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Hindawi Publishing Corporation

Journal of Applied Mathematics

Volume 2012, Article ID 691981, 12 pages

doi:10.1155/2012/691981

Research Article

Stability of Jensen-Type Functional

Equations on Restricted Domains in a Group and

Their Asymptotic Behaviors

Jae-Young Chung,1Dohan Kim,2and John Michael Rassias3

1Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea

2Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea

3Section of Mathematics and Informatics, Pedagogical Department E. E.,

National and Kapodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi,

15342 Athens, Greece

Correspondence should be addressed to Jae-Young Chung, jychung@kunsan.ac.kr

Received 20 June 2011; Accepted 5 September 2011

Academic Editor: Junjie Wei

Copyright q 2012 Jae-Young Chung et al. 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 consider the Hyers-Ulam stability problems for the Jensen-type functional equations in general

restricted domains. The main purpose of this paper is to find the restricted domains for which the

functional inequality satisfied in those domains extends to the inequality for whole domain. As

consequences of the results we obtain asymptotic behavior of the equations.

1. Introduction

The Hyers-Ulam stability problems of functional equations was originated by Ulam in 1960

when he proposed the following question ?1?.

Let f be a mapping from a group G1to a metric group G2with metric d?·,·? such that

d?f?xy?,f?x?f?y??≤ ε.

Then does there exist a group homomorphism h and δ?> 0 such that

?1.1?

d?f?x?,h?x??≤ δ?

?1.2?

for all x ∈ G1?

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2Journal of Applied Mathematics

One of the first assertions to be obtained is the following result, essentially due to

Hyers ?2?, that gives an answer for the question of Ulam.

Theorem 1.1. Suppose that S is an additive semigroup, Y is a Banach space, ? ≥ 0, and f : S → Y

satisfies the inequality

??f?x ? y?− f?x? − f?y???≤ ?

?1.3?

for all x, y ∈ S. Then there exists a unique function A : S → Y satisfying

A?x ? y?? A?x? ? A?y?

for which

?1.4?

??f?x? − A?x???≤ ?

?1.5?

for all x ∈ S.

We call the functions satisfying ?1.4? additive functions. Perhaps Aoki in 1950 was the

first author treating the generalized version of Hyers’ theorem ?3?. Generalizing Hyers’ result

he proved that if a mapping f : X → Y between two Banach spaces satisfies

??f?x ? y?− f?x? − f?y???≤ Φ?x,y?for x,y ∈ X

with Φ?x,y? ? ???x?p? ?y?p? ?? ≥ 0, 0 ≤ p < 1?, then there exists a unique additive function

A : X → Y suchthat?f?x?−A?x?? ≤ 2??x?p/?2−2p?forallx ∈ X.In1951Bourgin?4,5?stated

that if Φ is symmetric in ?x? and ?y? with?∞

all x ∈ X. Unfortunately, there was no use of these results until 1978 when Rassias ?6? dealt

with the inequality of Aoki ?3?. Following Rassias’ result, a great number of papers on the

subject have been published concerning numerous functional equations in various directions

?6–15?. Among the results, stability problem in a restricted domain was investigated by Skof,

who proved the stability problem of inequality ?1.3? in a restricted domain ?16, 17?. Develop-

ing this result, Jung, Rassias, and M. J. Rassias considered the stability problems in restricted

domains for some functional equations including the Jensen functional equation ?9? and

Jensen-type functional equations ?13?. We also refer the reader to ?18–27? for some related

results on Hyers-Ulam stabilities in restricted conditions. The results can be summarized as

follows. Let X and Y be a real normed space and a real Banach space, respectively. For fix-

ed d ≥ 0, if f : X → Y satisfies the functional inequalities ?such as that of Cauchy, quadratic,

Jensen, and Jensen type? for all x,y ∈ X with ?x? ? ?y? ≥ d ?which is the case where the

inequalities are given by two indeterminate variables x and y?, the inequalities hold for all

x,y ∈ X. Following the approach in ?28? we consider the Jensen-type equation in various

restricted domains in an Abelian group. As applications, we obtain the stability problems for

the above equations in more general restricted domains than that of the form {?x,y? ∈ X :

?x???y? ≥ d}, which generalizes and refines the stability theorems in ?13?. As an application

we obtain asymptotic behaviors of the equations.

?1.6?

j?1Φ?2jx,2jx?/2j< ∞ for each x ∈ X, then there

existsauniqueadditivefunctionA : X → Y suchthat?f?x?−A?x?? ≤?∞

j?1Φ?2jx,2jx?/2jfor

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Journal of Applied Mathematics3

2. Stability of Jensen-Type Functional Equations

Throughout this section, we denote by G, X, and Y, an Abelian group, a real normed space,

and a Banach space, respectively. In this section we consider the Hyers-Ulam stability of the

Jensen and Jensen-type functional inequalities for the functions f : G → Y

??f?x ? y?? f?x − y?− 2f?x???≤ ?,

?2.1?

??f?x ? y?− f?x − y?− 2f?y???≤ ?

?2.2?

in restricted domains U ⊂ G × G.

Inequalities ?2.1? and ?2.2? were previously treated by J. M. Rassias and M. J. Rassias

?13?, who proved the Hyers-Ulam stability of the inequalities in the restricted domain U ?

{?x,y? : ?x? ? ?y? ≥ d}, d ≥ 0, for the functions f : X → Y:

Theorem 2.1. Let d ≥ 0 and ? > 0 be fixed. Suppose that f : X → Y satisfies the inequality

??f?x ? y?? f?x − y?− 2f?x???≤ ?

for all x,y ∈ X, with ?x? ? ?y? ≥ d. Then there exists a unique additive function A : X → Y such

that

?2.3?

??f?x? − A?x? − f?0???≤5

2?

?2.4?

for all x ∈ X.

Theorem 2.2. Let d ≥ 0 and ? > 0 be fixed. Suppose that f : X → Y satisfies the inequality

??f?x ? y?− f?x − y?− 2f?y???≤ ?

for all x,y ∈ X, with ?x? ? ?y? ≥ d and

??f?x? ? f?−x???≤ 3?

for all x ∈ X, with ?x? ≥ d. Then there exists a unique additive function A : X → Y such that

?2.5?

?2.6?

??f?x? − A?x???≤33

2?

?2.7?

for all x ∈ X.

We use the following usual notations. We denote by G × G ? {?a1,a2? : a1,a2 ∈ G}

the product group; that is, for a ? ?a1,a2?, b ? ?b1,b2? ∈ G × G, we define a ? b ? ?a1? b1,

a2? b2?, a − b ? ?a1− b1,a2− b2?. For a subset H of G × G and a,b ∈ G × G, we define

a ? H ? {a ? h : h ∈ H}.

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For given x,y ∈ G we denote by Px,y, Qx,ythe subsets of points of the forms ?not nec-

essarily distinct? in G × G, respectively,

Px,y???0,0?,?x,−x?,?y,y?,?x ? y,−x ? y??,

Qx,y???−x,x?,?y,y?,?−x ? y,x ? y??.

The set Px,ycan be viewed as the vertices of rectangles in G × G, and Qx,ycan be viewed as a

subset of the vertices of rectangles in G × G.

Definition 2.3. Let U ⊂ G×G. One introduces the following conditions ?J1? and ?J2? on U. For any

x,y ∈ G, there exists a z ∈ G such that

?0,z? ? Px,y???0,z?,?x,−x ? z?,?y,y ? z?,?x ? y,−x ? y ? z??⊂ U,

?J2?

?2.8?

?J1?

?z,0? ? Qx,y???−x ? z,x?,?y ? z,y?,?−x ? y ? z,x ? y??⊂ U,

?2.9?

respectively.

The sets ?0,z??Px,y, ?z,0??Qx,ycan be understood as the translations of Px,yand Qx,y

by ?0,z? and ?z,0?, respectively.

There are many interesting examples of the sets U satisfying some of the conditions

?J1? and ?J2?. We start with some trivial examples.

Example 2.4. Let G be a real normed space. For d ≥ 0, x0,y0∈ G, let

U ???x,y?∈ G × G : k?x? ? s??y??≥ d?,

V ???x,y?∈ G × G :??kx ? sy??≥ d?.

?2.10?

Then U satisfies ?J1? if s > 0, ?J2? if k > 0 and V satisfies ?J1? if s/ ?0, ?J2? if k/ ?0.

Example 2.5. Let G be a real inner product space. For d ≥ 0, x0,y0∈ G

U ???x,y?∈ G × G : ?x0,x? ??y0,y?≥ d?.

Then U satisfies ?J1? if y0 / ?0, ?J2? if x0 / ?0.

?2.11?

Example 2.6. Let G be the group of nonsingular square matrices with the operation of matrix

multiplication. For k,s ∈ R, δ,d ≥ 0, let

?

V ?

?P1,P2? ∈ G × G : |detP1|k|detP2|s≥ d

U ?

?P1,P2? ∈ G × G : |detP1|k|detP2|s≤ δ

?

?

,

?

.

?2.12?

Then both U and V satisfy ?J1? if s/ ?0, ?J2? if k/ ?0.

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Journal of Applied Mathematics5

In the following one can see that if Px,yand Qx,yare replaced by arbitrary subsets of

four points ?not necessarily distinct? in G×G, respectively, then the conditions become strong-

er; that is, there are subsets U1and U2which satisfy the conditions ?J1? and ?J2?, respectively,

but U1and U2fail to fulfill the following conditions ?2.13? and ?2.14?, respectively. For any

subset {X1,X2,X3,X4} of points ?not necessarily distinct? in G × G, there exists a z ∈ G such

that

?0,z? ? {X1,X2,X3,X4} ⊂ U1,

?z,0? ? {X1,X2,X3,X4} ⊂ U2,

?2.13?

?2.14?

respectively.

Now we give examples of U1and U2which satisfy ?J1? and ?J2?, respectively, but not

?2.13? and ?2.14?, respectively.

Example 2.7. Let G ? Z be the group of integers. Enumerate

Z × Z ? {?a1,b1?,?a2,b2?,...,?an,bn?,...} ⊂ R2

?2.15?

such that

|a1| ? |b1| ≤ |a2| ? |b2| ≤ ··· ≤ |an| ? |bn| ≤ ··· ,

?2.16?

and let Pn? {?0,0?,?an,−an?,?bn,bn?,?an? bn,−an? bn?}, n ? 1,2,.... Then it is easy to see

that U ??∞

for any choices of ?xn,yn? ∈ Pn??0,2n?, n ? 1,2,..., we have ym−yn> |xm−xn| for all m > n,

m,n ? 1,2,....Thus,if?0,z??P ⊂ U forsomez ∈ Z,thenP??0,z? ⊂ ?0,2n??Pnforsomen ∈ N.

Thus, it follows from the condition q2−q1≤ |p2−p1| that the line segment joining the points of

P ? ?−z,z? intersects the line x ? 0 in R2, which contradicts the condition p1p2> 0. Similarly,

let Qn? {?−an,an?,?bn,bn?,?−an?bn,an?bn?}. Then it is easy to see that U ??∞

n?1??0,2n? ? Pn? satisfies the condition ?J1?. Now let P ? {?p1,q1?,?p2,q2?} ⊂ Z × Z

with |q2− q1| ≤ |p2− p1|, p1p2> 0. Then ?0,z? ? P is not contained in U for all z ∈ Z. Indeed,

n?1??2n,0??Qn?

satisfies the condition ?J2? but not ?2.14?.

Theorem 2.8. Let U ⊂ G×G satisfy the condition ?J1? and ? ≥ 0. Suppose that f : G → Y satisfies

?2.1? for all ?x,y? ∈ U. Then there exists an additive function A : G → Y such that

??f?x? − A?x? − f?0???≤ 2?

for all x ∈ G.

Proof. For given x,y ∈ G, choose a z ∈ G such that ?0,z? ? Px,y⊂ U. Replacing x by x ? y, y

by −x ? y ? z; x by x, y by −x ? z; x by y, y by y ? z; x by 0, y by z in ?2.1?, respectively, we

have

?2.17?

??f?2y ? z?? f?2x − z? − 2f?x ? y???≤ ?,

??f?z? ? f?2x − z? − 2f?x???≤ ?,

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??f?2y ? z?? f?−z? − 2f?y???≤ ?,

??f?z? ? f?−z? − 2f?0???≤ ?.

?2.18?

From ?2.18?, using the triangle inequality and dividing the result by 2, we have

??f?x ? y?− f?x? − f?y?? f?0???≤ 2?

?2.19?

for all x,y ∈ G. From ?2.19?, using Theorem 1.1, we get the result.

Let d ≥ 0, s ∈ R, and let U ? {?x,y? : ?x? ? s?y? ≥ d}. Then U satisfies the condition

?J1?. Thus, as a direct consequence of Theorem 2.8, we obtain the following ?cf. Theorem 2.1?.

Corollary 2.9. Let d ≥ 0, s ∈ R. Suppose that f : X → Y satisfies inequality ?2.1? for all x,y ∈ X,

with ?x? ? s?y? ≥ d. Then there exists a unique additive function A : X → Y such that

??f?x? − A?x? − f?0???≤ 2?

for all x ∈ X.

Theorem 2.10. Let U ⊂ G×G satisfy the condition ?J2? and ? ≥ 0. Suppose that f : G → Y satisfies

?2.2? for all ?x,y? ∈ U. Then there exists a unique additive function A : G → Y such that

?2.20?

??f?x? − A?x???≤3

2?

?2.21?

for all x ∈ G.

Proof. For given x,y ∈ G, choose z ∈ G such that ?z,0??Qx,y⊂ U. Replacing x by −x?y?z, y

by x ? y; x by −x ? z, y by x; x by y ? z, y by y in ?2.2?, respectively, we have

??f?2y ? z?− f?−2x ? z? − 2f?x ? y???≤ ?,

??f?2y ? z?− f?z? − 2f?y???≤ ?.

From ?2.22?, using the triangle inequality and dividing the result by 2, we have

??f?z? − f?−2x ? z? − 2f?x???≤ ?,

?2.22?

??f?x ? y?− f?x? − f?y???≤3

2?.

?2.23?

Now by Theorem 1.1, we get the result.

Let d ≥ 0, k ∈ R, and let U ? {?x,y? : k?x? ? ?y? ≥ d}. Then U satisfies the condition

?J2?. Thus, as a direct consequence of Theorem 2.10, we generalize and refine Theorem 2.2 as

follows.

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Corollary 2.11. Let d ≥ 0, k ∈ R. Suppose that f : X → Y satisfies inequality ?2.2? for all x,y,

with k?x? ? ?y? ≥ d. Then there exists a unique additive function A : X → Y such that

??f?x? − A?x???≤3

for all x ∈ X.

Remark 2.12. Corollary 2.11 refines Theorem 2.2 in both the bounds and the condition ?2.6?.

2?

?2.24?

Now we discuss other possible restricted domains. We assume that G is a 2-divisible

Abelian group. For given x,y ∈ G, we denote by Rx,y, Sx,y⊂ G × G,

?

?

Rx,y?

?x,x?,?x,y?,

?x,x?,?y,x?,

?x − y

?x − y

2

,x − y

2

?

?

,

?x − y

?−x ? y

2

,−x ? y

2

??

??

,

Sx,y?

2

,x − y

2

,

2

,x − y

2

.

?2.25?

One can see that Rx,yand Sx,yconsist of the vertices of parallelograms in G×G, respectively.

Definition 2.13. Let U ⊂ G × G. One introduces the following conditions ?J3?,?J4? on U. For any

x,y ∈ G, there exists a z ∈ G such that

?

?x − y

?

?−x ? y

?J3??z,−z? ? Rx,y?

?x ? z,x − z?,?x ? z,y − z?,

? z,−x ? y

2

?x − y

2

? z,x − y

2

− z

?

,

2

− z

??

⊂ U,

?J4??z,−z? ? Sx,y?

?x ? z,x − z?,?y ? z,x − z?,

? z,x − y

?x − y

2

? z,x − y

2

− z

?

,

22

− z

??

⊂ U,

?2.26?

respectively.

Example 2.14. Let G be a real normed space. For k,s,d ∈ R, let

U ???x,y?∈ G × G : k?x? ? s??y??≥ d?,

V ???x,y?∈ G × G :??kx ? sy??≥ d?.

?2.27?

Then U satisfies ?J3? and ?J4? if k ? s > 0, and V satisfies ?J3? and ?J4? if k/ ?s.

Example 2.15. Let G be a real inner product space. For d ≥ 0, x0,y0∈ G,

U ???x,y?∈ G × G : ?x0,x? ??y0,y?≥ d?.

Then U satisfies ?J3?,?J4? if x0 / ?y0.

?2.28?

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Example 2.16. Let G be the group of nonsingular square matrices with the operation of matrix

multiplication. For k,s ∈ R, δ,d ≥ 0, let

?

V ?

?P1,P2? ∈ G × G : |detP1|k|detP2|s≥ d

U ?

?P1,P2? ∈ G × G : |detP1|k|detP2|s≤ δ

?

?

,

?

.

?2.29?

Then U and V satisfy both ?J3? and ?J4? if k/ ?s.

From now on, we assume that G is a 2-divisible Abelian group.

Theorem 2.17. Let U ⊂ G×G satisfy the condition ?J3? and ? ≥ 0. Suppose that f : G → Y satisfies

?2.1? for all ?x,y? ∈ U. Then there exists a unique additive function A : G → Y such that

??f?x? − A?x? − f?0???≤ 4?

for all x ∈ G.

Proof. For given x,y ∈ G, choose a z ∈ G such that ?z,−z? ? Rx,y⊂ U. Replacing x by x ? z, y

by x −z; x by x ?z, y by y −z; x by ?x −y?/2?z, y by ?x −y?/2−z; x by ?x −y?/2?z, y by

?−x ? y?/2 − z in ?2.1?, respectively, we have

??f?2x? ? f?2z? − 2f?x ? z???≤ ?,

????f?x − y?? f?2z? − 2f

?2.30?

??f?x ? y?? f?x − y ? 2z?− 2f?x ? z???≤ ?,

?x − y

?x − y

2

? z

?????≤ ?,

? z

????f?0? ? f?x − y ? 2z?− 2f

2

?????≤ ?.

?2.31?

From ?2.31?, using the triangle inequality, we have

??f?2x? − f?x ? y?− f?x − y?? f?0???≤ 4?

?2.32?

for all x,y ∈ G. Replacing x by ?x ? y/2?, y by ?x − y/2? in ?2.32?, we have

??f?x ? y?− f?x? − f?y?? f?0???≤ 4?

for all x,y ∈ G. From ?2.33?, using Theorem 1.1, we get the result.

Let d ≥ 0, k,s ∈ R with k ? s > 0, and let U ? {?x,y? : k?x? ? s?y? ≥ d}. Then U sat-

isfies the conditions ?J3? and ?J4?. Thus, as a direct consequence of Theorem 2.17 we gener-

alize Theorem 2.1 as follows.

?2.33?

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Journal of Applied Mathematics9

Corollary 2.18. Let d ≥ 0, k,s ∈ R with k ?s > 0. Suppose that f : X → Y satisfies the inequality

?2.1? for all x,y, with k?x? ? s?y? ≥ d. Then there exists a unique additive function A : X → Y

such that

??f?x? − A?x? − f?0???≤ 4?

?2.34?

for all x ∈ X.

Theorem 2.19. Let U ⊂ G×G satisfy the condition ?J4? and ? ≥ 0. Suppose that f : G → Y satisfies

?2.2? for all ?x,y? ∈ U. Then there exists a unique additive function A : G → Y such that

??f?x? − A?x???≤ 4?

?2.35?

for all x ∈ G.

Proof. For given x,y ∈ G, choose z ∈ G such that ?z,−z? ? Sx,y⊂ U. Replacing x by x ? z, y

by x − z; x by y ? z, y by x − z; x by ?x − y?/2 ? z, y by ?x − y?/2 − z; x by ?−x ? y?/2 ? z, y

by ?x − y?/2 − z in ?2.2?, respectively, we have

??f?2x? − f?2z? − 2f?x − z???≤ ?,

????f?x − y?− f?2z? − 2f

??f?x ? y?− f?−x ? y ? 2z?− 2f?x − z???≤ ?,

?x − y

?x − y

2

? z

?????≤ ?,

????f?0? − f?−x ? y ? 2z?− 2f

2

? z

?????≤ ?.

?2.36?

From ?2.36?, using the triangle inequality, we have

??f?2x? − f?x ? y?− f?x − y?? f?0???≤ 4?

?2.37?

for all x,y ∈ G. Replacing x by ?x ? y?/2, y by ?x − y?/2 in ?2.37? and using Theorem 1.1, we

get the result.

As a direct consequence of Theorem 2.19, we have the following.

Corollary 2.20. Let d ≥ 0, k,s ∈ R with k ?s > 0. Suppose that f : X → Y satisfies the inequality

?2.2? for all x,y, with k?x? ? s?y? ≥ d. Then there exists a unique additive function A : X → Y

such that

??f?x? − A?x???≤ 4?

?2.38?

for all x ∈ X.

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3. Asymptotic Behavior of the Equations

In this section we discuss asymptotic behaviors of the equations which gives refined versions

of the results in ?13?.

Using Theorems 2.8 and 2.17, we have the following ?cf. ?13??.

Theorem 3.1. Let U satisfy ?J1? or ?J3?. Suppose that f : X → Y satisfies the asymptotic condition

??f?x ? y?? f?x − y?− 2f?x???−→ 0

as ?x? ? ?y? → ∞,?x,y? ∈ U. Then there exists a unique additive function A : X → Y such that

?3.1?

f?x? ? A?x? ? f?0?

?3.2?

for all x ∈ X.

Proof. By the condition ?3.1?, for each n ∈ N, there exists dn> 0 such that

??f?x ? y?? f?x − y?− 2f?x???≤1

n

?3.3?

for all ?x,y? ∈ U with ?x???y? ≥ dn. Let U0? U∩{?x,y? : ?x???y? ≥ dn}. Then U0satisfies

both the conditions ?J1? and ?J3?. By Theorems 2.8 and 2.17, there exists a unique additive

function An: X → Y such that

??f?x? − An?x? − f?0???≤2

for all x ∈ X. Putting n ? m in ?3.4? and using the triangle inequality, we have

n

or

4

n

?3.4?

?An?x? − Am?x?? ≤ 8?3.5?

for all x ∈ X. Using the additivity of An, Am, we have An ? Amfor all n,m ∈ N. Letting

n → ∞ in ?3.4?, we get the result.

Corollary 3.2. Let k,s ∈ R satisfy one of the conditions: s > 0, k ? s > 0. Suppose that f : X → Y

satisfies the condition

??f?x ? y?? f?x − y?− 2f?x???−→ 0?3.6?

as k?x? ? s?y? → ∞. Then there exists a unique additive function A : X → Y such that

f?x? ? A?x? ? f?0?

?3.7?

for all x ∈ X.

Using Theorems 2.10 and 2.19, we have the following ?cf. ?13??.

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Journal of Applied Mathematics 11

Theorem 3.3. Let U satisfy ?J2? or ?J4?. Suppose that f : X → Y satisfies the condition

??f?x ? y?− f?x − y?− 2f?y???−→ 0

as ?x? ? ?y? → ∞,?x,y? ∈ U. Then f is an additive function.

Corollary 3.4. Let k,s ∈ R satisfy one of the conditions: k > 0, k ? s > 0. Suppose that f : X → Y

satisfies the condition

?3.8?

??f?x ? y?− f?x − y?− 2f?y???−→ 0?3.9?

as k?x? ? s?y? → ∞. Then f is an additive function.

Acknowledgments

The first author was supported by Basic Science Research Program through the National Re-

search Foundation of Korea ?NRF? funded by the Ministry of Education, Science and Techno-

logy ?MEST? ?no. 2011-0003898?, and the second author was partially supported by the Re-

search Institute of Mathematics, Seoul National University.

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