Author:
Long Jianren,Qin Hongyan,Tao Lei
Abstract
AbstractThe fast growing solutions of the following linear differential equation $$(*)$$
(
∗
)
is investigated by using a more general scale $${[p,q]_{,\varphi }}$$
[
p
,
q
]
,
φ
-order, $$\begin{aligned} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdot \cdot \cdot +A_0(z)f=0,\qquad (*) \end{aligned}$$
f
(
k
)
+
A
k
-
1
(
z
)
f
(
k
-
1
)
+
·
·
·
+
A
0
(
z
)
f
=
0
,
(
∗
)
where $$A_i(z)$$
A
i
(
z
)
are entire functions in the complex plane, $$i=0,1,\ldots ,k-1$$
i
=
0
,
1
,
…
,
k
-
1
. The growth relationships between entire coefficients and solutions of the equation $$(*)$$
(
∗
)
is found by using the concepts of $${[p,q]_{,\varphi }}$$
[
p
,
q
]
,
φ
-order and $${[p,q]_{,\varphi }}$$
[
p
,
q
]
,
φ
-type, which extend and improve some previous results.
Funder
The National Natural Science Foundation of China
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference21 articles.
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2. Belaïdi, B.: Fast growing solutions to linear differential equations with entire coefficients having the same $$\rho _\varphi$$-order. J. Math. Appl. 42, 63–77 (2019)
3. Cao, T.B., Xu, J.F., Chen, Z.X.: On the meromorphic solutions of linear differential equations on the complex plane. J. Math. Anal. Appl. 364, 130–142 (2010)
4. Chyzhykov, I., Semochko, N.: Fast growing entire solutions of linear differential equations. Math. Bull. Shevchenko Sci. Soc. 13, 68–83 (2016)
5. Clunie, J.: On integral functions having prescribed asymptotic growth. Can. J. Math. 17, 396–404 (1965)
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