Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 898274, 11 pages
doi:10.1155/2010/898274
Research Article
Approximation of Analytic Functions by
Kummer Functions
Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of
Korea
Correspondence should be addressed to Soon-Mo Jung, smjung@hongik.ac.kr
Received 3 February 2010; Revised 27 March 2010; Accepted 31 March 2010
Academic Editor: Alberto Cabada
Copyright q 2010 Soon-Mo Jung. 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.
We
the inhomogeneous Kummer differential equation of the form xy β − xy − αy ∞ solve m
a
x
and apply this result to the proof of a local Hyers-Ulam stability of the Kummer
m
m0
differential equation in a special class of analytic functions.
1. Introduction
Assume that X and Y are a topological vector space and a normed space, respectively, and
that I is an open subset of X. If for any function f : I → Y satisfying the differential inequality
an xyn x an−1 xyn−1 x · · · a1 xy x a0 xyx hx ≤ ε
1.1
for all x ∈ I and for some ε ≥ 0, there exists a solution f0 : I → Y of the differential equation
an xyn x an−1 xyn−1 x · · · a1 xy x a0 xyx hx 0
1.2
such that fx − f0 x ≤ Kε for any x ∈ I, where Kε depends on ε only, then we say that
the above differential equation satisfies the Hyers-Ulam stability or the local Hyers-Ulam
stability if the domain I is not the whole space X. We may apply this terminology for other
differential equations. For more detailed definition of the Hyers-Ulam stability, refer to 1–6.
Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of
linear differential equations see 7, 8. Here, we will introduce a result of Alsina and Ger
see 9. If a differentiable function f : I → R is a solution of the differential inequality
2
Journal of Inequalities and Applications
|y x − yx| ≤ ε, where I is an open subinterval of R, then there exists a solution f0 : I → R
of the differential equation y x yx such that |fx − f0 x| ≤ 3ε for any x ∈ I.
This result of Alsina and Ger has been generalized by Takahasi et al.. They proved
in 10 that the Hyers-Ulam stability holds true for the Banach space valued differential
equation y x λyx see also 11.
Using the conventional power series method, the author 12 investigated the general
solution of the inhomogeneous Legendre differential equation of the form
∞
am xm
1 − x2 y x − 2xy x p p 1 yx 1.3
m0
under some specific conditions, where p is a real number and the convergence radius of the
power series is positive. Moreover, he applied this result to prove that every analytic function
can be approximated in a neighborhood of 0 by the Legendre function with an error bound
expressed by Cx2 /1 − x2 see 13–16.
In Section 2 of this paper, employing power series method, we will determine the
general solution of the inhomogeneous Kummer differential equation
∞
am xm ,
xy x β − x y x − αyx 1.4
m0
where α and β are constants and the coefficients am of the power series are given such that the
radius of convergence is ρ > 0, whose value is in general permitted to be infinite. Moreover,
using the idea from 12, 13, 15, we will prove the Hyers-Ulam stability of the Kummer’s
equation in a class of special analytic functions see the class CK in Section 3.
In this paper, N0 and Z denote the set of all nonnegative integers and the set of all
integers, respectively. For each real number α, we use the notation α
to denote the ceiling of
α, that is, the least integer not less than α.
2. General Solution of 1.4
The Kummer differential equation
xy x β − x y x − αyx 0,
2.1
which is also called the confluent hypergeometric differential equation, appears frequently in
practical problems and applications. The Kummer’s equation 2.1 has a regular singularity
at x 0 and an irregular singularity at ∞. A power series solution of 2.1 is given by
∞
αm m
M α, β, x x ,
m0 m! β m
2.2
where αm is the factorial function defined by α0 1 and αm αα1α2 · · · αm−1
for all m ∈ N. The above power series solution is called the Kummer function or the confluent
Journal of Inequalities and Applications
3
hypergeometric function. We know that if neither α nor β is a nonpositive integer, then the
power series for Mα, β, x converges for all values of x.
Let us define
M 1 α − β, 2 − β, x
M α, β, x
π
1−β
.
U α, β, x −x
sin βπ Γ 1 α − β Γ β
ΓαΓ 2 − β
2.3
We know that if β /
1 then Mα, β, x and Uα, β, x are independent solutions of the
Kummer’s equation 2.1. When β > 1, Uα, β, x is not defined at x 0 because of the
factor x1−β in the above definition of Uα, β, x.
By considering this fact, we define
⎧
⎨ −ρ, ρ ,
for β < 1 ,
Iρ ⎩ −ρ, 0 ∪ 0, ρ, for β > 1,
2.4
for any 0 < ρ ≤ ∞. It should be remarked that if β /
∈ Z and both α and 1 α − β are not
nonpositive integers, then Mα, β, x and Uα, β, x converge for all x ∈ I∞ see 17, Section
13.1.3.
Theorem 2.1. Let α and β be real constants such that β /
∈ Z and neither α nor 1α−β is a nonpositive
m
integer. Assume that the radius of convergence of the power series ∞
m0 am x is ρ > 0 and that there
exists a real number μ ≥ 0 with
m−1
m − 1!β am i! β i ai m
≤ μ
i0 αi1 αm1
2.5
for all sufficiently large integers m. Let us define ρ0 min{ρ, 1/μ} and 1/0 ∞. Then, every
solution y : Iρ0 → C of the inhomogeneous Kummer’s equation 1.4 can be represented by
∞ m−1
i!αm β i ai
yx yh x xm ,
m!α
i1 β m
m1 i0
2.6
where yh x is a solution of the Kummer’s equation 2.1.
Proof. Assume that a function y : Iρ0 → C is given by 2.6. We first prove that the function
yp x, defined by yx − yh x, satisfies the inhomogeneous Kummer’s equation 1.4. Since
yp x
∞ m i!α
i!αm β i ai
m1 β i ai m
m−1
x ,
x
m0 i0 m!αi1 β m1
m1 i0 m − 1!αi1 β m
m
∞ i!αm1 β i ai
yp x xm−1 ,
−
1!α
β
m
i1
m1 i0
m1
∞ m−1
2.7
4
Journal of Inequalities and Applications
we have
xyp x
β−x
yp x
m i!α
∞ m1 β i m β ai m
x
− αyp x a0 m!αi1 β m1
m1 i0
∞ m−1
i!αm β i m αai
−
xm
m!α
β
i1
m1 i0
m
a0 ∞
2.8
am xm ,
m1
which proves that yp x is a particular solution of the inhomogeneous Kummer’s equation
1.4.
We now apply the ratio test to the power series expression of yp x as follows:
∞ m−1
∞
i!αm β i ai
cm x m .
yp x xm m!α
β
i1
m1 i0
m1
m
2.9
Then, it follows from 2.5 that
⎡
−1 ⎤
m − 1!β am m−1
cm1 α m
i!
β i ai 1
m
m
⎣
≤ lim lim ⎦
m → ∞ cm m → ∞ β m m 1
m1
αm1
α
i1
i0
2.10
≤ μ.
Therefore, the power series expression of yp x converges for all x ∈ I1/μ . Moreover, the
convergence region of the power series for yp x is the same as those of power series for
yp x and yp x. In this paper, the convergence region will denote the maximum open set
where the relevant power series converges. Hence, the power series expression for xy p x β − xyp x − αyp x has the same convergence region as that of yp x. This implies that
yp x is well defined on Iρ0 and so does for yx in 2.6 because yh x converges for all
x ∈ I∞ under our hypotheses for α and β see above Theorem 2.1.
Since every solution to 1.4 can be expressed as a sum of a solution yh x of the
homogeneous equation and a particular solution yp x of the inhomogeneous equation, every
solution of 1.4 is certainly in the form of 2.6.
Remark 2.2. We fix α 1 and β 10/3, and we define
a0 10
,
3
am 1 4m2 − 6m − 3
3m2 m 1
2.11
Journal of Inequalities and Applications
5
for every m ∈ N. Then, since limm → ∞ am /am−1 1, there exists a real number μ > 1 such that
m − 1!β am 10 · 13 · 16 · · · 3m 4
3m 7 am
m
m
·
am−1 ·
·
m−1
3m
am−1 m 1
αm1
m3
m − 1! β m−1 am−1 3m 7 am
m
·
·
·
3m
am−1 m 1
αm
m − 1! β m−1 am−1
≤μ
αm
m−1
i! β i ai ≤ μ
i0 αi1 2.12
for all sufficiently large integers m. Hence, the sequence {am } satisfies condition 2.5 for all
sufficiently large integers m.
3. Hyers-Ulam Stability of 2.1
In this section, let α and β be real constants and assume that ρ is a constant with 0 < ρ ≤ ∞.
For a given K ≥ 0, let us denote CK the set of all functions y : Iρ → C with the properties a
and b:
a yx is represented by a power series
least ρ;
∞
m0
bm xm whose radius of convergence is at
∞
m
m
b it holds true that ∞
m0 |am x | ≤ K|
m0 am x | for all x ∈ Iρ , where am m βm 1bm1 − m αbm for each m ∈ N0 .
m
It should be remarked that the power series ∞
m0 am x in b has the same radius of
∞
m
convergence as that of m0 bm x given in a.
In the following theorem, we will prove a local Hyers-Ulam stability of the Kummer’s
equation under some additional conditions. More precisely, if an analytic function satisfies
some conditions given in the following theorem, then it can be approximated by a
“combination” of Kummer functions such as Mα, β, x and M1 α − β, 2 − β, x see the
first part of Section 2.
Theorem 3.1. Let α and β be real constants such that β /
∈ Z and neither α nor 1α−β is a nonpositive
m
integer. Suppose a function y : Iρ → C is representable by a power series ∞
m0 bm x whose radius
of convergence is at least ρ > 0. Assume that there exist nonnegative constants μ /
0 and ν satisfying
the condition
m−1
m 1!β am m − 1!β am i! β i ai m
m
≤ μ
≤ ν
i0 αi1 αm1
αm1
3.1
6
Journal of Inequalities and Applications
for all m ∈ N0 , where am m βm 1bm1 − m αbm . Indeed, it is sufficient for the first
inequality in 3.1 to hold true for all sufficiently large integers m. Let us define ρ0 min{ρ, 1/μ}. If
y ∈ CK and it satisfies the differential inequality
xy x β − x y x − αyx ≤ ε
3.2
for all x ∈ Iρ0 and for some ε ≥ 0, then there exists a solution yh : I∞ → C of the Kummer’s equation
2.1 such that
⎧ ν 2α − 1
⎪
·
Kε
⎪
⎪
⎪
α
⎨μ
yx − yh x ≤
m −1
⎪
0
m 1 m 2 m0 1
⎪
ν
⎪
−
⎪
Kε
⎩
μ m0 m α m 1 α m0 α
for α > 1,
3.3
for α ≤ 1,
for any x ∈ Iρ0 , where m0 max{0, −α
}.
Proof. By the definition of am , we have
xy x β − x y x − αyx
∞
m β m 1bm1 − m αbm xm
m0
∞
3.4
am xm
m0
for all x ∈ Iρ . So by 3.2 we have
∞
m
am x ≤ ε
m0
3.5
for any x ∈ Iρ0 . Since y ∈ CK , this inequality together with b yields
∞
m
|am x | ≤ K am x ≤ Kε
m0
m0
∞
m
3.6
for each x ∈ Iρ0 .
By Abel’s formula see 18, Theorem 6.30, we have
m 1
|am x |
m α
m0
n n
m m 1 m 2 i n 2 i ai x ai x m α − m 1 α
n 1 α
n
m
i0
m0
i0
3.7
Journal of Inequalities and Applications
7
for any x ∈ Iρ0 and n ∈ N. With m0 max{0, −α
} −α
is the ceiling of −α, we know that
if α > 1, then
m2
m1
<
mα m1α
m1
m2
if α ≤ 1, then
≥
mα m1α
for m ≥ 0;
3.8
for m ≥ m0 .
Due to 3.4, it follows from Theorem 2.1 and 2.6 that there exists a solution yh x of
the Kummer’s equation 2.1 such that
∞ m−1
i!αm β i ai
yx yh x xm
β m
m!α
i1
m0 i0
3.9
for all x ∈ Iρ0 . By using 3.1, 3.6, 3.7, and 3.8, we can estimate
∞ m−1
αm1
i! β i ai yx − yh x ≤
am xm m 1 m α m 1! β m am i0 αi1 m0
n
m 1
ν
≤ lim
|am xm |
μ n → ∞m0
m α
⎧
n
n2 ⎪
m
1
m
2
ν
⎪
⎪
lim Kε
−
Kε
⎪
⎪
⎪
μn→∞
n 1 α m0
m1α mα
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
0 −1
n 2 m
m 1 m 2 ν
≤
−
Kε
lim
Kε
⎪
n 1 α
m α m 1 α ⎪
μn→∞
⎪
m0
⎪
⎪
⎪
⎪
⎪
n
⎪
⎪
m
2
m
1
⎪
⎪
−
Kε
⎪
⎩
mα m1α
mm0
⎧ ν 2α − 1
⎪
·
Kε
⎪
⎪
⎪μ
α
⎨
m −1
⎪
0
m 1 m 2 m0 1
⎪
ν
⎪
−
⎪
Kε
⎩
μ m0 m α m 1 α m0 α
for α > 1,
for α ≤ 1
for α > 1,
for α ≤ 1
3.10
for all x ∈ Iρ0 .
We now assume a stronger condition, in comparison with 3.1, to approximate the
given function yx by a solution yh x of the Kummer’s equation on a larger punctured
interval.
Corollary 3.2. Let α and β be real constants such that β /
∈ Z and neither α nor 1α−β is a nonpositive
m
which
integer. Suppose a function y : I∞ → C is representable by a power series ∞
m0 bm x
8
Journal of Inequalities and Applications
converges for all x ∈ I∞ . For every m ∈ N0 , let us define am m βm 1bm1 − m αbm .
Moreover, assume that
m − 1! β m am
lim
0,
m→∞
αm1
∞
i! β i ai 0<
<∞
i0 αi1 3.11
and there exists a nonnegative constant ν satisfying
m 1!β am m−1
i! β i ai m
≤ ν
i0 αi1 αm1
3.12
for all m ∈ N0 . If y ∈ CK and it satisfies the differential inequality 3.2 for all x ∈ I∞ and for some
ε ≥ 0, then there exists a solution yn : I∞ → C of the Kummer’s equation 2.1 such that
⎧
2α − 1
⎪
⎪
Kε
ν·
⎪
⎪
α
⎨
yx − yn x ≤
m −1
0
⎪
m 1 m 2 m0 1
⎪
⎪
⎪
⎩ν
m α − m 1 α m α Kε
0
m0
for α > 1,
3.13
for α ≤ 1
for any x ∈ In , where m0 max{0, −α
} and n is a sufficiently large integer.
Proof. In view of 3.11 and 3.12, we can choose a sufficiently large integer n with
m − 1!β am 1 m−1
i! β i ai ν m 1! β m am m
≤ ≤ ,
n i0 αi1 n αm1
αm1
3.14
where the first inequality holds true for all sufficiently large m, and the second one holds true
for all m ∈ N0 .
If we define ρ0 n, then Theorem 3.1 implies that there exists a solution yn : I∞ → C
of the Kummer’s equation such that the inequality given for |yx − yn x| holds true for any
x ∈ In .
4. An Example
We fix α 1, β 10/3, ε > 0, and 0 < ρ < 1. And we define
b0 0,
bm ε 1
·
s m2
4.1
Journal of Inequalities and Applications
9
for all m ∈ N, where we set s 5/32 − ρ/1 − ρ. We further define
yx ∞
bm xm
4.2
m0
for any x ∈ Iρ .
Then, we set am m βm 1bm1 − m αbm , that is,
10 ε
· ,
a0 3 s
4m2 − 6m − 3
1
3m2 m 1
am ε 5 ε
≤ ·
s 3 s
4.3
for every m ∈ N. Obviously, all am s are positive, and the sequence {am } is strictly monotone
decreasing, from the 4th term on, to ε/s. More precisely, a0 > a1 < a2 < a3 < a4 > a5 >
a6 > · · · .
Since
a0 10 ε 1 ε 41 ε
· > · · a1 a3 ,
3 s 6 s 36 s
4.4
we get
∞
m
am x a0 a1 x a2 x2 a3 x3 a4 x4 a5 x5 a6 x6 a7 x7 · · · m0
≥ a0 a1 x a2 x2 a3 x3 4.5
≥ a0 − a1 − a3
73 ε
·
36 s
for each x ∈ Iρ and
∞
m
|am x | ≤
m0
∞
m
am ρ ≤
m0
∞
5 m
10 ρ
3 m1 3
ε
ε
s
4.6
for all x ∈ Iρ . Hence, we obtain
∞
m
|am x | ≤ K am x m0
m0
∞
m
for any x ∈ Iρ , where K 60/73 · 2 − ρ/1 − ρ, implying that y ∈ CK .
4.7
10
Journal of Inequalities and Applications
We will now show that {am } satisfies condition 3.1. For any m ∈ N, we have
m−1
m−1
10 · 13 · 16 · · · 3i 7
i! β i ai ai
a0 i0 αi1 i 13i
i1
10 · 13 · 16 · · · 3i 7 5 ε
10 m−1
,
≤
·
3
3 s
i 13i
i1
m 1!β am 10 · 13 · 16 · · · 3m 7 1 ε
m
· · ,
≥
3m
6 s
αm1
4.8
since limm → ∞ am ε/s.
It follows from 4.8 that
m−1
10 · 13 · 16 · · · 3i 7 1 ε
1 m−1
i! β i ai ·
≤ 10
i0 αi1 3 i1
6 s
i 13i
1 ε
3m−i
1
1 10 · 13 · · · 3m 7 m−1
·
·
10
3
3m
3i 10 · · · 3m 7 i 1 6 s
i1
1
1 ε
1 10 · 13 · 16 · · · 3m 7 m−1
·
≤ 10
2 6 s
3
3m
i1 i 1
10 · 13 · 16 · · · 3m 7 1
1
ε
≤ 10
−
1
ζ2
3m
10 6
s
4.9
5π 2 − 12 10 · 13 · 16 · · · 3m 7 1 ε
·
· ·
3
3m
6 s
5π 2 − 12 m 1! β m am ≤
.
3
αm1
We know that the inequality 4.9 is also true for m 0.
On the other hand, in view of Remark 2.2, there exists a constant μ > 1 such that
inequality 2.12 holds true for all sufficiently large integers m. By 2.12 and 4.9, we
conclude that {am } satisfies condition 3.1 with ν 5π 2 − 12μ/3.
Finally, it follows from 4.6 that
∞
∞
m
xy x β − x y x − αyx am x ≤
|a xm | ≤ ε
m0
m0 m
for all x ∈ Iρ0 with ρ0 min{ρ, 1/μ}.
4.10
Journal of Inequalities and Applications
11
According to Theorem 3.1, there exists a solution yh : I∞ → C of the Kummer’s
equation 2.1 such that
2
yx − yh x ≤ 100π − 240 · 2 − ρ ε
73
1−ρ
4.11
for all x ∈ Iρ0 .
Acknowledgments
The author would like to express his cordial thanks to the referee for his/her useful
comments. This work was supported by National Research Foundation of Korea Grant
funded by the Korean Government No. 2009-0071206.
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