Volume 2010, Article ID 428936,8pages doi:10.1155/2010/428936
Research Article
On the Exponent of Convergence for the Zeros of
the Solutions of
y
Ay
By
0
Abdullah Alotaibi
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Abdullah Alotaibi,[email protected]
Received 1 July 2010; Accepted 12 September 2010
Academic Editor: P. J. Y. Wong
Copyrightq2010 Abdullah Alotaibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetBandCbe entire functions of order less than 1 withC /≡0 andBtranscendental. We prove that every solutionf /≡0 of the equationyAyBy0,Az Czeαz,α∈C\ {0}being has zeros
with infinite exponent of convergence.
1. Introduction
It is assumed that the reader of this paper is familiar with the basic concepts of Nevanlinna theory1,2such asTr, f, mr, f, Nr, f, andSr, f. Suppose thatfis a meromorphic function, then the order of growth of the functionfand the exponent of convergence of the zeros offare defined, respectively, as
ρflim sup
r→ ∞
logTr, f
logr , λ
flim sup
r→ ∞
logNr,1/f
logr . 1.1
Let E be a measurable subset of 1,∞. The lower logarithmic density and the upper logarithmic density ofEare defined, respectively, by
log densE lim inf
r→ ∞
r
1
χtdt/t
logr , log densE lim supr→ ∞
r
1
χtdt/t
whereχtis the characteristic function ofEdefined as
χt ⎧ ⎨ ⎩
1, ift∈E,
0, ift /∈E. 1.3
Now let us recall some of the previous results on the linear differential equation
ye−zyBzy0, 1.4
whereBzis an entire function of finite order, WhenBzis polynomial, many authors3–6
have studied the properties of the solutions of1.4. IfBzis a transcendental entire function withρB/1, Gundersen7proved that every nontrivial solution of1.4has infinite order of growth. In8, Wang and Laine considered the nonhomogeneous equation of type
yA1zeazyA0zebzyHz, 1.5
where A0z, A1z, Hz are entire functions of order less than one anda, b are complex
numbers. In fact, they have proved the following theorem.
Theorem 1.1. Suppose that A0/≡0, A1/≡0, H are entire functions of order less than one, and suppose thata, b ∈ C withab /0and a /b. Then every nontrivial solution of 1.5 is of infinite order.
Corollary 1.2. Suppose thatBz hzebz, wherehis a nonvanishing entire function withρh<
1andb∈Cwithb /0,−1. Then every nontrivial solution of 1.4is of infinite order.
2. Results
We observe that all the above results concern the order of growth only. In this paper, we are going to prove the following theorem which concerns the exponent of convergence.
Theorem 2.1. Let B and C be entire functions of order less than 1 with C /≡0 and B being transcendental. Then every solutionf /≡0of the equation
yAyBy0,
Az Czeαz, α∈C\ {0}, 2.1
has zeros with infinite exponent of convergence.
The hypothesis thatBis transcendental is not redundant since Frei4has shown that
ye−zyKy0 2.2
We notice thatTheorem 2.1fails forρB ≥ 1. For any entire functionDthe function
feDsolves2.1with
−B f
f A
f
f D
D2
AD. 2.3
3. Some Lemmas
Throughout this paper we need the following lemmas. In 1965, Hayman 9 proved the following lemma.
Lemma 3.1. Let the functiongbe meromorphic of finite orderρin the plane and let0< δ <1. Then
T2r, g≤Cρ, δTr, g 3.1
for allroutside a setEof upper logarithmic densityδ, where the positive constantCρ, δdepends only onρandδ.
In 1962, Edrei and Fuchs10proved the following lemma.
Lemma 3.2. Letg be a meromorphic function in the complex plane and letI Ir⊆0,2πhave measureμμr. Then
1 2π
I
log greiθ dθ≤22μ
1log1
μ
T2r, g. 3.2
In 2007, Bergweiler and Langley11proved the following lemma.
Lemma 3.3. LetHbe a transcendental entire function of orderρ <∞. For larger >0defineθr
to be the length of the longest arc of the circle|z| r on which|Hz| > 1, withθr 2π if the minimum modulusm0r, H min{|Hz|:|z|r}satisfiesm0r, H>1. Then at least one of the following is true:
ithere exists a setF ⊆1,∞of positive upper logarithmic density such thatm0r, H>1 forr∈F;
iifor eachτ ∈0,1the setFr {r : θr > 2π1−τ}has lower logarithmic density at
least1−2ρ1−τ/τ.
We deduce the following.
Lemma 3.4. Let 0 < < π/4, letN be a positive integer, and letG ⊆ 1,∞ have logarithmic density1. LetF be a transcendental entire function such that|Fz| ≤ |z|N on a pathγ tending to
infinity and for allzwith|z| ∈Gand|argz| ≤π/2−. ThenFhas order at leastπ/π2.
Proof. Assume thatρF ρ <∞and choose a polynomialP of degree at mostN−1 such that
Hz Fz−Pz
is transcendental entire. Then we have|Hz| ≤1 for allz∈γand for allzwith|z| ∈Gand |argz| ≤π/2−. With the notation ofLemma 3.3, we see thatm0r, H≤1 for all larger, and
so we must have caseii, as well asθr≤π2forr∈G. Defineτby
2π1−τ π2. 3.4
SinceGhas logarithmic density 1 this gives
2ρ1−τ≥1, ρ≥ 1
21−τ π
π2. 3.5
4. Proof of
Theorem 2.1
LetA,BandCbe as in the hypotheses. We can assume thatα1. Suppose thatfis a solution of2.1having zeros with finite exponent of convergence. Then we can write
f Πeh, 4.1
whereΠandhare entire functions withρΠ < ∞. We can assume thath/≡0, since if his constant we can replacehzbyhz zandΠzbyΠze−z. Using2.1and4.1, we get
Π
Π 2 Π
Πhhh
2
A
Π
Π h
B0. 4.2
Lemma 4.1. One hasρh≤1.
Proof. Suppose that|hz| ≥1. Dividing4.2byh, we get
hz ≤ Πz
Πz
2 Πz
Πz
hz
hz
|Az| Πz
Πz 1
|Bz|. 4.3
Hence, providedrlies outside a set of finite measure,
Tr, hmr, h≤OlogrTr, A Tr, B oTr, h, 4.4
and so, using the fact thatBandChave order less than 1, we obtain
Tr, hOr. 4.5
withr≤s≤2rso that
Tr, h≤Ts, hOs Or. 4.6
Lemma 4.1is proved.
LetM1, M2, . . .denote large positive constants. Chooseσwith
maxρB, ρC< σ <1. 4.7
There exists anR-setU2, page 84such that for all largezoutsideU, we have
ΠΠzz Πz Πz
hz
hz
≤ |z|M1, 4.8
and using the Poisson-Jensen formula,
log |Cz| ≤ |z|σ. 4.9
Moreover, there exists a setG ⊆1,∞of logarithmic density 1 such that forr ∈Gthe circle |z|rdoes not meet theR-setU.
Lemma 4.2. The functionshandhAare both transcendental.
Proof. Letbe small and positive and suppose thathorhAis a polynomial. Letzbe large withz /∈Uand|argz−π| ≤π/2−. SinceAzis small it follows from4.2and4.8that
Bz O|z|M2. Chooseθwith|θ−π|< such that the intersection ofUwith the rayLgiven by argz θis bounded. ApplyingLemma 3.4to the functionB−z, withγa subpath ofL, givesρB≥π/π2, butmay be chosen arbitrarily small, and this contradicts4.7.
The next step is to estimatehin the right half-plane.
Lemma 4.3. LetNbe a large positive integer and let0< <1/2. Then for largezwith
argz ≤ π
2 −, z /∈U 4.10
one has, either
hz ≤ |z|N 4.11
or
Proof. Letz be large and satisfy 4.10, and assume that 4.11 does not hold. Then4.8
implies that
ΠΠzzhz ≥1. 4.13
Also,4.7, and4.9give
log|Bz| ≤ |z|σ, log|Az| ≥Rez− |z|σ≥ |z|
2 cos
π
2 −
c1|z|. 4.14
Herec1, c2, . . . denote positive constants which may depend onbut not onz. Using4.8,
4.12and4.14we get, from4.2,
log hz ≥c2|z|. 4.15
Now divide4.2byhz. We obtain, using4.15,
hz Az
⎛ ⎜
⎝1
O|z|M1
hz ⎞ ⎟
⎠O|z|M10 4.16
which gives|hz| ∼ |Az|and4.12. This provesLemma 4.3.
Lemma 4.4. LetNandbe as inLemma 4.3. Chooseθ0∈−π/4, π/4such that the rayargzθ0 has bounded intersection with theR-setU. Let V be the union of the rayargz θ0 and the arcs
|z|r, r ∈G, |argz| ≤π/2−, whereG ⊆1,∞is the set chosen following4.9. Then one of the following holds:
ione has4.11for all largezinV;
iione has4.12for all largezinV.
Proof. This follows simply from continuity. For each largezinVwe have4.11or4.12, but we cannot have both because of4.14. This provesLemma 4.4.
Lemma 4.5. Let0< <1/2. Then for largez /∈Uwith|argz−π| ≤π/2−, one has
log hz O|z|σ, log hz Az O|z|σ. 4.17
Proof. Letzbe as in the hypotheses. SinceAz o1we only need to prove4.17for|hz|. Assume that|hz| ≥1. Then dividing4.2byhgives
hz ≤ |Bz|O|z|M1 4.18
Lemma 4.6. If conclusion (i) ofLemma 4.4holds thenρh<1, while if conclusion (ii) ofLemma 4.4
holds thenρhA<1.
Proof. Suppose that conclusioniofLemma 4.4holds. Chooseδ1>0 such that
σ1δ1<1 4.19
and letδ >0 be small compared toδ1. Assume thatinLemma 4.4is small compared toδ,
in particular so small that
88
1log 1 4
Cρh, δ≤ 1
2, 4.20
whereCρh, δis the positive constant fromLemma 3.1. Let
Ir π
2 −,
π
2
∪
3π
2 −, 3π
2
, 4.21
and letEbe the exceptional set ofLemma 3.1, withg h. Then for larger ∈G\Ewe have, using4.20and Lemmas3.1,3.2, and4.5,
Tr, hmr, h≤Orσ Ologr 1
2π
Irlog
hreiθ dθ
≤Orσ 88
1log 1 4
T2r, h
≤Orσ 88
1log 1 4
Cρh, δTr, h
≤Orσ 1
2T
r, h.
4.22
We then have
Tr, hOrσ 4.23
for larger ∈G\E. Now take any larger. SinceGhas logarithmic density 1, whileEhas upper logarithmic density at mostδ, and sinceδ/δ1is small, there existsswith
r≤s≤r1δ1, s∈G\E, Tr, h≤Ts, hOsσ O
rσ1δ1
, 4.24
To finish the proof suppose first that conclusion ii of Lemma 4.4 holds. Then
Lemma 3.4implies thathhas order at least π/π2. Since may be chosen arbitrarily small, this contradictsLemma 4.6. The same contradiction, withhreplaced byhA, arises if conclusioniofLemma 4.4holds, and the proof of the theorem is complete.
Acknowledgment
The author thanks Professor J. K. Langley for the invaluable discussions on the results of this paper during his visit in summer 2008 and summer 2010 to the University of Nottingham in the U.K.
References
1 W. K. Hayman,Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.
2 I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, W. de Gruyter, Berlin, Germany, 1993.
3 I. Amemiya and M. Ozawa, “Nonexistence of finite order solutions ofw e−zwQzw 0,”
Hokkaido Mathematical Journal, vol. 10, pp. 1–17, 1981.
4 M. Frei, “ ¨Uber die subnormalen L ¨osungen der Differentialgleichungw e−zwKonst.w 0,”
Commentarii Mathematici Helvetici, vol. 36, pp. 1–8, 1961.
5 J. K. Langley, “On complex oscillation and a problem of Ozawa,”Kodai Mathematical Journal, vol. 9, no. 3, pp. 430–439, 1986.
6 M. Ozawa, “On a solution ofwe−z azbw0,”Kodai Mathematical Journal, vol. 3, no. 2, pp.
295–309, 1980.
7 G. G. Gundersen, “On the question of whetherfe−zfBzf 0 can admit a solutionf /≡0 of
finite order,”Proceedings of the Royal Society of Edinburgh. Section A, vol. 102, no. 1-2, pp. 9–17, 1986.
8 J. Wang and I. Laine, “Growth of solutions of second order linear differential equations,”Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 39–51, 2008.
9 W. K. Hayman, “On the characteristic of functions meromorphic in the plane and of their integrals,”
Proceedings of the London Mathematical Society. Third Series, vol. 3, no. 14, pp. 93–128, 1965.
10 A. Edrei and W. H. J. Fuchs, “Bounds for the number of deficient values of certain classes of meromorphic functions,”Proceedings of the London Mathematical Society. Third Series, vol. 12, pp. 315– 344, 1962.
