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Functional Equalities and Some Mean Values
Journal of Mathematical Analysis and Applications 250, 181᎐186 Ž2000. doi:10.1006rjmaa.2000.7066, available online at http:rrwww.idealibrary.com on
Functional Equalities and Some Mean Values Shoshana Abramovich Department of Mathematics, Uni¨ ersity of Haifa, Haifa 31905, Israel E-mail: [email protected]
and Josip Pecaric ˘ ´ Faculty of Textile Technology, Uni¨ ersity of Zagreb, Peirottije¨ a 6, 10000 Zagreb, Croatia E-mail: [email protected] Submitted by Themistocles M. Rassias Received November 24, 1999
1. INTRODUCTION Means as a generalization of arithmetic, geometric, and harmonic means are dealt with extensively in mathematical literature; see w1᎐8x to quote a few. In this paper we define a mean value of a, and b, as a function which satisfies the following: ŽA. M: Rq= Rqª R, ŽB. M Ž a, b . s M Ž b, a., ŽC. M Ž a, a. s a, ŽD. 0 - a - M Ž a, b . - b for 1 - bra. Some interesting mean values were considered recently by Haruki and Rassias w2᎐4x. In w5x the following classes of means were considered:
M Ž a, b; h . s
1 H Ž a, b .
hy1
1
žH
/
h Ž s . dt ,
0
H Ž a, b . s
2 ab aqb
181 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
ABRAMOVICH AND PECARIC ˘ ´
182
hŽ x . s x n, n s "1, "2, and hŽ x . s log x, in Theorem 3.1 in the case that s s Žsin 2 2 tra2 q cos 2 2 trb 2 .y1 and in Theorem 3.2 in the case that s s a2 sin 2 2 t q b 2 cos 2 2 t. Note that his means for h s x 2 in Theorem 3.1 and for hŽ x . s x, x 2 , log x in Theorem 3.2 satisfy conditions ŽA., ŽB., and ŽC. only. In this paper we present generalizations of his results. 1 We introduce a function M Ž a, b, hŽ s .. s M Ž a, b . hy1 Ž H01 hŽ s . dt . where s s sŽ a, b, t . and M Ž a, b . is a given mean value Žthat is, a mean value satisfying ŽA. to ŽD... We deal with a functional equation which leads to a characterization of our means. We also show that this mean and Kim’s means are not means in classical sense since they do not satisfy the very important condition ŽD.. In fact, we show that instead of condition ŽD., such means satisfy ŽD⬘. 0 - a F M Ž a, b, hŽ s .. F b for 1 F bra F T, so we shall say that these means are on a limited interval.
2. THE RESULTS THEOREM 1. Let g: Rqª R, h: Rqª R be strictly monotonic functions with continuous second deri¨ ati¨ es. Let gy1 and hy1 be the in¨ erse functions of g and h, respecti¨ ely. Let sŽ a, b, t . be a positive continuous function in a, b, and t, with continuous first and second derivatives with respect to a, where sŽ c, c, t . s ⭸ sŽ c, c, t . DŽ c . is independent of t, and s sXaŽ c, c, t . is not independent of t. ⭸a Then, hy1
1
žH
h Ž s Ž a, b, t . . dt s gy1
/
0
1
žH
/
g Ž s Ž a, b, t . . dt ,
0
Ž 1.
for all real a, b iff g Ž x . s c1 hŽ x . q c 2 , c1 / 0. Proof. We will show that the technique used in w4, 5x for specific means works in the much more general case stated here. Instead of proving Ž1. we will prove that 1
H0
h Ž s Ž a, b, t . . dt s h gy1
1
ž žH
g Ž s Ž a, b, t . . dt
0
We will denote f Ž a, b . s
1
H0
h Ž s Ž a, b, t . . dt
k Ž a, b . s h gy1
1
ž žH
0
g Ž s Ž a, b, t . . dt
//
.
//
.
Ž 2.
183
FUNCTIONAL EQUALITIES AND MEANS
It is easy to see that f Ž c, c . s k Ž c, c . s h Ž D Ž c . . .
Ž 3.
Differentiating f Ž a, b . and k Ž a, b . with respect to a and setting a s c, b s c, we get
⭸f ⭸a
⭸k
1
Ž c, c . s h⬘ Ž D . H sXa Ž c, c, t . dt s
⭸a
0
Ž c, c . .
Ž 4.
Computing Ž ⭸ 2 fr⭸ a2 .Ž a, b . and Ž ⭸ 2 kr⭸ a2 .Ž a, b . we get that
⭸ 2f ⭸a ⭸ k
2
1
1
Ž c, c . s h⬙ Ž D . H Ž sXa Ž c, c, t . . dt q h⬘ Ž D . H sYaa Ž c, c . dt, 2
0
0
2
⭸ a2
Ž c, c . s h⬙ Ž D .
ž
2
1 X
s a Ž c, c, t . dt
H0
q h⬘ Ž D .
/
0
1
=
1 Y
½H
H0
Ž sXa Ž c, c, t . .
2
s aa Ž c, c, t . dt q
dt y
g⬙ Ž D. g ⬘Ž D . 2
1 X
žH
0
s a Ž c, c, t . dt
5
/
.
Therefore, in order for
⭸ 2k ⭸ a2
Ž c, c, t . s
⭸ 2f ⭸ a2
Ž c, c, t .
Ž 5.
to be satisfied, we get that h⬙ Ž D .
žH
0
s
2
1 X
s a Ž c, c, t . dt
g⬙ Ž D. g ⬘Ž D .
h⬘ Ž D .
ž
y
/
1
H0
Ž sXa Ž c, c, t . . 2
1 X
H0
s a Ž c, c, t . dt
/
y
2
dt 1
X a
H0 Ž s Ž c, c, t . .
2
dt . Ž 6 .
According to the Cauchy Schwarz inequality,
ž
1 X
H0
2
s a Ž c, c, t . dt
/
-
1
H0
Ž sXa Ž c, c, t . .
2
dt
as long as sXaŽ c, c, t . is not independent of t. Therefore we get from Ž6. h⬙ Ž D . h⬘ Ž D .
s
g⬙ Ž D. g ⬘Ž D .
ABRAMOVICH AND PECARIC ˘ ´
184 and hence
g Ž D . s c1 h Ž D . q c 2 .
Ž 7.
Thus we proved that if f Ž a, b . s k Ž a, b ., Ž7. holds. To prove that if Ž7. holds then Ž1. holds is trivial. Let n q 1 be a positi¨ e integer, then for e¨ ery a, b ) 0
THEOREM 2. Inq 1 s
dt
1
H0
Ž a sin 2 2 t q b cos 2 2 t .
n
s
Ý ks0
1 n!2
2n
ž /Ž n k
nq1
2 k . !Ž 2 n y 2 k . ! k! Ž n y k . !
ayŽ2 kq1.r2 byŽ2 ny2 kq1.r2 . Ž 8 .
Proof. It is easy to verify that I1 s
dt
1
H0
a sin 2 t q b cos 2 t 2
2
s
1
Ž 9.
'ab
and that Inq 1 s
1 n
ž
⭸ In ⭸a
q
⭸ In ⭸b
/
.
Ž 10 .
The proof is by induction using Ž9. and Ž10.. We will omit it. Result. Replacing a with ay2 , b with by2 in Ž8., we get that Jnq 1 s
n
s
dt
1
H0
Ý ks0
ž
sin 2 2 t a2 1 n!2
2n
q
ž /Ž n k
cos 2 2 t b2
nq1
/
2 k . !Ž 2 n y 2 k . ! k! Ž n y k . !
a2 kq1 b 2 ny2 kq1 ,
where n is a positive integer. THEOREM 3. Let g: Rqª R, h: Rqª R be strictly monotonic functions in Rq with continuous second deri¨ ati¨ es. Let gy1 and hy1 be the inverse functions of g and h, respectively. Then M Ž a, b, g Ž s . . s s
aqb 2 ab aqb 2 ab
gy1
1
žH žH 0
hy1
1
0
g Ž s . dt
/ /
h Ž s . dt s M Ž a, b; h Ž s . . ,
Ž 11 .
185
FUNCTIONAL EQUALITIES AND MEANS
where ss
ž
sin k 2 2 t a2
cos 2 2 t
q
b2
y1
/
Ž 12 .
holds for all nonzero a and b iff g Ž x . s c1 hŽ x . q c 2 , where c1 / 0 and c 2 are arbitrary numbers. Proof. This theorem follows from Theorem 1 by replacing sŽ a, b, t . by Ž12.. THEOREM 4. Then
Let n q 1 be an integer, and let h and s be as in Theorem 3.
M Ž a, b; h Ž s . . s s
aqb 2 ab aqb 2 ab
hy1
1
žH
h Ž s . dt
0
n
žÝ
ks0
1 n!2 2 n
/
n Ž 2 k . ! Ž 2 n y 2 k . ! 2 kq1 2 ny2 kq1 a b k k! Ž n y k . !
ž /
1r Ž nq1 .
/ Ž 13 .
iff hŽ x . s c1 x nq 1 q c 2 . Proof. The theorem follows easily from Theorem 1 and the result of Theorem 2. THEOREM 5. Let h: Rqª R be a strictly monotonic function in Rq with a continuous deri¨ ati¨ e. ⭸M Let M Ž a, b . be a mean value such that ⭸ a Ž c, c . s q, 0 - q - 1. Then there exists a number T ) 1 such that
a-
1 M Ž a, b .
hy1
1
žH
h Ž s . dt - b
/
0
for 1 - bra - T, s s Žsin 2 2 tra2 q cos 2 2 trb 2 .y1 , and in this interval M Ž a, b; h . s
1 M Ž a, b .
hy1
1
žH
0
h Ž s . dt
/
Ž 14 .
is a mean value on a limited interval. ⭸M ⭸M Proof. Computing ⭸ x Ž a, x; h. we get that ⭸ x Ž a, a; h. s yq q 1 ) 0. Hence M Ž a, x; h. is increasing in x in the neighborhood of x s a. Therefore a s M Ž a, a; h. - M Ž a, b; h. in some interval 1 - bra - R.
ABRAMOVICH AND PECARIC ˘ ´
186 Because ⭸M ⭸y
M Ž a, b . s M Ž b, a., we get from
⭸M ⭸y
Ž y, b; h. also that
Ž b, b; h. s yq q 1 ) 0. Therefore M Ž a, b; h. - M Ž b, b; h. s b in
some interval 1 - bra - Q. Hence there is an interval 1 - bra - T such that a - M Ž a, b; h . - b,
Ž 15 .
and condition ŽD. in the definition of mean value is satisfied in the interval 1 - bra - T Ž0 - a.. It is obvious that M Ž a, b; h. also satisfies conditions ŽA., ŽB., and ŽC. there. Therefore M Ž a, b; h. is a mean value in the interval 1 - bra - T. Now we show that inequality Ž15. for hŽ x . s x n, n s 2, 3 . . . , holds only for 1 - bra - T, where T is a finite real number. THEOREM 6.
For 1 - bra, where bra is large enough M Ž a, b; h . ) b,
Ž 16 .
2 ab where hŽ x . s x n, n s 2, 3, . . . , M Ž a, b . s a q b , and s s Žsin 2 2 tra2 q cos 2 2 trb 2 .y1 .
Proof. We proved in Theorem 4 that M Ž a, b, x nq 1 . satisfies equality Ž13.. It is easy to see that when bra ª ⬁, then M Ž a, b, x nq 1 .rb ª ⬁ too. Therefore Ž16. holds for bra large enough. Hence in these cases M Ž a, b; h. is a mean value only on a limited interval.
REFERENCES 1. P. S. Bullen, D. S. Mitrinovic, ´ and P. S. Vasic, ´ ‘‘Means and Their Inequalities,’’ Reidel, DordrechtrBostonrLancasterrTokyo, 1988. 2. H. Haruki, New characterization of arithmetic᎐geometric mean of Gauss and other well known mean values, Publ. Math. Debrecen 38 Ž1991., 323᎐332. 3. H. Haruki and T. M. Rassias, A new analogue of Gauss’ functional equation, Internat. J. Math. Sci. 18 Ž1995., 749᎐756. 4. H. Haruki and T. M. Rassias, New characterization of some mean values, J. Math. Anal. Appl. 202 Ž1996., 333᎐348. 5. Y.-H. Kim, On some further extensions of the characterizations of mean values by H. Haruki and Th. Rassias, J. Math. Anal. Appl. 235 Ž1999., 598᎐607. 6. D. S. Mitrinovic, ´ J. E. Pecaric, ˘ ´ and A. M. Fink, ‘‘Classical and New Inequalities in Analysis,’’ Kluwer Academic, DordrechtrBostonrLondon, 1993. 7. Gh. Toader, Some mean values related to the arithmetic᎐geometric mean, J. Math. Anal. Appl. 218 Ž1998., 358᎐368. 8. Gh. Toader and Th. Rassias, New properties of some mean values, J. Math. Anal. Appl. 232 Ž1999., 376᎐383.
Functional Equalities and Some Mean Values Shoshana Abramovich Department of Mathematics, Uni¨ ersity of Haifa, Haifa 31905, Israel E-mail: [email protected]
and Josip Pecaric ˘ ´ Faculty of Textile Technology, Uni¨ ersity of Zagreb, Peirottije¨ a 6, 10000 Zagreb, Croatia E-mail: [email protected] Submitted by Themistocles M. Rassias Received November 24, 1999
1. INTRODUCTION Means as a generalization of arithmetic, geometric, and harmonic means are dealt with extensively in mathematical literature; see w1᎐8x to quote a few. In this paper we define a mean value of a, and b, as a function which satisfies the following: ŽA. M: Rq= Rqª R, ŽB. M Ž a, b . s M Ž b, a., ŽC. M Ž a, a. s a, ŽD. 0 - a - M Ž a, b . - b for 1 - bra. Some interesting mean values were considered recently by Haruki and Rassias w2᎐4x. In w5x the following classes of means were considered:
M Ž a, b; h . s
1 H Ž a, b .
hy1
1
žH
/
h Ž s . dt ,
0
H Ž a, b . s
2 ab aqb
181 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
ABRAMOVICH AND PECARIC ˘ ´
182
hŽ x . s x n, n s "1, "2, and hŽ x . s log x, in Theorem 3.1 in the case that s s Žsin 2 2 tra2 q cos 2 2 trb 2 .y1 and in Theorem 3.2 in the case that s s a2 sin 2 2 t q b 2 cos 2 2 t. Note that his means for h s x 2 in Theorem 3.1 and for hŽ x . s x, x 2 , log x in Theorem 3.2 satisfy conditions ŽA., ŽB., and ŽC. only. In this paper we present generalizations of his results. 1 We introduce a function M Ž a, b, hŽ s .. s M Ž a, b . hy1 Ž H01 hŽ s . dt . where s s sŽ a, b, t . and M Ž a, b . is a given mean value Žthat is, a mean value satisfying ŽA. to ŽD... We deal with a functional equation which leads to a characterization of our means. We also show that this mean and Kim’s means are not means in classical sense since they do not satisfy the very important condition ŽD.. In fact, we show that instead of condition ŽD., such means satisfy ŽD⬘. 0 - a F M Ž a, b, hŽ s .. F b for 1 F bra F T, so we shall say that these means are on a limited interval.
2. THE RESULTS THEOREM 1. Let g: Rqª R, h: Rqª R be strictly monotonic functions with continuous second deri¨ ati¨ es. Let gy1 and hy1 be the in¨ erse functions of g and h, respecti¨ ely. Let sŽ a, b, t . be a positive continuous function in a, b, and t, with continuous first and second derivatives with respect to a, where sŽ c, c, t . s ⭸ sŽ c, c, t . DŽ c . is independent of t, and s sXaŽ c, c, t . is not independent of t. ⭸a Then, hy1
1
žH
h Ž s Ž a, b, t . . dt s gy1
/
0
1
žH
/
g Ž s Ž a, b, t . . dt ,
0
Ž 1.
for all real a, b iff g Ž x . s c1 hŽ x . q c 2 , c1 / 0. Proof. We will show that the technique used in w4, 5x for specific means works in the much more general case stated here. Instead of proving Ž1. we will prove that 1
H0
h Ž s Ž a, b, t . . dt s h gy1
1
ž žH
g Ž s Ž a, b, t . . dt
0
We will denote f Ž a, b . s
1
H0
h Ž s Ž a, b, t . . dt
k Ž a, b . s h gy1
1
ž žH
0
g Ž s Ž a, b, t . . dt
//
.
//
.
Ž 2.
183
FUNCTIONAL EQUALITIES AND MEANS
It is easy to see that f Ž c, c . s k Ž c, c . s h Ž D Ž c . . .
Ž 3.
Differentiating f Ž a, b . and k Ž a, b . with respect to a and setting a s c, b s c, we get
⭸f ⭸a
⭸k
1
Ž c, c . s h⬘ Ž D . H sXa Ž c, c, t . dt s
⭸a
0
Ž c, c . .
Ž 4.
Computing Ž ⭸ 2 fr⭸ a2 .Ž a, b . and Ž ⭸ 2 kr⭸ a2 .Ž a, b . we get that
⭸ 2f ⭸a ⭸ k
2
1
1
Ž c, c . s h⬙ Ž D . H Ž sXa Ž c, c, t . . dt q h⬘ Ž D . H sYaa Ž c, c . dt, 2
0
0
2
⭸ a2
Ž c, c . s h⬙ Ž D .
ž
2
1 X
s a Ž c, c, t . dt
H0
q h⬘ Ž D .
/
0
1
=
1 Y
½H
H0
Ž sXa Ž c, c, t . .
2
s aa Ž c, c, t . dt q
dt y
g⬙ Ž D. g ⬘Ž D . 2
1 X
žH
0
s a Ž c, c, t . dt
5
/
.
Therefore, in order for
⭸ 2k ⭸ a2
Ž c, c, t . s
⭸ 2f ⭸ a2
Ž c, c, t .
Ž 5.
to be satisfied, we get that h⬙ Ž D .
žH
0
s
2
1 X
s a Ž c, c, t . dt
g⬙ Ž D. g ⬘Ž D .
h⬘ Ž D .
ž
y
/
1
H0
Ž sXa Ž c, c, t . . 2
1 X
H0
s a Ž c, c, t . dt
/
y
2
dt 1
X a
H0 Ž s Ž c, c, t . .
2
dt . Ž 6 .
According to the Cauchy Schwarz inequality,
ž
1 X
H0
2
s a Ž c, c, t . dt
/
-
1
H0
Ž sXa Ž c, c, t . .
2
dt
as long as sXaŽ c, c, t . is not independent of t. Therefore we get from Ž6. h⬙ Ž D . h⬘ Ž D .
s
g⬙ Ž D. g ⬘Ž D .
ABRAMOVICH AND PECARIC ˘ ´
184 and hence
g Ž D . s c1 h Ž D . q c 2 .
Ž 7.
Thus we proved that if f Ž a, b . s k Ž a, b ., Ž7. holds. To prove that if Ž7. holds then Ž1. holds is trivial. Let n q 1 be a positi¨ e integer, then for e¨ ery a, b ) 0
THEOREM 2. Inq 1 s
dt
1
H0
Ž a sin 2 2 t q b cos 2 2 t .
n
s
Ý ks0
1 n!2
2n
ž /Ž n k
nq1
2 k . !Ž 2 n y 2 k . ! k! Ž n y k . !
ayŽ2 kq1.r2 byŽ2 ny2 kq1.r2 . Ž 8 .
Proof. It is easy to verify that I1 s
dt
1
H0
a sin 2 t q b cos 2 t 2
2
s
1
Ž 9.
'ab
and that Inq 1 s
1 n
ž
⭸ In ⭸a
q
⭸ In ⭸b
/
.
Ž 10 .
The proof is by induction using Ž9. and Ž10.. We will omit it. Result. Replacing a with ay2 , b with by2 in Ž8., we get that Jnq 1 s
n
s
dt
1
H0
Ý ks0
ž
sin 2 2 t a2 1 n!2
2n
q
ž /Ž n k
cos 2 2 t b2
nq1
/
2 k . !Ž 2 n y 2 k . ! k! Ž n y k . !
a2 kq1 b 2 ny2 kq1 ,
where n is a positive integer. THEOREM 3. Let g: Rqª R, h: Rqª R be strictly monotonic functions in Rq with continuous second deri¨ ati¨ es. Let gy1 and hy1 be the inverse functions of g and h, respectively. Then M Ž a, b, g Ž s . . s s
aqb 2 ab aqb 2 ab
gy1
1
žH žH 0
hy1
1
0
g Ž s . dt
/ /
h Ž s . dt s M Ž a, b; h Ž s . . ,
Ž 11 .
185
FUNCTIONAL EQUALITIES AND MEANS
where ss
ž
sin k 2 2 t a2
cos 2 2 t
q
b2
y1
/
Ž 12 .
holds for all nonzero a and b iff g Ž x . s c1 hŽ x . q c 2 , where c1 / 0 and c 2 are arbitrary numbers. Proof. This theorem follows from Theorem 1 by replacing sŽ a, b, t . by Ž12.. THEOREM 4. Then
Let n q 1 be an integer, and let h and s be as in Theorem 3.
M Ž a, b; h Ž s . . s s
aqb 2 ab aqb 2 ab
hy1
1
žH
h Ž s . dt
0
n
žÝ
ks0
1 n!2 2 n
/
n Ž 2 k . ! Ž 2 n y 2 k . ! 2 kq1 2 ny2 kq1 a b k k! Ž n y k . !
ž /
1r Ž nq1 .
/ Ž 13 .
iff hŽ x . s c1 x nq 1 q c 2 . Proof. The theorem follows easily from Theorem 1 and the result of Theorem 2. THEOREM 5. Let h: Rqª R be a strictly monotonic function in Rq with a continuous deri¨ ati¨ e. ⭸M Let M Ž a, b . be a mean value such that ⭸ a Ž c, c . s q, 0 - q - 1. Then there exists a number T ) 1 such that
a-
1 M Ž a, b .
hy1
1
žH
h Ž s . dt - b
/
0
for 1 - bra - T, s s Žsin 2 2 tra2 q cos 2 2 trb 2 .y1 , and in this interval M Ž a, b; h . s
1 M Ž a, b .
hy1
1
žH
0
h Ž s . dt
/
Ž 14 .
is a mean value on a limited interval. ⭸M ⭸M Proof. Computing ⭸ x Ž a, x; h. we get that ⭸ x Ž a, a; h. s yq q 1 ) 0. Hence M Ž a, x; h. is increasing in x in the neighborhood of x s a. Therefore a s M Ž a, a; h. - M Ž a, b; h. in some interval 1 - bra - R.
ABRAMOVICH AND PECARIC ˘ ´
186 Because ⭸M ⭸y
M Ž a, b . s M Ž b, a., we get from
⭸M ⭸y
Ž y, b; h. also that
Ž b, b; h. s yq q 1 ) 0. Therefore M Ž a, b; h. - M Ž b, b; h. s b in
some interval 1 - bra - Q. Hence there is an interval 1 - bra - T such that a - M Ž a, b; h . - b,
Ž 15 .
and condition ŽD. in the definition of mean value is satisfied in the interval 1 - bra - T Ž0 - a.. It is obvious that M Ž a, b; h. also satisfies conditions ŽA., ŽB., and ŽC. there. Therefore M Ž a, b; h. is a mean value in the interval 1 - bra - T. Now we show that inequality Ž15. for hŽ x . s x n, n s 2, 3 . . . , holds only for 1 - bra - T, where T is a finite real number. THEOREM 6.
For 1 - bra, where bra is large enough M Ž a, b; h . ) b,
Ž 16 .
2 ab where hŽ x . s x n, n s 2, 3, . . . , M Ž a, b . s a q b , and s s Žsin 2 2 tra2 q cos 2 2 trb 2 .y1 .
Proof. We proved in Theorem 4 that M Ž a, b, x nq 1 . satisfies equality Ž13.. It is easy to see that when bra ª ⬁, then M Ž a, b, x nq 1 .rb ª ⬁ too. Therefore Ž16. holds for bra large enough. Hence in these cases M Ž a, b; h. is a mean value only on a limited interval.
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