Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 107, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SUMS OF ZEROS OF SOLUTIONS TO SECOND ORDER ODE
WITH NON-POLYNOMIAL COEFFICIENTS
MICHAEL I. GIL’
Abstract. We consider the equation y 00 = F (z)y (z ∈ C) with an entire
function F satisfying the condition
` |z|ρ ´
(ρ ≥ 1, A = const > 0).
|F (z)| ≤ A exp
ρ
Let zk (y), k = 1, 2, . . . be the zeros of a solution y(z) to the above equation.
Bounds for the sums
j
X
1
(j = 1, 2, . . . )
|z
(y)|
k
k=1
are established. Some applications of these bounds are also considered.
1. Introduction and statement of the main result
In the present article, we consider linear differential equations with non-polynomial
coefficients in the complex domain. The literature devoted to the zeros of solutions
of such equations is very rich. Here the main tool is the Nevanlinna theory. An
excellent exposition of the Nevanlinna theory and its applications to differential
equations is given in the book [11]. In that book, in particular, the well-known
results of Bank, Brűggemann, Hellerstein, Rossi, Huang and other mathematicians
are featured. In connection with recent results see the very interesting papers [2][5], [8], [12]-[17]. In particular, in the paper [15], the authors study the convergence
of the zeros of a non-trivial (entire) solution to the linear differential equation
f 00 + Q1 (z)eP1 (z) + Q2 (z)eP2 (z) + Q3 (z)eP3 (z) f = 0
where Pj are polynomials of degree n ≥ 1 and Qj (Qj 6≡ 0) are entire functions
of order less than n (j = 1, 2, 3). The remarkable results on the zeros of a wide
class of ordinary differential equations with polynomial coefficients whose solutions
are classical orthogonal polynomials was established by Anghel [1]. Besides, he
had derived the important results connected with the equations of mathematical
physics.
Certainly we could not survey the whole subject here and we refer the reader to
the above listed publications and references given therein.
2000 Mathematics Subject Classification. 34C10, 34A30.
Key words and phrases. Complex differential equation; zeros of solutions.
c
2012
Texas State University - San Marcos.
Submitted September 8, 2011. Published June 25, 2012.
1
2
M. I. GIL’
EJDE-2012/107
In the above cited works mainly the asymptotic distributions of zeros and counting functions of zeros are investigated. At the same time, bounds for the zeros
of solutions are very important in various applications. But to the best of our
knowledge, they have been investigated considerably less than the asymptotic distributions. In the paper [10], bounds for the sums of the zeros of solutions are
established for the second order equations with polynomial coefficients. In this
paper, we obtain such bounds in the case of non-polynomial coefficients. Besides,
below we estimate the zero free domains. That estimation supplements the wellknown results of Eloe and Henderson [7] on the positivity of solutions for higher
order ordinary differential equations, since the positivity of solutions implies the
absence of zeros. Note that, the proof of the main result of the present paper is
considerably different from the proof of the paper [10].
Consider the equation
d2 y(z)
= F (z)y(z)
(1.1)
dz 2
with an entire function F , satisfying the condition
|z|ρ |F (z)| ≤ A exp
(ρ ≥ 1; A = const > 0; z ∈ C).
(1.2)
ρ
In Section 3 below we check that to inequality (1.2) can be reduced the formally
more general inequality
|F (z)| ≤ A exp[B|z|ρ ]
(B = const > 0)
(1.3)
by the substitution
w
(1.4)
(ρB)1/ρ
into (1.1). Everywhere below, y(z) is a solution of (1.1) with y(0) = 1. Enumerate
the zeros zk (y) of y(z), with multiplicities taken into account, in order of increasing
modulus: |zk (y)| ≤ |zk+1 (y)| (k = 1, 2, . . . ). Put
z=
vn = n
e1+2/ρ
.
ρ
Theorem 1.1. Let y(z) be a solution of (1.1) with y(0) = 1 and let condition (1.2)
hold. Then
j
j
X
X
1
1
√
< θ0 + ζ0
(j = 1, 2, . . . ),
ρ
|zk (y)|
ln vk
k=1
k=1
where θ0 and ζ0 are positive constants defined by
r
e
θ0 = 2
(1 + |y 0 (0)|) exp[A2 e2/ρ ] and
3
ζ0 = 2e1/2 (3/ρ)1/ρ .
The proof of this theorem is presented in the next section. Let us point some
corollaries of Theorem 1.1.
Denote by ν(y, r) (r > 0) the counting function of the zeros of y in |z| ≤ r.
Theorem 1.1 implies
Corollary 1.2. With the notation
ηj (y) :=
θ0 + ζ0
j
Pj
k=1
1
√
ρ
ln vk
(j = 1, 2, . . . ),
EJDE-2012/107
ZEROS OF SOLUTIONS TO SECOND ORDER ODE
3
the inequality |zj (y)| > ηj (y) holds and thus ν(u, r) ≤ j − 1 for any r ≤ ηj (y)
(j ≥ 2).
Furthermore, put
ϑ1 = θ0 +
ζ0
ln1/ρ v1
,
ϑk =
ζ0
ln1/ρ vk
(k = 2, 3, . . . ).
Theorem 1.1 and [9, Lemma 1.2.1] yield the following result.
Corollary 1.3. Under the hypothesis of Theorem 1.1, let φ(t) (0 ≤ t < ∞) be a
continuous convex scalar-valued function, such that φ(0) = 0. Then
j
X
φ(|zk (y)|−1 ) ≤
k=1
j
X
φ(ϑk )
(j = 1, 2, . . . ).
k=1
In particular, take
φ(t) = tρ+1 exp[−
ζ0ρ
].
tρ
Then
φ(ϑk ) =
ζ0ρ+1
1+1/ρ
ln
vk
exp[− ln vk ] ≤
const
(ln k)1+1/ρ k
(k > 1).
By the previous corollary we get the following result.
Corollary 1.4. Under the hypothesis of Theorem 1.1, we have
∞
X
k=1
ρ
1
e−(ζ0 |zk (y)|) < ∞.
ρ+1
|zk (y)|
In addition, in light of Theorem 1.1 and [9, Lemma 1.2.2] we obtain our next
result.
Corollary 1.5. Let Φ(t1 , t2 , . . . , tj ) be a function with an integer j defined on the
domain
0 < tj ≤ tj−1 · · · ≤ t2 ≤ t1 < ∞
and satisfying the condition
∂Φ
∂Φ
∂Φ
>
> ··· >
>0
∂t1
∂t2
∂tj
for t1 > t2 > · · · > tj > 0.
Then
Φ
1
1 ,...,
≤ Φ(ϑ1 , . . . , ϑj ).
|z1 (y)|
|zj (y)|
In particular, let {dk }∞
k=1 be a decreasing sequence of positive numbers. Then
the previous corollary yields the inequality
j
X
k=1
j
X
dk
≤
dk ϑk
|zk (y)|
k=1
(j = 1, 2, . . . ).
4
M. I. GIL’
EJDE-2012/107
2. Proof of Theorem 1.1
Consider the entire function
f (z) =
∞
X
ck z k
(c0 = 1)
(2.1)
k−0
satisfying the condition
|f (z)| ≤ (1 + qr) exp[A exp(rρ /ρ)]
(q = const > 0; z ∈ C, r = |z|).
(2.2)
Put
1
C = (1 + q) exp[ e2/ρ A2 ].
2
Lemma 2.1. Let condition (2.2) hold. Then the Taylor coefficients of f are subjected to the inequality
3 (n−1)/ρ
|cn | ≤ Cen/2
.
ρ ln vn
Proof. Let
Mf (r) = max |f (z)|.
|z|=r
By the well-known inequality for the coefficients of a power series,
Mf (r)
|cn | ≤
(r > 0).
rn
Take into account that
a2
ab ≤
+ b2 c (a, b, cpositive constants).
4c
Then for a constant µ > 0,
A2
+ µ exp(2rρ /ρ).
4µ
A exp(rρ /ρ) ≤
Due to (2.2),
ρ
A2
exp[µe2r /ρ ]
]h(r) where h(r) :=
4µ
rn
Let us use the usual method for finding extrema. Clearly,
|cn | ≤ (1 + qr) exp[
2r ρ /ρ
r2n h0 (r) = eµe
[2µe2r
ρ
/ρ n+ρ−1
r
(n = 1, 2, . . . ).
(2.3)
− nrn−1 ].
Thus the zero r0 = r0 (n) of h0 (r) is defined by
ρ
2µr0ρ e2r0 /ρ = n.
Take
µ=
Then
(2.4)
1 −2/ρ
e
.
2
ρ
r0ρ e2(r0 −1)/ρ = n.
So for n ≥ 1 we have r0 ≥ 1. Hence by (2.4),
ρ
µe2r0 /ρ ≤ n/2.
3r0ρ /ρ
(2.5)
(2.6)
Since x ≤ ex−1 (x > 0), by (2.5) we have e
≥ ne1+2/ρ /ρ = vn , and therefore,
ρ
r0 ≥ ( ln vn )1/ρ .
(2.7)
3
EJDE-2012/107
ZEROS OF SOLUTIONS TO SECOND ORDER ODE
Since r0 ≥ 1, we obtain
1+qr0
r0
5
≤ 1 + q. Now (2.3) and (2.6) imply
|cn | ≤ exp[
en/2
A2
](1 + q) n−1 .
4µ
r0
Hence, (2.7) proves the lemma.
√
Put D = eC, and
τn = 2e1/2
3
ρ ln vn
1/ρ
.
Then according to Lemma 2.1,
τnn−1
.
2n−1
Denote ψ1 = 1, ψn = τnn−1 (n > 1); an = cn /ψn , and mn+1 = ψn+1 /ψn . As it is
proved in [9, Theorem 5.1.1], the inequality
|cn | ≤ D
j
X
k=1
j
X
1
≤ θ(f ) +
mk+1
|zk (f )|
k=1
is valid, where zk (f ) are the zeros of f (z), with multiplicities taken into account,
enumerated in order of increasing modulus, and
∞
hX
i1/2
|ak |2
.
θ(f ) :=
k=1
n−1
But |a1 | = |c1 | ≤ D; |an | ≤ D/2
, n ≥ 2. Moreover, since τn+1 ≤ τn , we get
mn+1 ≤ τn (n = 1, 2, . . . ), and θ(f ) ≤ θ̃0 , where
θ̃02 = D2
∞
X
1
= 4D2 /3.
4k
k=0
We thus have proved the following result.
Lemma 2.2. Let an entire function f satisfy condition (2.2). Then
j
X
k=1
j
j
k=1
k=1
X
X
1
1
≤ θ̃0 +
τn = θ̃0 + ζ0
1/ρ
|zk (f )|
ln vk
(j = 1, 2, . . . ).
Lemma 2.3. A solution y of (1.1) with the conditions (1.2) and y(0) = 1 is an
entire function satisfying the inequality
|y(z)| ≤ (1 + |y 0 (0)|r) exp[Aer
ρ
/ρ
]
(z ∈ C).
Proof. From (1.1) for a z = reit with a fixed argument t we have e−2it d2 y(z)/dr2 =
F (z)y(z). Hence, putting q = |y 0 (0)|, g(r) = |y(reit )|, and taking into account (1.2)
we obtain
Z r
Z r
g(r) ≤ 1 + qr +
(r − s)|F (seit )|g(s)ds ≤ 1 + qr + A
(r − s) exp[sρ /ρ]g(s)ds.
0
0
By [6, Lemma III.2.1] we have g(r) ≤ m(r), where m(r) is a solution of the equation
Z r
m(r) = 1 + qr + A
(r − s) exp[sρ /ρ]m(s)ds.
0
6
M. I. GIL’
EJDE-2012/107
However,
Z
r
r
Z
s
Z
(r − s)f (s)ds =
f (τ )dτ ds
0
0
0
for any integrable function f . Thus,
Z
r
s
Z
exp[τ ρ /ρ]m(τ )dτ ds.
m(r) = 1 + qr + A
0
0
Clearly the derivative of m is positive. So
Z r
Z
m(r) ≤ 1 + qr + A
m(τ )
0
τ
exp[sρ /ρ]ds dτ.
0
But for r ≤ 1,
r
Z
exp[sρ /ρ]ds ≤ exp[rρ /ρ],
0
and for an r ≥ 1,
Z
Z r
sρ /ρ
e
ds ≤
0
1
sρ /ρ
e
r
Z
sρ−1 es
ds +
0
ρ
/ρ
ds ≤ e1/ρ + (er
ρ
/ρ
− e1/ρ ) = er
ρ
/ρ
.
1
Thus,
r
Z
exp[sρ /ρ]ds ≤ exp[rρ /ρ].
(2.8)
Rr
Consequently, m(r) ≤ 1 + qr + A 0 m(s) exp[sρ /ρ]ds. By the Gronwall inequality,
Z r
ρ
m(r) ≤ (1 + qr) exp A
es /ρ ds .
0
0
Now (2.8) implies the required result.
Then the assertion of Theorem 1.1 follows from Lemmas 2.2 and 2.3.
3. Example
In this section we consider an example that illustrates Theorem 1.1. First substitute (1.4) into (1.1). Then we arrive at the equation
d2 x(w)
= F1 (w),
dw2
where F1 (w) =
1
w F
2/ρ
(ρB)
(ρB)1/ρ
and x(w) = y(w/(ρB)1/ρ ). If condition (1.3) holds, then
|F1 (w)| ≤ A1 exp
|w|ρ ,
ρ
where A1 = A/(ρB)2/ρ . By Theorem 1.1,
j
X
k=1
j
X
1
1
< θ1 + ζ0
1/ρ
|zk (x)|
ln vk
(j = 1, 2, . . . ),
(3.1)
k=1
where
r r e
dx(0) e
1
dy(0) 2 2/ρ
1+|
| exp[A1 e ] = 2
1+
|
| exp[A21 e2/ρ ].
θ1 = 2
3
dw
3
(Bρ)1/ρ dz
EJDE-2012/107
ZEROS OF SOLUTIONS TO SECOND ORDER ODE
7
But zk (y) = zk (x)(ρB)1/ρ . Now (3.1) implies
j
X
k=1
j
h
X
1 i
1
< (ρB)1/ρ θ1 + ζ0
|zk (y)|
ln1/ρ vk
(j = 1, 2, . . . ).
(3.2)
k=1
So the following result is holds.
Corollary 3.1. Let y(z) be a solution of (1.1) with y(0) = 1 and condition (1.3)
hold. Then inequality (3.2) is valid.
Furthermore, it can be directly checked that the function
y(z) = ce−z/2 sin(ez )
(3.3)
with c = 1/ sin(1) is a solution of the equation
1
y 00 (z) = ( − e2z )y(z)
(3.4)
4
Besides, y(0) = 1. Clearly, the zeros of y are ln πk (k = 0, ±1, ±2, . . . ). Hence, for
a sufficiently large j we have
2j
X
k=1
j
j
k=1
k=1
X
1
1 X 1
1
1
=
.
=
+
+
|zk (y)|
|z2k−1 (y)| |z2k (y)|
ln πk | ln(−πk)|
(3.5)
On the other hand, due to (3.4),F (z) = 14 − e2z and therefore, |F (z)| ≤ (1 + 41 )e2|z| .
By (3.2) with B = 2, A = 1 + 1/4, we have
j
X
k=1
j
X
1
1
< 2θ1 + 2ζ0
|zk (y)|
ln(ke3 )
(j = 1, 2, . . . ).
k=1
This result is rather close to (3.5).
Note that, if F (z) is of infinite order, then the problem considered in this paper
is much more complicated.
References
[1] N. Anghel; Stieltjes-Calogero-Gil relations associated to entire functions of finite order, Journal of Mathematical Physics, 51, no. 5 (2010) 251-262.
[2] B. Belaii, A. E. Farissi; Differential polynomials generated by some complex linear differential
equations with meromorphic coefficients, Glasnik Matematicki, 43, no. 2 (2008) 363-373.
[3] T. B. Cao, L. M. Li; Oscillation results on meromorphic solutions of second order differential
equations in the complex plane, E. J. Qualitative Theory of Diff. Eqs., no. 68 (2010), 1-13.
[4] T. B. Cao, J. F. Xu and Z. X. Chen; On the meromorphic solutions of linear differential
equations on the complex plane, J. Math. Anal. Appl., 364, no. 1, (2010), 130-142.
[5] W. J. Chen, J. F. Xu; Growth of meromorphic solutions of higher-order linear differential
equations, E. J. Qualitative Theory of Diff. Equ. 2009 (2009), no. 1, 1-13.
[6] Yu L. Daleckii, M.G. Krein; Stability of Solutions of Differential Equations in Banach Space,
Amer. Math. Soc., Providence, R. I. 1971.
[7] P. W. Eloe, J. Henderson; Positive solutions for higher order ordinary differential equations.
Electron. J. Differ. Equ., 1995 (1995), no. 03, 1-8.
[8] A. E. Farissi, B. Belaii; On oscillation theorems for differential polynomials, E. J. Qualitative
Theory of Diff. Equ., 2009 (2009), no. 22, 1-10.
[9] M. I. Gil’; Localization and Perturbation of Zeros of Entire Functions, CRC Press, Taylor
and Francis Group, New York, 2009.
[10] M. I. Gil’, Bounds for zeros of solutions of second order differential equations with polynomial
coefficients, Results Math., 59, (2011), 115-124.
[11] I. Laine; Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter Berlin,
1993
8
M. I. GIL’
EJDE-2012/107
[12] C. H. Lin, Y. Sibuya, T. Tabara; Zeros of solutions of a second order linear differential
equation with polynomial coefficients. Funkcial. Ekvac. 36 (1993), no. 2, 375-384.
[13] G. M. Muminov; On the zeros of solutions of the differential equation ω (2m) + p(z)ω = 0.
Demonstr. Math., 35 (2002), no. 1, 41-48.
[14] F. Peng, Z. X. Chen; On the growth of solutions of some second-order differential equations,
J. Ineq. Appl., 2011 (2011), 1-9.
[15] J. Tu, Z. X. Chen; Zeros of solutions of certain second order linear differential equation, J.
Math. Anal. Appl. 332 (2007), no. 1, 279-291.
[16] J. Wang, I. Laine; Growth of solutions of second order linear differential equations, J. Math.
Anal. Appl. 342 (2008), 39-51.
[17] J. F. Xu, H. X. Yi; Solutions of higher order linear differential equations in an angle, Appl.
Math. Letters, 22 (2009), no. 4, 484-489.
Michael I. Gil’
Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, BeerSheva 84105, Israel
E-mail address: gilmi@bezeqint.net