Stability of a Jensen Type Logarithmic

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doi:10.1155/2010/432796

Research Article

Stability of a Jensen Type Logarithmic

Functional Equation on Restricted Domains and Its Asymptotic Behaviors

Jae-Young Chung

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

Correspondence should be addressed to Jae-Young Chung,[email protected] Received 28 June 2010; Revised 30 October 2010; Accepted 25 December 2010 Academic Editor: Roderick Melnik

Copyrightq2010 Jae-Young Chung. 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.

LetRbe the set of positive real numbers,Ba Banach space,f :R → B,and >0,p, q, P, Q∈ RwithpqP Q /0. We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequalityfx^py^q−P fx−Qfy ≤in restricted domains of the form{x, y: x >0, y >

0, x^ky^s ≥d}for fixedk, s ∈Rwithk /0 ors /0 andd > 0. As consequences of the results we obtain asymptotic behaviors of the inequality asx^ky^s → ∞.

1. Introduction

The stability problems of functional equations have been originated by Ulam in 1940see 1. 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 thatS, is an additive semigroup,Bis a Banach space,≥0, andf :S → Bsatisfies the inequality

f

xy

−fx−f

y≤ 1.1

for allx, y∈S. Then there exists a unique functionA:S → Bsatisfying

A xy

Ax A y

1.2

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for which

fx−Ax≤ 1.3

for allx∈S.

In 1950-1951 this result was generalized by the authors Aoki3and Bourgin4,5.

Unfortunately, no results appeared until 1978 when Th. M. Rassias generalized the Hyers’

result to a new approximately linear mappings 6. Following the Rassias’ result, a great number of the papers on the subject have been published concerning numerous functional equations in various directions 6–16. For more precise descriptions of the Hyers-Ulam stability and related results, we refer the reader to the paper of Moszner 17. Among the results, the stability problem in a restricted domain was investigated by Skof, who proved the stability problem of the inequality 1.1 in a restricted domain 16. Developing this result, Jung considered the stability problems in restricted domains for the Jensen functional equation11and Jensen type functional equations14. The results can be summarized as follows: letXandY be a real normed space and a real Banach space, respectively. For fixed d > 0, iff :X → Y satisfies the functional inequalitiessuch as that of Cauchy, Jensen and Jensen type, etc.for allx, y∈Xwithxy ≥d, the inequalities hold for allx, y∈X. We also refer the reader to18–26for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions.

Throughout this paper, we denote byR the set of positive real numbers,Ba Banach space,f :R → B, andp, q, P, Q∈ RwithpqP Q /0. We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality

f

x^py^q

−P fx−Qf

y≤ 1.4

in the restricted domains of the formU_k,s{x, y:x >0, y >0, x^ky^s≥d}for fixedk, s∈R withk /0 ors /0, andd > 0. As a result, we prove that if the inequality1.4holds for all x, y∈Uk,s, there exists a unique functionL:R → Bsatisfying

L xy

−Lx−L y

0, x, y >0 1.5

for which

fx−Lx−f1≤4 1.6

for allx >0 ifk/p /s/q,

fx−Lx−f1≤ 4

|P| 1.7

for allx >0 ifs /0, and

fx−Lx−f1≤ 4

|Q| 1.8

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for allx >0 ifk /0. As a consequence of the result we obtain the stability of the inequality

f

pxqy

−P fx−Qf

y≤ 1.9

in the restricted domains of the form{x, y∈R² :kxsy≥d}for fixedk, s∈Rwithk /0 ors /0, andd∈R. Also we obtain asymptotic behaviors of the inequalities1.4and1.9as x^ky^s → ∞andkxsy → ∞, respectively.

2. Hyers-Ulam Stability in Restricted Domains

We call the functions satisfying 1.5 logarithmic functions. As a direct consequence of Theorem 1.1, we obtain the stability of the logarithmic functional equation, viewingR,× as a multiplicative groupsee also the result of Forti9.

Theorem A. Suppose thatf:R → B,≥0, and

f

xy

−fx−f

y≤ 2.1

for allx, y >0. Then there exists a unique logarithmic functionL:R → Bsatisfying

fx−Lx≤ 2.2

for allx >0.

We first consider the usual logarithmic functional inequality 2.1 in the restricted domainsUk,s.

Theorem 2.1. Let, d >0,k, s∈Rwithk /0 ors /0. Suppose thatf :R → Bsatisfies

f

xy

−fx−f

y≤ 2.3

for allx, y >0, withx^ky^s ≥ d. Then there exists a unique logarithmic functionL :R → Bsuch that

fx−Lx≤3 2.4

for allx∈R.

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Proof. From the symmetry of the inequality we may assume thats /0. For givenx, y ∈ R, choose az >0 such thatx^ky^kz^s≥d,x^ky^sz^s≥d, andy^kz^s≥d. Then we have

f

xy

−fx−f

y≤−f xyz

f xy

fz f

xyz

−fx−f yz f

yz

−f y

−fz

≤3.

2.5

This completes the proof.

Now we consider the Hyers-Ulam stability of the Jensen type logarithmic functional inequality1.4in the restricted domainsU_k,s.

Theorem 2.2. Let, d >0, k, s∈R, k/p /s/q. Suppose thatf:R → Bsatisfies

f

x^py^q

−P fx−Qf

y≤ 2.6

fx−Lx−f1≤4 2.7

for allx∈R.

Proof. Replacingxbyx^1/p,ybyy^1/qin2.6we have f

xy

−P f x^1/p

−Qf

y^1/q≤ 2.8

for allx, y >0, withx^k/py^s/q≥d.

For givenx, y ∈ R, choose az > 0 such thatx^k/py^s/qz^s/q−k/p ≥ d,x^k/pz^s/q−k/p ≥ d, y^s/qz^s/q−k/p≥d, andz^s/q−k/p≥d. Replacingxbyxz⁻¹,ybyyz;xbyxz⁻¹,ybyz;xbyz⁻¹,y byyz;xbyz⁻¹,ybyzin2.8we have

f

xy

−fx−f y

f1≤f xy

−P f

x^1/pz^−1/p

−Qf

yz_1/q −fx P f

x^1/pz^−1/p Qf

z^1/q −f

y P f

z^−1/p

Qf

yz1/q f1−P f

z^−1/p

−Qf z^1/q

≤4.

2.9

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Now by TheoremA, there exists a unique logarithmic functionL:R → Bsuch that

fx−Lx−f1≤4 2.10

for allx∈R. This completes the proof.

As a matter of fact, we obtain thatL0 inTheorem 2.2provided thatp /PandporP is a rational number, orq /QandqorQis a rational number.

Theorem 2.3. Let, d >0,k, s∈R,k/p /s/q. Suppose thatp /PandporPis a rational number, orq /QandqorQis a rational number, andf :R → Bsatisfies

f

x^py^q

−P fx−Qf

y≤ 2.11

for allx, y >0, withx^ky^s≥d. Then one has

fx−f1≤4 2.12

for allx∈R.

Proof. We prove2.12only for the case thatp /P andporP is a rational number since the other case is similarly proved. From2.7and2.11, using the triangle inequality we have

L

x^py^q

−P Lx−QL

y≤M 2.13

for allx, y >0, withx^ky^s≥d, whereM54|P|4|Q| |f11−P−Q|. Ifk /0, putting y1 in2.13we have

Lx^p−P Lx ≤M 2.14

for allx > 0, with x^k ≥ d. It is easy to see thatLx^r rLxfor allx > 0 and all rational numbersr. Thus ifpis a rational number, it follows from2.14that

Lx ≤ M

p−P 2.15

for allx > 0, withx^k ≥d. If there existsx0 >0 such thatLx0/0, we can choose a rational numberrsuch thatx^rk₀ ≥dandrLx0> M/|p−P|it is realized whenris large ifx^k₀ >1, and when−ris large ifx^k₀ <1. Now we have

pM−P <rLx0L

x₀^r≤ M

p−P. 2.16

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Thus it follows thatL0. IfP is a rational number, it follows from2.14that

L

x^p−P≤M 2.17

for allx >0, withx^k≥d, which implies

Lx ≤M 2.18

for allx > 0, with x^k/p−P ≥ d. Similarly, using 2.18 we can show thatL 0. Ifk 0, choosingy₀ > 0 such thaty₀^s ≥d, puttingy y₀ in2.13and using the triangle inequality we have

Lx^p−P Lx ≤ML y^q₀

−QL

y0 2.19

for allx >0. Similarly, using2.19we can show thatL0. Thus the inequality2.12follows from2.7. This completes the proof.

Theorem 2.4. Let, d >0, k, s∈Rwithk /0 ors /0. Suppose thatf:R → Bsatisfies

f

x^py^q

−P fx−Qf

y≤ 2.20

fx−Lx−f1≤ 4

|P| 2.21

for allx∈Rifs /0, and

fx−Lx−f1≤ 4

|Q| 2.22

for allx∈Rifk /0.

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Proof. Assume thats /0. For givenx, y∈R, choose az >0 such thatx^ky^kz^s≥d,x^ky^ps/qz^s≥ d,y^kz^s ≥dandy^ps/qz^s ≥d. Replacingxbyxy,ybyz;xbyx,ybyy^p/qz;xbyy,ybyz;x by 1,ybyy^p/qzin2.20we have

P f xy

−P fx−P f y

P f1≤−f xy_p

z^q P f

xy

Qfz f

xy_p z^q

−P fx−Qf

y^p/qz f

y^pz^q

−P f y

−Qfz −f

y^pz^q

P f1 Qf

y^p/qz

≤4.

2.23

Dividing2.23by|P|and using TheoremA, we obtain that there exists a unique logarithmic functionL:R → Bsuch that

fx−Lx−f1≤ 4

|P| 2.24

for all x ∈ R. Assume thatk /0. For givenx, y ∈ R, choose az > 0 such thatx^sy^sz^k ≥ d, x^qk/py^sz^k≥d, x^sz^k≥dandx^qk/pz^k ≥d. Replacingybyxy,xbyz;ybyy,xbyx^q/pz;y byx,xbyz;yby 1,xbyx^q/pzin2.20we have

Qf

xy

−Qfx−Qf y

Qf1≤−f xy_q

z^p

P fz Qf xy f

xy_q z^p

−P f x^q/pz

−Qf y fx^qz^p−P fz−Qfx

−fx^qz^p P f x^q/pz

Qf1

≤4.

2.25

Dividing2.25by|Q|and using TheoremA, we obtain that there exists a unique logarithmic functionL:R → Bsuch that

fx−Lx−f1≤ 4

|Q| 2.26

for allx∈R. This completes the proof.

FromTheorem 2.4, using the same approach as in the proof ofTheorem 2.3we have the following.

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Theorem 2.5. Let, d >0,k, s∈Rwithk /0 ors /0. Suppose thatp /P andporP is a rational number, orq /QandqorQis a rational number, andf:R → Bsatisfies

f

x^py^q

−P fx−Qf

y≤ 2.27

for allx, y >0, withx^ky^s≥d. Then one has

fx−f1≤ 4

|P| 2.28

fx−f1≤ 4

|Q| 2.29

We callA:R → Ban additive function provided that A

xy

Ax A y

2.30

for allx, y∈R. UsingTheorem 2.2we have the following.

Corollary 2.6see22. Let >0,d, k, s∈Rwithk/p /s/q. Suppose thatg:R → Bsatisfies

g

pxqy

−P gx−Qg

y≤ 2.31

for allx, y∈R, withkxsy≥d. Then there exists a unique additive functionA:R → Bsuch that

gx−Ax−g0≤4 2.32

for allx∈R.

Proof. Replacingxby lnu,yby lnvin2.31and settingfx glnxwe have

fu^pv^q−P fu−Qfv≤ 2.33

for allu, v∈R, withu^kv^s≥e^d. UsingTheorem 2.2, we have

fx−Lx−f1≤4 2.34

for allx∈R, which implies

gx−Le^x−g0≤4 2.35

for allx∈R. LettingAx Le^xwe get the result.

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UsingTheorem 2.3, we have the following.

Corollary 2.7. Let > 0,d, k, s ∈ Rwithk/p /s/q. Suppose thatp /P andporP is a rational number, orq /QandqorQis a rational number, andg:R → Bsatisfies

g

pxqy

−P gx−Qg

y≤ 2.36

for allx, y∈R, withkxsy≥d. Then one has

gx−g0≤4 2.37

for allx∈R.

Corollary 2.8. Let >0,d, k, s∈Rwithk /0 ors /0. Suppose thatg:R → Bsatisfies

g

pxqy

−P gx−Qg

y≤ 2.38

for allx, y∈R, withkxsy≥d. Then there exists a unique additive functionA:R → Bsuch that

gx−Ax−g0≤ 4

|P| 2.39

gx−Ax−g0≤ 4

|Q| 2.40

Corollary 2.9. Let >0,d, k, s∈Rwithk /0 ors /0. Suppose thatp /PandporP is a rational number, orq /QandqorQis a rational number, andg:R → Bsatisfies

g

pxqy

−P gx−Qg

y≤ 2.41

for allx, y∈R, withkxsy≥d. Then one has

gx−g0≤ 4

|P| 2.42

gx−g0≤ 4

|Q| 2.43

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3. Asymptotic Behavior of the Inequality

In this section, we consider asymptotic behaviors of the inequalities1.4and2.1.

Theorem 3.1. Letk, s ∈ Rsatisfy one of the conditions; k /0, s /0. Suppose thatf : R → B satisfies the asymptotic condition

f

xy

−fx−f

y−→0 3.1

asx^ky^s → ∞. Thenfis a logarithmic function.

Proof. By the condition3.1, for eachn∈N, there existsd_n>0 such that

f

xy

−fx−f

y≤ 1

n 3.2

for allx, y >0, withx^ky^s ≥ d_n. ByTheorem 2.1, there exists a unique logarithmic function L_n:R → Bsuch that

fx−L_nx≤ 3

n 3.3

for allx∈R. From3.4we have

Lnx−L_mx ≤ 3 n 3

m ≤6 3.4

for allx∈Rand all positive integersn, m. Now, the inequality3.4impliesLn Lm. Indeed, for allx >0 and rational numbersr >0 we have

Lnx−Lmx 1

rLnx^r−Lmx^r ≤ 6

r. 3.5

Lettingr → ∞in3.5, we haveLnLm. Thus, lettingn → ∞in3.3, we get the result.

Theorem 3.2. Letk, s ∈ Rsatisfy one of the conditions;k /0,s /0,k/p /s/q. Suppose thatf : R → Bsatisfies the asymptotic condition

f

x^py^q

−P fx−Qf

y−→0 3.6

asx^ky^s → ∞. Then there exists a unique logarithmic functionL:R → Bsuch that

fx Lx f1 3.7

for allx∈R.

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Proof. By the condition3.6, for eachn∈N, there existsdn>0 such that

f

x^py^q

−P fx−Qf

y≤ 1

n 3.8

for allx, y > 0, withx^ky^s ≥ d_n. By Theorems2.2and2.4, there exists a unique logarithmic functionLn :R → Bsuch that

fx−L_nx−f1≤ 4

n 3.9

ifk/p /s/q,

fx−L_nx−f1≤ 4

n|P| 3.10

ifs /0, and

fx−L_nx−f1≤ 4

n|Q| 3.11

ifk /0. For all cases3.9,3.10, and3.11, there existsM >0 such that

Lnx−Lmx ≤M 3.12

for allx ∈R and all positive integersn, m. Now as in the proof ofTheorem 3.1, it follows from3.12thatLnLmfor alln, m∈N. Lettingn → ∞in3.9,3.10, and3.11we get the result.

Similarly using Theorems2.3and2.5, we have the following.

Theorem 3.3. Letk, s∈Rsatisfy one of the conditions;k /0,s /0,k/p /s/q. Suppose thatp /P andporP is a rational number, orq /QandqorQis a rational number, andf :R → Bsatisfies the asymptotic condition

f

x^py^q

−P fx−Qf

y−→0 3.13

asx^ky^s → ∞. Thenfis a constant function.

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Using Corollaries2.6and2.8we have the following.

Corollary 3.4. Let >0,k, s∈ Rsatisfy one of the conditionsk /0,s /0, ork/p /s/q. Suppose thatg:R → Bsatisfies

g

pxqy

−P gx−Qg

y−→0 3.14

askxsy → ∞. Then there exists a unique additive functionA:R → Bsuch that

gx Ax g0 3.15

for allx∈R.

Using Corollaries2.7and2.9we have the following.

Corollary 3.5. Let >0,k, s∈ Rsatisfy one of the conditionsk /0,s /0, ork/p /s/q. Suppose thatp /PandporPis a rational number, orq /QandqorQis a rational number, andg:R → B satisfies

g

pxqy

−P gx−Qg

y−→0 3.16

askxsy → ∞. Thengis a constant function.

Acknowledgments

The author expresses his sincere gratitude to a referee of the paper for many useful comments and introducing the interesting related recent results including the papers17–26. This work was supported by Basic Science Research Program through the National Research Foundation of KoreaNRFfunded by the Ministry of Education, Science and TechnologyMEST no.

2010-0016963.

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