof would apply only to meromorphic functions / of order > 1. In the next section we will show that Q(f) is a homogeneous differential polynomial in L(f) of degree n with coefficients rational near infinity, a(f) = Q(F), F = L(f).

(10)

Assuming (10), the proof of Theorem 1 runs as follows. Either Q(/) = 0 and so (locally) (11) which is a rational function of yx, ..., yq, or Q(/) =j= 0, which gives

The first term on the right-hand side of (12) is S(r,f) by Nevanlinna's theorem on the proximity function of the logarithmic derivative, while the second term is estimated by the First Main Theorem

(for the last inequality, see [1] or [5]). The result is

m{r, j^j ^ m(r, L(/)) + (l + ^JN(r, f) + Sir,/),

(13)

and to complete the proof we have only to choose s in such a way that k/n < 1 +e, which is possible since infs k/n = 1 (see [5]). 7. Completion of the proof of Theorem 1 Applying the transformation formula

w{gx,...,gm)

530

NORBERT STEINMETZ

(see Muir [4, p. 663]) to (9), we find ^Kf) =

J7^—~~L

T^

>

(14)

where

is a linear differential operator of order k having coefficients which are rational near infinity. This follows by repeated application of Theorem 2. Now define new operators K{y): = K(u,y)/ur

(16)

Then the coefficients of Kj are linear combinations of u'j/up u"ilup ..., over the field of functions rational near infinity and every solution of L{y) = 0 is also a solution of Kj(y) = 0 by definition of K. Thus the elimination process described in Ince [3, p. 126] leads to operators L] having coefficients of the same type such that K} = L;OL.

(17)

We thus have • • • « .

which is a homogeneous differential polynomial of degree n in F = L(f), = Q(L(f)).

(19)

We remark that the operators Lt depend on up but Q and Q are independent of the special base w1; ..., un. We have to show that the coefficients of Q are rational near infinity. To this end let argz = 6 be an arbitrary direction, let S be a corresponding sector and let yl, ...,yq be the distinguished fundamental set of L(y) = 0 (analogous to (c) in Section 4). We choose bases of Ms and M g+1 whose elements are monomials in yv ...,yQ and thus have a similar asymptotic behaviour. In particular, the second factor on the right-hand side of (18) and the logarithmic derivatives u'Ju^u'Ju^ {u'}/u}y + (u'j/uj)2, ... are functions of type R{z, logz) described in Section 4. This gives the desired result. The coefficients a,b,c,... of Q(L(f)) satisfy a growth condition a(z) = O(\zn ...

(20)

as z -> oo in |argz — 6\ < \d say, where m may depend on 6, and an easy compactness argument shows that (20) is true for some m without restriction of argz, and so the coefficients of Q (and, of course, of Q) are rational near infinity.

References 1. G. FRANK and G. WEISSENBORN, 'On the zeros of linear differential polynomials of meromorphic functions', to appear. 2. W. K.. HAYMAN, Meromorphic functions (Oxford University Press, 1964). 3. E. L. INCE, Ordinary differential equations (Dover, New York, 1956). 4. T. MUIR, A treatise on the theory of determinants (Dover, New York, 1960). 5. N. STEINMETZ, 'Eine Verallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes', J. Reine Angew. Math. 368(1986) 134-141.

ON THE ZEROS OF A CERTAIN WRONSKIAN

531

6. W. STERNBERG, 'Uber die asymptotische Integration von Differentialgleichungen', Math. Ann. 81 (1920) 119-186. 7. W. WASOW, Asymptotic expansions for ordinary differential equations (Wiley & Sons, 1965).

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