Growth estimates of solutions of linear differential equations with
dominant coefficient of lower (α, β, γ)-order
BENHARRAT BELAÏDI
Department of Mathematics, Laboratory of Pure and Applied Mathematics
University of Mostaganem (UMAB)
B. P. 227 Mostaganem
ALGERIA
Abstract: - In this paper, we deal with the growth and oscillation of solutions of higher order linear differential
equations. Under the conditions that there exists a coefficient which dominates the other coefficients by its lower
(α, β, γ)-order and lower (α, β, γ)-type, we obtain some growth and oscillation properties of solutions of such
equations which improve and extend some recently results of the author and Biswas [4].
Key-Words: - Differential equations, (α, β, γ)-order, lower (α, β, γ)-order, (α, β, γ)-type, lower (α, β, γ)-type,
growth of solutions.
Received: March 13, 2024. Revised: September 15, 2024. Accepted: October 14, 2024. Published: November 22, 2024.
1 Introduction
Throughout this paper, we assume that the reader is
familiar with the fundamental results and the standard
notations of the Nevanlinna value distribution theory
of meromorphic functions [11, 15, 45].
Nevanlinna theory has appeared to be a powerful
tool in the field of complex differential equations. For
an introduction to the theory of differential equations
in the complex plane by using the Nevanlinna theory
see [25]. Active research in this field was started by
Wittich [42, 43] and his students in the 1950’s and
1960’s. After their many authors have investigated
the complex differential equations
f(k)(z) + Ak−1(z)f(k−1)(z) + · · · +A1(z)f′(z)
+A0(z)f(z) = 0,
(1)
f(k)(z) + Ak−1(z)f(k−1)(z) + · · · +A1(z)f′(z)
+A0(z)f(z) = F,
(2)
and achieved many valuable results when the coeffi-
cients A0(z), ..., Ak−1(z),(k≥2) and F(z)in (1)
and (2) are entire or meromorphic functions of finite
order or finite iterated p-order or (p, q)-th order or
(p, q)-φorder; see ([5], [8], [14], [18], [23], [25],
[27], [28], [29], [37], [39], [40], [44]).
Chyzhykov and Semochko [9] showed that both
definitions of iterated p-order ([20], [23], [34], [35])
and the (p, q)-th order ([21], [22]) have the disadvan-
tage that they do not cover arbitrary growth (see [9,
Example 1.4]). They used more general scale, called
the φ-order (see [9], [36]) and the concept of φ-order
is used to study the growth of solutions of complex
differential equations in the whole complex plane and
in the unit disc which extend and improve many pre-
vious results see ([1, 9, 36]). Extending this notion,
Long et al. [30] recently introduce the concepts of
[p, q],φ-order and [p, q],φ-type (see [30]) and obtain
some interesting results which considerably extend
and improve some earlier results. For details one may
see [30].
The concept of generalized order (α, β)of an en-
tire function was introduced by Sheremeta [38]. Sev-
eral authors made close investigations on the prop-
erties of entire functions related to generalized order
(α, β)in some different direction [6, 7]. On the other
hand, Mulyava et al. [31] have used the concept of
(α, β)-order of an entire function in order to inves-
tigate the properties of solutions of a heterogeneous
differential equation of the second order and obtained
several remarkable results. For details about (α, β)-
order one may see [31, 38].
Now, let Lbe a class of continuous non-negative
on (−∞,+∞)function αsuch that α(x)=α(x0)≥
0for x≤x0and α(x)↑+∞as x0≤x→+∞. We
say that α∈L1, if α∈Land α(a+b)≤α(a) +
α(b) + cfor all a, b ≥R0and fixed c∈(0,+∞).
Further, we say that α∈L2, if α∈Land α(x+
O(1)) = (1 + o(1))α(x)as x→+∞. Finally, α∈
L3, if α∈Land α(a+b)≤α(a) + α(b)for all
a, b ≥R0,i.e., α is subadditive. Clearly L3⊂L1.
Particularly, when α∈L3, then one can easily
verify that α(mr)≤mα(r), m ≥2is an integer. Up
to a normalization, subadditivity is implied by con-
cavity. Indeed, if α(r)is concave on [0,+∞)and
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satisfies α(0) ≥0, then for t∈[0,1],
α(tx) = α(tx + (1 −t)·0) ≥tα(x) + (1 −t)α(0)
≥tα(x),
so that by choosing t=a
a+bor t=b
a+b,
α(a+b) = a
a+bα(a+b) + b
a+bα(a+b)
≤αa
a+b(a+b)
+αb
a+b(a+b)
=α(a) + α(b),a, b ≥0.
As a non-decreasing, subadditive and unbounded
function, α(r)satisfies
α(r)≤α(r+R0)≤α(r) + α(R0)
for any R0≥0. This yields that α(r)∼α(r+R0)
as r→+∞.
Let α, β and γsatisfy the following two condi-
tions : (i) Always α∈L1, β ∈L2and γ∈L3; and
(ii) α(log[p]x) = o(β(log γ(x))), p ≥2, α(log x) =
o(α(x)) and α−1(kx) = oα−1(x)(0 < k < 1) as
x→+∞.
Throughout this paper, we assume that α, β and γ
always satisfy the above two conditions unless other-
wise specifically stated.
Recently, Heittokangas et al. [19] have introduced
a new concept of φ-order of entire and meromorphic
functions considering φas subadditive function. For
details one may see [19]. Extending this notion, re-
cently the author and Biswas [2] introduce the defini-
tion of the (α, β, γ)-order of a meromorphic function.
The main aim of this paper is to study the growth
and oscillation of solutions of higher order linear
differential equations using the concepts of lower
(α, β, γ)-order and lower (α, β, γ)-type. In fact,
some works relating to study the growth of solutions
of higher order linear differential equations using the
concepts of (α, β, γ)-order have been explored in [2],
[3] and [4]. In this paper, we obtain some results
which improve and generalize some previous results
of the author and Biswas [4].
For x∈[0,+∞)and k∈Nwhere Nis the
set of all positive integers, define iterations of the
exponential and logarithmic functions as exp[k]x=
exp(exp[k−1] x)and log[k]x=log(log[k−1] x)with
convention that log[0] x=x, log[−1] x=exp x,
exp[0] x=xand exp[−1] x=log x.
Definition 1.1. ([2]) The (α, β, γ)-order denoted by
ρ(α,β,γ)[f]of a meromorphic function fis defined by
ρ(α,β,γ)[f] = lim sup
r→+∞
α(log T(r, f))
β(log γ(r)) ,
and for an entire function f, we define
ρ(α,β,γ)[f] = lim sup
r→+∞
α(log T(r, f))
β(log γ(r))
=lim sup
r→+∞
α(log[2] M(r, f))
β(log γ(r)) .
Similar to Definition 1.1, one can also define the
lower (α, β, γ)-order of a meromorphic function fin
the following way:
Definition 1.2. The lower (α, β, γ)-order denoted by
µ(α,β,γ)[f]of a meromorphic function fis defined by
µ(α,β,γ)[f] = lim inf
r→+∞
α(log T(r, f))
β(log γ(r)) ,
for an entire function f, one can easily by Theorem
7.1 in [11] verify that
µ(α,β,γ)[f] = lim inf
r→+∞
α(log T(r, f))
β(log γ(r))
=lim inf
r→+∞
α(log[2] M(r, f))
β(log γ(r)) .
Proposition 1.3. ([2]) If fis an entire function, then
ρ(α(log),β,γ)[f] = lim sup
r→+∞
α(log[2] T(r, f))
β(log γ(r))
=lim sup
r→+∞
α(log[3] M(r, f))
β(log γ(r)) ,
and also by Theorem 7.1 in [11], one can easily verify
that
µ(α(log),β,γ)[f] = lim inf
r→+∞
α(log[2] T(r, f))
β(log γ(r))
=lim inf
r→+∞
α(log[3] M(r, f))
β(log γ(r)) ,
where (α(log), β, γ)-order denoted by ρ(α(log),β,γ)[f]
and lower (α(log), β, γ)-order denoted by
µ(α(log),β,γ)[f].
Now to compare the relative growth of two
meromorphic functions having same non zero finite
(α, β, γ)-order or non zero finite lower (α, β, γ)-
order, one may introduce the definitions of (α, β, γ)-
type and lower (α, β, γ)-type in the following man-
ner:
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Definition 1.4. ([4]) The (α, β, γ)-type denoted by
τ(α,β,γ)[f]of a meromorphic function fwith 0<
ρ(α,β,γ)[f]<+∞is defined by
τ(α,β,γ)[f] = lim sup
r→+∞
exp(α(log T(r, f)))
(exp (β(log γ(r))))ρ(α,β,γ)[f].
If fis an entire function with ρ(α,β,γ)[f]∈(0,+∞),
then the (α, β, γ)-type of fis defined by
τ(α,β,γ),M [f] = lim sup
r→+∞
exp(α(log[2] M(r, f)))
(exp (β(log γ(r))))ρ(α,β,γ)[f].
Definition 1.5. The lower (α, β, γ)-type denoted by
τ(α,β,γ)[f]of a meromorphic function fwith 0<
µ(α,β,γ)[f]<+∞is defined by
τ(α,β,γ)[f] = lim inf
r→+∞
exp(α(log T(r, f)))
(exp (β(log γ(r))))µ(α,β,γ)[f].
If fis an entire function with µ(α,β,γ)[f]∈(0,+∞),
then the lower (α, β, γ)-type of fis defined by
τ(α,β,γ),M [f] = lim inf
r→+∞
exp(α(log[2] M(r, f)))
(exp (β(log γ(r))))µ(α,β,γ)[f].
In order to study the oscillation properties of solu-
tions of (1) and (2), we define the (α, β, γ)-exponent
convergence of the zero-sequence of a meromorphic
function fin the following way:
Definition 1.6. ([2]) The (α, β, γ)-exponent conver-
gence of the zero-sequence denoted by λ(α,β,γ)[f]of
a meromorphic function fis defined by
λ(α,β,γ)[f] = lim sup
r→+∞
α(log n(r, 1/f))
β(log γ(r))
=lim sup
r→+∞
α(log N(r, 1/f))
β(log γ(r)) .
Analogously, the (α, β, γ)-exponent convergence of
the distinct zero-sequence denoted by λ(α,β,γ)[f]of f
is defined by
λ(α,β,γ)[f] = lim sup
r→+∞
α(log n(r, 1/f))
β(log γ(r))
=lim sup
r→+∞
α(log N(r, 1/f))
β(log γ(r)) .
Accordingly, the values
λ(α(log),β,γ)[f] = lim sup
r→+∞
α(log[2] n(r, 1/f))
β(log γ(r))
=lim sup
r→+∞
α(log[2] N(r, 1/f))
β(log γ(r))
and
λ(α(log),β,γ)[f] = lim sup
r→+∞
α(log[2] n(r, 1/f))
β(log γ(r))
=lim sup
r→+∞
α(log[2] N(r, 1/f))
β(log γ(r))
are respectively called as (α(log), β, γ)-exponent
convergence of the zero-sequence and (α(log), β, γ)-
exponent convergence of the distinct zero-sequence
of a meromorphic function f.
Similar to Definition 1.6, one can also define the
lower (α, β, γ)-exponent convergence of the zero-
sequence of a meromorphic function fin the follow-
ing way:
Definition 1.7. The lower (α, β, γ)-exponent conver-
gence of the zero-sequence denoted by λ(α,β,γ)[f]of
a meromorphic function fis defined by
λ(α,β,γ)[f] = lim inf
r→+∞
α(log n(r, 1/f))
β(log γ(r))
=lim inf
r→+∞
α(log N(r, 1/f))
β(log γ(r)) .
Analogously, the lower (α, β, γ)-exponent conver-
gence of the distinct zero-sequence denoted by
λ(α,β,γ)[f]of fis defined by
λ(α,β,γ)[f] = lim inf
r→+∞
α(log n(r, 1/f))
β(log γ(r))
=lim inf
r→+∞
α(log N(r, 1/f))
β(log γ(r)) .
Accordingly, the values
λ(α(log),β,γ)[f] = lim inf
r→+∞
α(log[2] n(r, 1/f))
β(log γ(r))
=lim inf
r→+∞
α(log[2] N(r, 1/f))
β(log γ(r))
and
λ(α(log),β,γ)[f] = lim inf
r→+∞
α(log[2] n(r, 1/f))
β(log γ(r))
=lim inf
r→+∞
α(log[2] N(r, 1/f))
β(log γ(r))
are respectively called as lower (α(log), β, γ)-
exponent convergence of the zero-sequence and
lower (α(log), β, γ)-exponent convergence of the
distinct zero-sequence of a meromorphic function f.
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Proposition 1.8. ([2]) Let f1(z), f2(z)be noncon-
stant meromorphic functions with ρ(α(log),β,γ)[f1]and
ρ(α(log),β,γ)[f2]as their (α(log), β, γ)-order. Then
(i) ρ(α(log),β,γ)[f1±f2]≤max{ρ(α(log),β,γ)[f1],
ρ(α(log),β,γ)[f2]};
(ii) ρ(α(log),β,γ)[f2·f2]≤max{ρ(α(log),β,γ)[f1],
ρ(α(log),β,γ)[f2]};
(iii) If ρ(α(log),β,γ)[f1]=ρ(α(log),β,γ)[f2], then
ρ(α(log),β,γ)[f1±f2]
=max{ρ(α(log),β,γ)[f1], ρ(α(log),β,γ)[f2]};
(iv) If ρ(α(log),β,γ)[f1]=ρ(α(log),β,γ)[f2], then
ρ(α(log),β,γ)[f2·f2]
=max{ρ(α(log),β,γ)[f1], ρ(α(log),β,γ)[f2]}.
By using the properties T(r, f) = T(r, 1
f) + O(1)
and T(r, af) = T(r, f) + O(1),a∈C\ {0}, one can
obtain the following result.
Proposition 1.9. ([4]) Let fbe a non-constant mero-
morphic function. Then
(i) ρ(α,β,γ)[1
f] = ρ(α,β,γ)[f] (f≡ 0) ;
(ii) ρ(α(log),β,γ)[1
f] = ρ(α(log),β,γ)[f] (f≡ 0) ;
(iii) If a∈C\{0}, then ρ(α,β,γ)[af ] = ρ(α,β,γ)[f]and
τ(α,β,γ)[af ] = τ(α,β,γ)[f]if 0< ρ(α,β,γ)[f]<+∞;
(iii) If a∈C\ {0}, then ρ(α(log),β,γ)[af ] =
ρ(α(log),β,γ)[f]and τ(α(log),β,γ)[af ] = τ(α(log),β,γ)[f]
if 0< ρ(α(log),β,γ)[f]<+∞.
Proposition 1.10. Let f, g be non-constant meromor-
phic functions with ρ(α(log),β,γ)[f]as (α(log), β, γ)-
order and µ(α(log),β,γ)[g]as lower (α(log), β, γ)-
order. Then
µ(α(log),β,γ)(f+g)
≤max ρ(α(log),β,γ)(f), µ(α(log),β,γ)(g)
and
µ(α(log),β,γ)(f g)
≤max ρ(α(log),β,γ)(f), µ(α(log),β,γ)(g).
Furthermore, if µ(α(log),β,γ)(g)> ρ(α(log),β,γ)(f),
then we obtain
µ(α(log),β,γ)(f+g) = µ(α(log),β,γ)(fg)
=µ(α(log),β,γ)(g).
Proof. Without loss of generality, we assume that
ρ(α(log),β,γ)(f)<+∞and µ(α(log),β,γ)(g)<+∞.
From the definition of the lower (α(log), β, γ)-order,
there exists a sequence rn−→ +∞(n−→ +∞)
such that
lim
n−→+∞
αlog[2] T(rn, g)
β(log γ(rn)) =µ(α(log),β,γ)(g).
Then, for any given ε > 0,there exists a positive in-
teger N1such that
T(rn, g)≤exp[2]{α−1µ(α(log),β,γ)(g) + ε
×β(log γ(rn)))}
holds for n > N1.From the definition of the
(α(log), β, γ)−order, for any given ε > 0,there ex-
ists a positive number Rsuch that
T(r, f)≤exp[2]{α−1ρ(α(log),β,γ)(f) + ε
×β(log γ(r)))}
holds for r≥R. Since rn−→ +∞(n−→ +∞),
there exists a positive integer N2such that rn> R,
and thus
T(rn, f)≤exp[2]{α−1ρ(α(log),β,γ)(f) + ε
×β(log γ(rn)))}
holds for n > N2.Note that
T(r, f +g)≤T(r, f) + T(r, g) + ln 2
and
T(r, fg)≤T(r, f) + T(r, g).
Then, for any given ε > 0,we have for n >
max {N1, N2}
T(rn, f +g)≤T(rn, f) + T(rn, g) + ln 2
≤exp[2] α−1ρ(α(log),β,γ)(f) + εβ(log γ(rn))
+exp[2]{α−1µ(α(log),β,γ)(g) + ε
×β(log γ(rn)))}+ln 2
≤3exp[2]{α−1max ρ(α(log),β,γ)(f),
µ(α(log),β,γ)(g)+εβ(log γ(rn))}(3)
and
T(rn, fg)≤T(rn, f ) + T(rn, g)
≤2exp[2]{α−1max ρ(α(log),β,γ)(f),
µ(α(log),β,γ)(g)+εβ(log γ(rn))}.(4)
Since ε > 0is arbitrary, then from (3) and (4), we
easily obtain
µ(α(log),β,γ)(f+g)
≤max ρ(α(log),β,γ)(f), µ(α(log),β,γ)(g)(5)
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and
µ(α(log),β,γ)(f g)
≤max ρ(α(log),β,γ)(f), µ(α(log),β,γ)(g).(6)
Suppose now that µ(α(log),β,γ)(g)> ρ(α(log),β,γ)(f).
Considering that
T(r, g) = T(r, f +g−f)
≤T(r, f +g) + T(r, f) + ln 2(7)
and
T(r, g) = Tr, fg
f≤T(r, fg) + Tr, 1
f
=T(r, fg) + T(r, f) + O(1) .(8)
By (7), (8) and the same method as above we obtain
that
µ(α(log),β,γ)(g)
≤max µ(α(log),β,γ)(f+g), ρ(α(log),β,γ)(f)
=µ(α(log),β,γ)(f+g)
(9)
and
µ(α(log),β,γ)(g)
≤max µ(α(log),β,γ)(f g), ρ(α(log),β,γ)(f)
=µ(α(log),β,γ)(f g).
(10)
By using (5) and (9) we obtain µ(α(log),β,γ)(f+g) =
µ(α(log),β,γ)(g)and by (6) and (10), we get
µ(α(log),β,γ)(f g) = µ(α(log),β,γ)(g).
2 Main Results
Very recently the author and Biswas have investigated
the growth of solutions of equation (1) and established
the following two results.
Theorem 2.1. ([4]) Let A0(z), A1(z), ..., Ak−1(z)
be entire functions such that ρ(α,β,γ)[A0]>
max{ρ(α,β,γ)[Aj], j = 1, ..., k −1}. Then every
solution f(z)≡ 0of (1) satisfies ρ(α(log),β,γ)[f] =
ρ(α,β,γ)[A0].
Theorem 2.2. ([4]) Let A0(z), A1(z), ..., Ak−1(z)be
entire functions. Assume that
max{ρ(α,β,γ)[Aj], j = 1, ..., k −1}
≤ρ(α,β,γ)[A0] = ρ0<+∞
and
max{τ(α,β,γ),M [Aj] : ρ(α,β,γ)[Aj] = ρ(α,β,γ)[A0]>0}
< τ(α,β,γ),M [A0] = τM.
Then every solution f(z)≡ 0of (1) satisfies
ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0].
Theorems 2.1 and 2.2 concerned the growth prop-
erties of solutions of (1), when A0is dominat-
ing the others coefficients by its (α, β, γ)-order and
(α, β, γ)-type. Thus, the natural question which
arises: If A0is dominating coefficient with its lower
(α, β, γ)-order and lower (α, β, γ)-type, what can we
say about the growth of solutions of (1)? The follow-
ing results give answer to this question.
Theorem 2.3. Let A0(z), ..., Ak−1(z)be entire
functions.Assume that max{ρ(α,β,γ)[Aj] : j=
1, ..., k −1}< µ(α,β,γ)[A0]≤ρ(α,β,γ)[A0]<+∞.
Then every solution f≡ 0of (1) satisfies
λ(α(log),β,γ)[f−g] = µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]
≤ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0] = λ(α(log),β,γ)[f−g],
where g≡ 0is an entire function satisfying
ρ(α(log),β,γ)[g]< µ(α,β,γ)[A0].
Theorem 2.4. Let A0(z), A1(z), ..., Ak−1(z)be en-
tire functions. Assume that
max{ρ(α,β,γ)[Aj] : j= 1, ..., k −1} ≤ µ(α,β,γ)[A0]
≤ρ(α,β,γ)[A0] = ρ < +∞(0 < ρ < +∞)
and
τ1=max{τ(α,β,γ),M [Aj] : ρ(α,β,γ)[Aj]
=µ(α,β,γ)[A0]>0}
< τ(α,β,γ),M [A0] = τ(0 < τ < +∞).
Then every solution f≡ 0of (1) satisfies
λ(α(log),β,γ)[f−g] = µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]
≤ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0] = λ(α(log),β,γ)[f−g],
where g≡ 0is an entire function satisfying
ρ(α(log),β,γ)[g]< µ(α,β,γ)[A0].
Theorem 2.5. Let A0(z), ..., Ak−1(z)be entire
functions.Assume that max{ρ(α,β,γ)[Aj] : j=
1, ..., k −1} ≤ µ(α,β,γ)[A0]<+∞and
lim sup
r→+∞
k−1
P
j=1
m(r, Aj)
m(r, A0)<1.
Then every solution f≡ 0of (1) satisfies
λ(α(log),β,γ)[f−g] = µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]
≤ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0] = λ(α(log),β,γ)[f−g],
where g≡ 0is an entire function satisfying
ρ(α(log),β,γ)[g]< µ(α,β,γ)[A0].
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Theorem 2.6. Let A0(z), ..., Ak−1(z)be entire
functions such that A0(z)is transcendental.Assume
that max{ρ(α,β,γ)[Aj] : j= 1, ..., k −1} ≤
µ(α,β,γ)[A0] = ρ(α,β,γ)[A0]<+∞and
lim inf
r→+∞
k−1
P
j=1
m(r, Aj)
m(r, A0)<1, r /∈E,
where Eis a set of rof finite linear measure. Then
every solution f≡ 0of (1) satisfies
λ(α(log),β,γ)[f−g] = µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]
=ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0] = λ(α(log),β,γ)[f−g],
where g≡ 0is an entire function satisfying
ρ(α(log),β,γ)[g]< µ(α,β,γ)[A0].
Remark 2.7. Nevanlinna theory, originally part of
complex analysis, has broad applications in both ap-
plied science and advanced mathematical methods.
In signal processing and control theory, Nevanlinna
theory helps analyze system stability and signal be-
havior by determining how often critical values are
reached by a system. It is particularly useful in de-
signing robust filtering systems and feedback con-
trols, minimizing noise, and ensuring stability. In
mathematical physics, Nevanlinna theory aids in un-
derstanding the behavior of complex systems like
wave propagation, quantum mechanics, and electro-
magnetic fields. Nevanlinna theory also plays a cru-
cial role in algebraic geometry and Diophantine ap-
proximation, where it helps study the distribution of
rational points on algebraic varieties. Its connections
to the Mordell conjecture (Faltings’ theorem) show
its relevance in the intersection of complex analysis
with modern topology and algebraic methods. In both
applied sciences and advanced mathematics, Nevan-
linna theory provides a powerful tool for analyzing
value distributions and system dynamics, please see,
[10], [12], [24], [26], [32], [33].
3 Preliminary Lemmas
In this section we present some lemmas which will be
needed in the sequel. First, we denote the Lebesgue
linear measure of a set E⊂[0,+∞)by m(E) =
R
F
dt, and the logarithmic measure of a set F⊂
(1,+∞)by ml(F) = R
F
dt
t.
The following result due to Gundersen [13] plays
an important role in the theory of complex differential
equations.
Lemma 3.1. ([13]) Let fbe a transcendental mero-
morphic function, and let χ > 1be a given constant.
Then there exist a set E1⊂(1,∞)with finite loga-
rithmic measure and a constant B > 0that depends
only on χand i, j (0 ≤i<j≤k), such that for all
zsatisfying |z|=r/∈[0,1] ∪E1, we have
f(j)(z)
f(i)(z)
≤BT(χr, f)
r(logχr)log T(χr, f)j−i
.
Lemma 3.2. Let fbe a meromorphic function with
µ(α(log),β,γ)[f] = µ < +∞. Then there exists a
set E2⊂(1,+∞)with infinite logarithmic measure
such that for r∈E2⊂(1,+∞),we have for any
given ε > 0
T(r, f)<exp[2] α−1((µ+ε)β(log γ(r))).
Proof. The definition of lower (α(log), β, γ)-order
implies that there exists a sequence {rn}+∞
n=1 tending
to ∞satisfying 1 + 1
nrn< rn+1 and
lim
rn→∞
α(log[2] T(rn, f))
β(log γ(rn)) =µ(α(log),β,γ)[f].
Then for any given ε > 0, there exists an integer n1
such that for all n≥n1,
T(rn, f)<exp[2] nα−1µ+ε
2β(log γ(rn))o.
Set E2=
+∞
S
n=n1hn
n+1 rn, rni.Then for r∈E2⊂
(1,+∞),by using γ(2r)≤2γ(r)and β(r+O(1)) =
(1 + o(1))β(r)as r→+∞,we obtain for any given
ε > 0
T(r, f)≤T(rn, f )
<exp[2] nα−1µ+ε
2β(log γ(rn))o
≤exp[2] α−1µ+ε
2
×βlog γn+ 1
nr
≤exp[2] nα−1µ+ε
2β(log γ(2r))o
≤exp[2] nα−1µ+ε
2β(log (2γ(r)))o
=exp[2] nα−1µ+ε
2β(log 2 + log γ(r))o
=exp[2] nα−1µ+ε
2(1 + o(1)) β(log γ(r))o
<exp[2] α−1((µ+ε)β(log γ(r))),
and lm (E2) =
+∞
P
n=n1
rn
R
n
n+1 rn
dt
t=
+∞
P
n=n1
log 1 + 1
n=
∞.Thus, Lemma 3.2 is proved.
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We can also prove the following result by using
similar reason as in the proof of Lemma 3.2.
Lemma 3.3. Let fbe an entire function with
µ(α,β,γ)[f] = µ < +∞. Then there exists a set
E3⊂(1,+∞)with infinite logarithmic measure
such that for r∈E3⊂(1,+∞),we have for any
given ε > 0
M(r, f)<exp[2] α−1((µ+ε)β(log γ(r))).
The following lemma gives the relation between
the maximum term and the central index of an entire
function f.
Lemma 3.4. ([17], Theorems 1.9 and 1.10, or [20],
Satz 4.3 and 4.4) Let f(z) =
+∞
P
n=0
anznbe an entire
function, µ(r)be the maximum term of f, i.e.,
µ(r) = max {|an|rn:n= 0,1,2, ...},
and ν(r, f) = νf(r)be the central index of f, i.e.,
ν(r, f) = max {m:µ(r) = |am|rm}.
Then
(i)
log µ(r) = log |a0|+
r
Z0
νf(t)
tdt,
here we assume that |a0| = 0.
(ii)For r < R
M(r, f)< µ (r)νf(R) + R
R−r.
Lemma 3.5. ([16, 20, 41]) Let fbe a transcendental
entire function. Then there exists a set E4⊂(1,+∞)
with finite logarithmic measure such that for all zsat-
isfying |z|=r/∈E4and |f(z)|=M(r, f),we have
f(n)(z)
f(z)=νf(r)
zn
(1 + o(1)),(n∈N).
Here, we give the generalized logarithmic deriva-
tive estimates for meromorphic functions of finite
(α(log), β, γ)−order.
Lemma 3.6. ([4]) Let fbe a meromorphic function
of order ρ(α(log),β,γ)[f] = ρ < +∞,k∈N. Then, for
any given ε > 0,
mr, f(k)
f
=Oexp α−1((ρ+ε)β(log γ(r))),
outside, possibly, an exceptional set E5⊂[0,+∞)of
finite linear measure.
Lemma 3.7. ([4]) Let A0(z), A1(z), ..., Ak−1(z)be
entire functions. Then every nontrivial solution fof
(1) satisfies
ρ(α(log),β,γ)[f]
≤max{ρ(α,β,γ)[Aj] : j= 0,1, ..., k −1}.
Lemma 3.8. ([4]) Let fbe an entire function with
ρ(α,β,γ)[f] = ρ∈(0,+∞)and τ(α,β,γ),M [f]∈
(0,+∞). Then for any given η < τ(α,β,γ),M [f], there
exists a set E6⊂(1,+∞)of infinite logarithmic
measure such that for all r∈E6, one has
exp nα(log[2] M(r, f))o> η (exp {β(log γ(r))})ρ.
Lemma 3.9. Let f2(z)be an entire function of lower
(α(log), β, γ)-order with µ(α(log),β,γ)[f2] = µ > 0,
and let f1(z)be an entire function of (α(log), β, γ)-
order with ρ(α(log),β,γ)[f1] = ρ < +∞. If
ρ(α(log),β,γ)[f1]< µ(α(log),β,γ)[f2],then we have
T(r, f1) = o(T(r, f2)) as r→+∞.
Proof. By definitions of (α(log), β, γ)-order and
lower (α(log), β, γ)-order, for any given εwith 0<
2ε < µ −ρand sufficiently large r, we have
T(r, f1)≤exp[2] α−1((ρ+ε)β(log γ(r)))
(11)
and
T(r, f2)≥exp[2] α−1((µ−ε)β(log γ(r))).
(12)
Now by (11) and (12), we get
T(r, f1)
T(r, f)≤exp[2] α−1((ρ+ε)β(log γ(r)))
exp[2] {α−1((µ−ε)β(log γ(r)))}
=exp{exp α−1((ρ+ε)β(log γ(r)))
−exp α−1((µ−ε)β(log γ(r)))}
=exp ( exp α−1((ρ+ε)β(log γ(r)))
exp {α−1((µ−ε)β(log γ(r)))}−1!
×exp α−1((µ−ε)β(log γ(r))))
=exp (
exp nα−1ρ+ε
µ−ε(µ−ε)β(log γ(r))o
exp {α−1((µ−ε)β(log γ(r)))}
−1!exp α−1((µ−ε)β(log γ(r))) ).
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Set
y=
exp nα−1ρ+ε
µ−ε(µ−ε)β(log γ(r))o
exp {α−1((µ−ε)β(log γ(r)))}−1
×exp α−1((µ−ε)β(log γ(r))).
Then by putting (µ−ε)β(log γ(r)) = x, ρ+ε
µ−ε=
k(0 < k < 1) and making use of the condition
α−1(kx) = oα−1(x)(0 < k < 1) as x→+∞,
we get
lim
r→+∞y
=lim
x→+∞ exp α−1(kx)
exp {α−1(x)}−1!exp α−1(x)
=lim
x→+∞ exp oα−1(x)
exp {α−1(x)}−1!exp α−1(x)
=lim
x→+∞exp (o(1) −1) α−1(x)−1
×exp α−1(x)=−∞,
this implies
lim
r→+∞exp y= 0.
Therefore yielding
lim
r→+∞
T(r, f1)
T(r, f2)= 0,
that is T(r, f1) = o(T(r, f2)) as r→+∞.
Lemma 3.10. Let F(z)≡ 0,Aj(z) (j= 0, ..., k −1)
be meromorphic functions, and let fbe a meromor-
phic solution of (2) satisfying
max{ρ(α(log),β,γ)[Aj], ρ(α(log),β,γ)[F] :
j= 0,1, ..., k −1}< µ(α(log),β,γ)[f].
Then we have
λ(α(log),β,γ)[f] = λ(α(log),β,γ)[f] = µ(α(log),β,γ)[f].
Proof. By (2), we get that
1
f=1
Ff(k)
f+Ak−1(z)f(k−1)
f+· · ·
+A1(z)f′
f+A0. (13)
Now, by (2) it is easy to see that if fhas a zero at z0
of order a(a > k), and if A0, ..., Ak−1are analytic at
z0, then F(z)must have a zero at z0of order a−k,
hence
nr, 1
f≤kn r, 1
f+nr, 1
F+
k−1
X
j=0
n(r, Aj)
(14)
and
Nr, 1
f≤kN r, 1
f+Nr, 1
F+
k−1
X
j=0
N(r, Aj).
(15)
By the lemma on logarithmic derivative ([15], p. 34)
and (13), we have
mr, 1
f≤mr, 1
F+
k−1
X
j=0
m(r, Aj)
+O(log T(r, f) + log r) (r/∈E7).
(16)
where E7⊂[0,+∞)is a set of rof finite linear mea-
sure. By (15) and (16), we obtain that
T(r, f) = Tr, 1
f+O(1) ≤kN r, 1
f
+T(r, F )+
k−1
X
j=0
T(r, Aj)+O(log(rT (r, f))) (r/∈E7).
(17)
Since max{ρ(α(log),β,γ)[Aj], ρ(α(log),β,γ)[F] : j=
0,1, ..., k −1}< µ(α(log),β,γ)[f], then by Lemma 3.9
T(r, F ) = o(T(r, f)), T (r, Aj) = o(T(r, f ))
(j= 0, ..., k −1) as r→+∞.
(18)
Since fis transcendental, then we have
O(log(rT (r, f))) = o(T(r, f)) as r→+∞. (19)
Therefore, by substituting (18) and (19) into (17), for
all |z|=r/∈E7, we get that
T(r, f)≤ONr, 1
f.
Hence from above we have
µ(α(log),β,γ)[f]≤λ(α(log),β,γ)[f].
Since λ(α(log),β,γ)[f]≤λ(α(log),β,γ)[f]≤
µ(α(log),β,γ)[f],then
λ(α(log),β,γ)[f] = λ(α(log),β,γ)[f] = µ(α(log),β,γ)[f].
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Lemma 3.11. ([3]) Let F(z)≡ 0,Aj(z)
(j= 0, ..., k −1) be entire functions.
Also let fbe a solution of (2) satisfying
max{ρ(α(log),β,γ)[Aj], ρ(α(log),β,γ)[F] : j=
0,1, ..., k −1}< ρ(α(log),β,γ)[f]. Then we have
λ(α(log),β,γ)[f] = λ(α(log),β,γ)[f] = ρ(α(log),β,γ)[f].
Lemma 3.12. Let fbe a transcendental entire func-
tion. Then ρ(α(log),β,γ)[f] = ρ(α(log),β,γ)[f(k)], k ∈N.
Proof. By Lemma 4.4 in ([3]), we have ρ(α(log),β,γ)[f]
=ρ(α(log),β,γ)[f′],so by using mathematical induc-
tion, we easily obtain the result.
Lemma 3.13. ([3]) Let fbe a meromorphic function.
If ρ(α,β,γ)[f] = ρ < +∞, then ρ(α(log),β,γ)[f] = 0.
Lemma 3.14. ([17]) Let Aj(z) (j= 0, ..., k −1)
be entire coefficients in (1), and at least one of
them is transcendental. If As(z) (0 ≤s≤k−1)
is the first one (according to the sequence of
A0(z), ..., Ak−1(z)) satisfying
lim inf
r→+∞
k−1
P
j=s+1
m(r, Aj)
m(r, As)<1, r /∈E8,
where E8is a set of rof finite linear measure. Then
(1) possesses at most slinearly independent entire so-
lutions satisfying
lim sup
r→+∞
log T(r, f)
m(r, As)= 0, r /∈E8.
4 Proof of the Main Results
Proof of Theorem 2.3. Suppose that f(≡ 0) is
a solution of equation (1). By Theorem 2.1, we
know that every solution f(≡ 0) of (1) satisfies
ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0]. So, we only need
to prove that every solution f(≡ 0) of (1) satisfies
µ(α(log),β,γ)[f] = µ(α,β,γ)[A0].First, we prove that
µ1=µ(α(log),β,γ)[f]≥µ(α,β,γ)[A0] = µ0. Suppose
the contrary. Set max{ρ(α,β,γ)[Aj] : j= 1, ..., k −
1, µ(α(log),β,γ)[f]}=ρ < µ(α,β,γ)[A0] = µ0.From
(1), we can write
|A0(z)| ≤
f(k)
f
+|Ak−1(z)|
f(k−1)
f
+· · ·
+|A1(z)|
f′
f
.
(20)
For any given ε(0 <2ε < µ0−ρ)and for suffi-
ciently large r, we have
|A0(z)|>exp[2] α−1((µ0−ε)β(log γ(r)))
(21)
and
|Aj(z)| ≤ exp[2] nα−1ρ+ε
2β(log γ(r))o,
j∈ {1,2, ..., k −1}.
(22)
By Lemma 3.1, there exist a constant B > 0and a
set E1⊂(1,+∞)having finite logarithmic measure
such that for all zsatisfying |z|=r/∈[0,1] ∪E1, we
have
f(j)(z)
f(z)
≤B[T(2r, f)]k+1 (j= 1,2, ..., k).
(23)
It follows by Lemma 3.2 and (23), that for sufficiently
large |z|=r∈E2\(E1∪[0,1])
f(j)(z)
f(z)
≤B[T(2r, f)]k+1
≤Bhexp[2] nα−1µ1+ε
2β(log γ(r))oik+1
(j= 1,2, ..., k),
(24)
where E2is a set of infinite logarithmic measure.
Hence, by substituting (21)-(24) into (20), for the
above ε(0 <2ε < µ0−ρ), we obtain for sufficiently
large |z|=r∈E2\(E1∪[0,1])
exp[2] α−1((µ0−ε)β(log γ(r)))
≤Bk exp[2] nα−1ρ+ε
2β(log γ(r))o
×[T(2r, f)]k+1
≤Bk exp[2] nα−1ρ+ε
2β(log γ(r))o
×hexp[2] nα−1µ1+ε
2β(log γ(r))oik+1
≤exp[2] α−1((ρ+ε)β(log γ(r))).(25)
Since E2\(E1∪[0,1]) is a set of infinite logarithmic
measure, then there exists a sequence of points |zn|=
rn∈E2\(E1∪[0,1]) tending to +∞.It follows by
(25) that
exp[2] α−1((µ0−ε)β(log γ(rn)))
≤exp[2] α−1((ρ+ε)β(log γ(rn)))(26)
holds for all znsatisfying |zn|=rn∈
E2\(E1∪[0,1]) as |zn| → +∞.By arbitrariness of
ε > 0and the monotony of the function α−1, from
(26) we obtain that ρ≥µ(α,β,γ)[A0] = µ0. This
contradiction proves the inequality µ(α(log),β,γ)[f]≥
µ(α,β,γ)[A0].
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Now, we prove µ(α(log),β,γ)[f]≤µ(α,β,γ)[A0] = µ0.
By (1), we have
f(k)
f
≤ |Ak−1(z)|
f(k−1)
f
+· · ·
+|A1(z)|
f′
f
+|A0(z)|.
(27)
By Lemma 3.5, there exists a set E4⊂(1,+∞)of
finite logarithmic measure such that the estimation
f(j)(z)
f(z)=νf(r)
zj
(1 + o(1)) (j= 1, ..., k)
(28)
holds for all zsatisfying |z|=r/∈E4, r →+∞and
|f(z)|=M(r, f). By Lemma 3.3, for any given ε >
0,there exists a set E3⊂(1,+∞)that has infinite
logarithmic measure, such that
|A0(z)| ≤ exp[2] nα−1µ0+ε
2β(log γ(r))o
(29)
and for sufficiently large r
|Aj(z)| ≤ exp[2] nα−1ρ+ε
2β(log γ(r))o
≤exp[2] nα−1µ0+ε
2β(log γ(r))o
(j= 1, ..., k −1) .
(30)
Substituting (28), (29) and (30) into (27), we obtain
νf(r)≤krk|1 + o(1)|exp[2] nα−1µ0+ε
2
×β(log γ(r)) o
≤exp[2] α−1((µ0+ε)β(log γ(r)))(31)
for all zsatisfying |z|=r∈E3\E4, r →+∞and
|f(z)|=M(r, f).By Lemma 3.4, from (31) we ob-
tain for each ε > 0
T(r, f)≤log M(r, f)<log [µ(r) (νf(2r) + 2)]
=log haνf(r)rνf(r)(νf(2r) + 2)i
< νf(r)log r+log (2νf(2r)) + log aνf(r)
≤exp[2] α−1((µ0+ε)β(log γ(r)))log r
+log 2exp[2] α−1((µ0+ε)β(log γ(2r)))
+log aνf(r)
≤exp[2] α−1((µ0+ 2ε)β(log γ(r)))+log 2
+exp α−1((µ0+ε)β(log γ(2r)))
+log aνf(r)
≤exp[2] α−1((µ0+ 3ε)β(log γ(r))).
Hence,
α(log[2] T(r, f))
β(log γ(r)) ≤µ0+ 3ε.
It follows
µ(α(log),β,γ)[f] = lim inf
r−→+∞
α(log[2] T(r, f))
β(log γ(r)) ≤µ0+3ε.
Since ε > 0is arbitrary, then we obtain
µ(α(log),β,γ)[f]≤µ0.Hence every solution
f≡ 0of equation (1) satisfies µ(α,β,γ)[A0] =
µ(α(log),β,γ)[f]≤ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0].
Secondly, we prove that λ(α(log),β,γ)[f−g] =
µ(α(log),β,γ)[f]and
λ(α(log),β,γ)[f−g] = ρ(α(log),β,γ)[f].
Set h=f−g. Since
ρ(α(log),β,γ)[g]< µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]
≤ρ(α(log),β,γ)[f],
it follows from Proposition 1.8 and Proposition 1.10
that ρ(α(log),β,γ)[h] = ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0]
and µ(α(log),β,γ)[h] = µ(α(log),β,γ)[f] = µ(α,β,γ)[A0].
By substituting f=g+h, f′=g′+h′, . . . , f (k)=
g(k)+h(k)into (1), we obtain
h(k)+Ak−1(z)h(k−1) +· · · +A0(z)h
=−(g(k)+Ak−1(z)g(k−1) +· · · +A0(z)g).(32)
If g(k)+Ak−1(z)g(k−1) +· · · +A0(z)g=G≡0,
then by the first part of the proof of Theorem 2.3 we
have ρ(α(log),β,γ)[g]≥µ(α,β,γ)[A0]which contradicts
the assumption ρ(α(log),β,γ)[g]< µ(α,β,γ)[A0].Hence
G≡ 0.By Proposition 1.8, Lemma 3.12 and Lemma
3.13, we get
ρ(α(log),β,γ)[G]
≤max{ρ(α(log),β,γ)[g], ρ(α(log),β,γ)(Aj)
(j= 0,1, ..., k −1)}
=ρ(α(log),β,γ)[g]< µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]
=µ(α(log),β,γ)[h]≤ρ(α(log),β,γ)[h] = ρ(α(log),β,γ)[f]
=ρ(α,β,γ)[A0].
Then, it follows from Lemma 3.10, Lemma 3.11
and (32) that λ(α(log),β,γ)[h] = λ(α(log),β,γ)[h] =
ρ(α(log),β,γ)(h) = ρ(α(log),β,γ)[f]and
λ(α(log),β,γ)[h] = λ(α(log),β,γ)[h] = µ(α(log),β,γ)[h]
=µ(α(log),β,γ)[f].
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Therefore, λ(α(log),β,γ)[f−g] = µ(α(log),β,γ)[f]and
λ(α(log),β,γ)[f−g] = ρ(α(log),β,γ)[f]
which completes the proof of Theorem 2.3.
Proof of Theorem 2.4. Suppose that f(≡ 0) is
a solution of equation (1). Then by Theorem 2.2,
we obtain ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0].Now, we
prove that µ1=µ(α(log),β,γ)[f]≥µ(α,β,γ)[A0] =
µ0. Suppose the contrary µ1=µ(α(log),β,γ)[f]<
µ(α,β,γ)[A0] = µ0. We set b=max{ρ(α,β,γ)[Aj] :
ρ(α,β,γ)[Aj]< µ(α,β,γ)[A0]}.If ρ(α,β,γ)[Aj]<
µ(α,β,γ)[A0],then for any given εwith 0<3ε <
min {µ0−b, τ −τ1}and for sufficiently large r, we
have
|Aj(z)| ≤ exp[2] α−1((b+ε)β(log γ(r)))
≤exp[2] α−1µ(α,β,γ)[A0]−2εβ(log γ(r)).
(33)
If ρ(α,β,γ)[Aj] = µ(α,β,γ)[A0],τ(α,β,γ),M [Aj]≤τ1<
τ(α,β,γ),M [A0] = τ, then for sufficiently large r, we
have
|Aj(z)| ≤ exp[2] α−1(log ((τ1+ε)
×(exp {β(log γ(r))})µ0))}
(34)
and
|A0(z)|>exp[2] α−1(log ((τ−ε)
×(exp {β(log γ(r))})µ0))}.
(35)
By Lemma 3.1 and Lemma 3.2, for any given εwith
0< ε < µ0−µ1and sufficiently large |z|=r∈
E2\(E1∪[0,1])
f(j)(z)
f(z)
≤B[T(2r, f)]k+1
≤Bhexp[2] α−1((µ1+ε)β(log γ(r)))ik+1
(j= 1,2, ..., k),
(36)
where E2is a set of infinite logarithmic measure.
Hence, by substituting (33)-(36) into (20), for the
above εwith 0< ε < min nµ0−b
3,τ−τ1
3, µ0−µ1o,
we obtain for sufficiently large |z|=r∈
E2\(E1∪[0,1])
exp[2] α−1(log ((τ−ε) (exp {β(log γ(r))})µ0))
≤Bk exp[2] α−1(log ((τ1+ε)
×(exp {β(log γ(r))})µ0))}[T(2r, f)]k+1
≤Bk exp[2] α−1(log ((τ1+ε)
×(exp {β(log γ(r))})µ0))}
×hexp[2] α−1((µ1+ε)β(log γ(r)))ik+1
≤exp[2] α−1(log ((τ1+ 2ε)
×(exp {β(log γ(r))})µ0))}.
(37)
Since E2\(E1∪[0,1]) is a set of infinite logarithmic
measure, then there exists a sequence of points |zn|=
rn∈E2\(E1∪[0,1]) tending to +∞.It follows by
(37) that
exp[2] α−1(log ((τ−ε) (exp {β(log γ(rn))})µ0))
≤exp[2] α−1(log ((τ1+ 2ε)
×(exp {β(log γ(rn))})µ0))}
holds for all znsatisfying |zn|=rn∈
E2\(E1∪[0,1]) as |zn| → +∞.By arbitrariness of
ε > 0and the monotonicity of the function α−1, we
obtain that τ1≥τ. This contradiction proves the in-
equality µ(α(log),β,γ)[f]≥µ(α,β,γ)[A0].
Now, we prove µ(α(log),β,γ)[f]≤µ(α,β,γ)[A0].By us-
ing similar arguments as in the proofs of Theorem 2.3,
we obtain µ(α(log),β,γ)[f]≤µ(α,β,γ)[A0].Hence, ev-
ery solution f≡ 0of equation (1) satisfies
µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]≤ρ(α(log),β,γ)[f]
=ρ(α,β,γ)[A0].
The second part of the proof of Theorem 2.3 com-
pletes the proof of Theorem 2.4.
Proof of Theorem 2.5. Suppose that f(≡ 0)
is a solution of equation (1). We divide the proof
into two parts: (i) ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0], (ii)
µ(α(log),β,γ)[f] = µ(α,β,γ)[A0].
(i) First, we prove that ρ1=ρ(α(log),β,γ)[f]≥
ρ(α,β,γ)[A0] = ρ0. Suppose the contrary ρ1=
ρ(α(log),β,γ)[f]< ρ(α,β,γ)[A0] = ρ0.From (1), we
can write
A0(z) = −f(k)
f+Ak−1(z)f(k−1)
f+· · ·
+A1(z)f′
f.(38)
By Lemma 3.6 and (38), we have
m(r, A0)≤
k−1
X
j=1
m(r, Aj) +
k
X
j=1
mr, f(j)
f+log k
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≤
k−1
X
j=1
m(r, Aj) + Oexp nα−1ρ1+ε
2
×β(log γ(r))) o (39)
holds possibly outside of an exceptional set E5⊂
(0,+∞)with finite linear measure. Suppose that
lim sup
r→+∞
k−1
P
j=1
m(r, Aj)
m(r, A0)=σ < κ < 1.
Then for sufficiently large r, we have
k−1
X
j=1
m(r, Aj)< κm (r, A0).(40)
By (39) and (40), we have
(1 −κ)m(r, A0)
≤Oexp nα−1ρ1+ε
2β(log γ(r))o,
r/∈E5.
It follows that
T(r, A0) = m(r, A0)
≤exp α−1((ρ1+ε)β(log γ(r))), r /∈E5.
(41)
Hence α(log T(r, A0))
β(log γ(r)) ≤ρ1+ε
and
ρ(α,β,γ)[A0] = lim sup
r−→+∞
α(log T(r, A0))
β(log γ(r)) ≤ρ1+ε.
Since ε > 0is arbitrary, then we obtain
ρ(α,β,γ)[A0]≤ρ1.This contradiction proves the in-
equality ρ(α(log),β,γ)[f]≥ρ(α,β,γ)[A0]. On the other
hand, by Lemma 3.7, we have
ρ(α(log),β,γ)[f]
≤max{ρ(α,β,γ)[Aj] : j= 0,1, ..., k −1}
=ρ(α,β,γ)[A0].
(42)
Hence every solution f≡ 0of equation (1) satisfies
ρ(α(log),β,γ)[f] = ρ(α,β,γ)[A0].
(ii) By using similar arguments as in the proofs
of Theorem 2.3, we obtain µ(α(log),β,γ)[f] =
µ(α,β,γ)[A0].Hence, every solution f≡ 0of equa-
tion (1) satisfies
µ(α,β,γ)[A0] = µ(α(log),β,γ)[f]≤ρ(α(log),β,γ)[f]
=ρ(α,β,γ)[A0].
The second part of the proof of Theorem 2.3 com-
pletes the proof of Theorem 2.5.
Proof of Theorem 2.6. By Lemma 3.14, we ob-
tain that every linearly independent solution of (1) sat-
isfies lim sup
r→+∞
log T(r,f)
m(r,A0)>0, r /∈E8. So, every so-
lution f(≡ 0) of (1) satisfies lim sup
r→+∞
log T(r,f)
m(r,A0)>0,
r/∈E8. Hence, there exist δ > 0and a sequence
{rn}+∞
n=1 tending to ∞such that for sufficiently large
rn/∈E8and for every solution f(≡ 0) of (1), we
have
log T(rn, f)> δm (rn, A0).(43)
Since µ(α,β,γ)[A0] = ρ(α,β,γ)[A0],then by (43), for
any given ε > 0and sufficiently large rn/∈E8,we
get
log T(rn, f)
> δ exp nα−1µ(α,β,γ)[A0]−ε
2β(log γ(rn))o
≥exp α−1µ(α,β,γ)[A0]−εβ(log γ(rn)),
which implies
ρ(α(log),β,γ)[f]≥µ(α,β,γ)[A0] = ρ(α,β,γ)[A0].(44)
On the other hand, by Lemma 3.7, we have
ρ(α(log),β,γ)[f]
≤max{ρ(α,β,γ)[Aj] : j= 0,1, ..., k −1}
=µ(α,β,γ)[A0] = ρ(α,β,γ)[A0].
(45)
By (44) and (45), we obtain ρ(α(log),β,γ)[f] =
µ(α,β,γ)[A0] = ρ(α,β,γ)[A0].
The second part of the proof of Theorem 2.3 com-
pletes the proof of Theorem 2.6.
5 Conclusion
Throughout this article, by using the concepts of
lower (α, β, γ)-order and lower (α, β, γ)-type, we
obtain some growth and oscillation properties of so-
lutions of higher order linear differential equations in
which the coefficients are entire functions. We im-
prove and extend some recently obtained results by
the author and Biswas [4]. Inspired by the results al-
ready established, one may explore analogous theo-
rems for differential equations in which the coeffi-
cients are meromorphic functions of (α, β, γ)-order.
Further, we can study differential polynomials gener-
ated by solutions of the differential equations (1) and
(2) when the coefficients of these equations are entire,
meromorphic or analytic functions in the unit disc.
Acknowledgment:
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The author would like to thank the anonymous
reviewers and the Assistant Editor for their valuable
comments and suggestions, which were very helpful
for revision and improvement of this paper.
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findings and solution.
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Conflicts of Interest
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