- Email: [email protected]
Sharpening Jensen's Inequality and a Majorization Theorem
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
214, 721]728 Ž1997.
AY975527
Sharpening Jensen’s Inequality and a Majorization Theorem S. Abramovich Department of Mathematics and Computer Science, Uni¨ ersity of Haifa, Haifa, Israel
B. Mond School of Mathematics, La Trobe Uni¨ ersity, Bundoora, Victoria, 3082, Australia
and J. E. Pecaric ˇ ´ Faculty of Textile Technology, Uni¨ ersity of Zagreb, Pierottije¨ a 6, 10000, Zagreb, Croatia Submitted by A. M. Fink Received October 10, 1995
1. INTRODUCTION In this paper we shall show that sharpening Holder’s Inequality can be ¨ generalized to sharpen Jensen’s inequality for concave and convex functions w1, 4x. We shall also sharpen a majorization theorem for concave functions and two sequences Žor functions. when only one of them is monotonic w6x.
2. SHARPENING JENSEN’S INEQUALITY Let us consider a concave function F of several variables defined on an open convex set u in R n w3, pp. 31, 32x. Let the real functions 0 - w Ž t ., and f i Ž t . w Ž t ., i s 1, . . . , n be integrable functions on the real interval 721 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
ABRAMOVICH, MOND, AND PECARIC ˇ ´
722
w a, b x. Then Jensen’s Inequality in its integral form w5, p. 10x is b
Ha
F Ž f 1 , . . . , f n . w Ž t . dt b
Ha
b
FF
w Ž t . dt
b
Ha
b
f 1w dt
Ha
Ha
,...,
b
w Ž t . dt
Ha
f n w dt
w Ž t . dt
0
. Ž 2.1.
Using this inequality we get the following Let Ž x 1 , . . . , x n . be a complex function in n complex ¨ ari-
THEOREM 1. ables and let
< F Ž x1 , . . . , x n . < F < F Ž < x1 < , . . . , < x n <. < .
Ž 2.2.
Let also < F Ž x 1 , . . . , x n .< be a conca¨ e function for x s Ž x 1 , . . . , x n . g R n. If f i Ž t ., i s 1, . . . , n, w Ž t . are complex functions of real ¨ ariables, and Ž f i t . w Ž t ., w Ž t . are integrable on w a, b x, then c
AŽ c . s
Ha w Ž t . F Ž f Ž t . , . . . , f Ž t . dt n
1
q
Hc
b
< w Ž t . < dt F
Hc
b
Hc
< w Ž t . f 1 < dt b
,...,
Hc
< w Ž t . < dt
b
Hc
< w Ž t . f n < dt b
< w Ž t . < dt
is a decreasing function in c, a F c F b. Analogous results for the discrete case are obvious. Proof. Let a F c - d F b, AŽ d . s
d
Ha
wF Ž f 1 , . . . , f n . dt b
b
Hd < w < dt F
q
F
Hd < wf < dt b
c n
q
b
Hd < w < dt F
Hc
Hd < wf < dt
d
b
0
< wF Ž < f 1 < , . . . , < f n < . < dt b
Hd < wf < dt n
1
Hd < w < dt
b
Hd < w < dt
b
q
Hd < wf < dt n
,...,
Hd < w < dt
Ha wF Ž f , . . . , f . dt 1
b
1
,...,
b
Hd < w < dt
0
0
Ž 2.3.
SHARPENING JENSEN’S INEQUALITY
F
c
Ha wF Ž f , . . . , f . dt n
1
q
Hc
d
< w < dt F
b
Hd < w < dt F
q
F
Hc
d
Hc
< wf 1 < dt
d
,..., < w < dt
b
b
Hc
d
< wf 1 < dt q
Hc
d
< w < dt q
q
< wf n < dt
d
< w < dt
Hd < wf < dt n
,...,
Hd < w < dt n
1
d
b
1
c
Hc
Hc
Hd < wf < dt
Ha wF Ž f , . . . , f . dt = F
s
723
b
Hd < w < dt
žH
d
c
< w < dt q
b
Hd < wf < dt 1
b
,...,
Hd < w < dt
0 0
b
Hd < w < dt
Hc
d
Hc
/ b
wf n < dt q
Hd < wf < dt
d
Hd < w < dt
< w < dt q
n
b
0
c
Ha wF Ž f , . . . , f . dt n
1
q
Hc
b
< w < dt F
Hc
b
Hc
< wf 1 < dt
b
,..., < w < dt
Hc
b
Hc
< wf n < dt
b
< w < dt
0
s AŽ c . .
The first inequality is a result of the given inequality Ž2.2.. The second inequality is Jensen’s inequality Ž2.1. applied on c F t F d to the given concave function < F Ž x 1 , . . . , x n .< on Ž x 1 , . . . , x n . g R n. The third inequality follows again from the concavity of < F Ž x 1 , . . . , x n .< on Ž x 1 , . . . , x n . g R n. The inequality AŽ d . F AŽ c . becomes equality only if the arguments of Hac wF Ž f 1 , . . . , f n . dt and of wF Ž f 1 , . . . , f n . on w c, d x are the same, < wf i <, i s 1, . . . , n, are constants on w c, d x, < F Ž f 1 , . . . , f n .< s < F Ž< f 1 <, . . . , < f n <.< on w c,d x, and if d - b, Hcd < f i w < dt s Hdb < f i w < dt, i s 1, . . . , n. EXAMPLE 1. Sharpening Holder’s Inequality. Let F Ž x 1 , . . . , x n . s ¨ n ai Ł is1 x i , x i g C, i s 1, . . . , n, where a i G 0, i s 1, . . . , n, Ý nis1 a i F 1, a0 s 1 y Ý nis1 a i .
ABRAMOVICH, MOND, AND PECARIC ˇ ´
724
Let f i Ž t ., i s 0, 1, . . . , n, be a complex integrable function on the real interval a F t F b. Then
AŽ c . s
s
n
c
fi Ž t .
ž /
Ha f Ž t . is1 Ł 0
c
ai
dt q
f0 Ž t .
n
n
Ha is0 Ł
Ž t . dt q Ł
f ia i
is0
žH c
b
Hc
b
n
< f 0 Ž t . < dt Ł is1
< f i Ž t . < dt
b
< f 0 Ž t . < dt
Hc Hc
ai
b
0
ai
< f i Ž t . < dt
/
n is a decreasing function in c, a F c F b, because F s Ł is1 x ia i , x i ) 0, n a i ) 0, i s 1, . . . , n, Ý is1 a i - 1, is a concave function. The case n s 1, a1 s 1r2 was considered in w4x, and the case n s 1, 0 - a1 - 1 was considered in w1x. Analogous results for the discrete case are obvious.
EXAMPLE 2. Sharpening Minkowski’s Inequality. Let F Ž x 1 , . . . , x n . s a. a Ž1 q Ý nis1 x 1r , a G 1. i n As F is a concave function for Ž x 1 , . . . , x n . g Rq , we get under the same conditions as in Example 1 on f i Ž t ., i s 1, . . . , n, and on w Ž t . s f 0 Ž t . that AŽ c . s
c
Hd f Ž t . 0
q
Hc
s
c
b
ž
n
1q
a
1ra
ž / / f0 Ž t .
n
Hc
b
dt 1ra
a
< f i Ž t . dt <
00
< f 0 Ž t . < dt 1 q
Ý
is1
n
ž
Ý is1
fi Ž t .
1ra
Hd Ý Ž f Ž t . . i
is0
a
Hc
b
< f 0 Ž t . dt <
n
dt q
ž Ý žH is0
c
b
a
1ra
< f i Ž t . < dt
/
/
is a decreasing function in c, d F c F b. Analogous results are obvious for the discrete case.
3. SHARPENING A MAJORIZATION THEOREM In w6x the following majorization theorem was proved. THEOREM A w6 Theorem 3x. Let f and grf be increasing functions on Ž0, 1.. Let w ) 0 and let g ) 0 or f ) 0. Let also fw and gw be integrable functions on w0, 1x and H01 fw dtrH01 gw dt G 0.
SHARPENING JENSEN’S INEQUALITY
725
Let F be a conca¨ e function. Then for e¨ ery k ) 0 1
1
H0
F Ž kg . w dt F
1
H0
H0
gw dt
0
F k
f w dt.
1
Ž 3.1.
fw dt
H0
Specifically, if g Ž t . is a conca¨ e function satisfying g Ž0. F 0 H01 g Ž t . w Ž t . dt G 0 we get the inequality 1
1
H0
F Ž g Ž t . . w Ž t . dt F
1
H0
F
H0
g Ž t . w Ž t . dt 1
H0
and
0
t w dt. tw Ž t . dt
In the following theorem we sharpen this Majorization Theorem as was done for Theorem 1. THEOREM 2. Let f ) 0 and grf be increasing functions on Ž a, b .. Let fw and gw be integrable on w a, b x. Also, let w ) 0 and Hab g Ž t . w Ž t . dt G 0. Let F Ž t . be a conca¨ e function and let
BŽ c. s
c
Ha
F Ž g . w Ž t . dt q
Hc
b
F
Hc
b
gw dt
0 0 Hc
b
f w Ž t . dt.
Ž 3.2.
fw dt
Then B Ž c . is a decreasing function of c, a F c F b, and
BŽ c. F
c
Ha F Ž g . w Ž t . dt q Hc
b
Hc
wŽ t. F
b
Hc
gw dt
b
dt.
Ž 3.3.
w dt
In the special case that g Ž t . is a con¨ ex function on w0, b x, g Ž0. F 0, and H0b g Ž t . w Ž t . dt G 0 we get that
BŽ c. s
c
H0
F Ž g . w Ž t . dt q
is a decreasing function of c, 0 F c F b.
Hc
b
F
Hc
b
gw dt
0 Hc
b
t w dt
tw dt
Ž 3.4.
ABRAMOVICH, MOND, AND PECARIC ˇ ´
726
Proof. Let a F c - d F b b
BŽ d. s
d
Ha
F Ž g . w Ž t . dt q
b
Hd F
Hd gw dt
0 b
f w dt
Hd fw dt
b
s
F
c
Ha F Ž g . w Ž t . dt q Hc c
Ha F Ž g . w Ž t . dt q Hc
d
b
F Ž g . w Ž t . dt q
F
Hc
b
b
Hd F
0 b
Hd
f w dt
fw dt
gw dt
0 Hc
Hd gw dt
b
f w dt s B Ž c . .
Ž 3.5.
fw dt
To prove Ž3.5., let us define
¡g Ž t . ,
cFt-d
~ Hd gw dt f, b fw dt H ¢d
d F t - b.
b
GŽ t . s
We shall see that GŽ t . plays the same role as g Ž t . in Theorem 4. In order to show that GŽ t .rf Ž t . is increasing on Ž c, b . it is enough to show that b
g Ž dq . f Ž dq .
F
Hd gw dt b
.
Hd fw dt
Let us assume that b
g Ž dq . f Ž dq .
)
Hd gw dt b
.
Hd fw dt
Because grf is increasing we therefore get g Ž t .Hdb fw dt ) f Ž t .Hdb gw dt, d F t F b which by integration on w d, b x leads to a contradiction. Also, Hcb GŽ t . w Ž t . dt s Hcb g Ž t . w Ž t . dt ) 0 and therefore we get the inequality in Ž3.5. by Theorem A. Thus the assertion that B Ž c . is decreasing is proved. By applying Jensen’s inequality to B Ž c . we get Ž3.3.. In case g Ž t . is a convex function for t G 0 satisfying g Ž0. F 0 we get that B Ž c . defined in Ž3.4. is decreasing because g Ž t .rt is increasing for t ) 0.
SHARPENING JENSEN’S INEQUALITY
727
By a similar proof as those of Theorems 1 and 2 we get THEOREM 3. Let F Ž t . be a complex function defined on C and let < F Ž t .< F < F Ž< t <.<. Let < f Ž t .<, < g Ž t .rf Ž t .< be increasing functions on the real inter¨ al a F t F b then c
Ha F Ž g Ž t . . w Ž t . dt
q
Hc
b
Hc
F
b
< gw < dt
b
< fw < dt
Hc
0
< f Ž t . < w Ž t . dt
is decreasing with c, a F c F b. Moreo¨ er, c
Ha
F Ž g Ž t . . w Ž t . dt q
c
F
Ha
Hc
b
F
F Ž g Ž t . . w Ž t . dt q
Hc
b
< gw < dt
b
< fw < dt
Hc Hc
b
0
< f Ž t . < w Ž t . dt
< w Ž t . dt < F
Hc
b
< gw < dt
0 Hc
b
.
< w < dt
EXAMPLE 3. In this example we use the obvious discrete version of Theorem 2. Let 0 - f i s d irwi and g irf i s x ird i , i s 1, . . . , n, be increasing sequences such that n
d i s x 1, n s
Ý
x irn,
wi ) 0, i s 1, . . . , n.
is1
Define n
x mq 2 , n s
xi
Ý ismq2
nymy1
.
Let F Ž t . be a concave function. Then, n m
B Ž m. s
Ý
F
is1 m
s
Ý is1
F
xi
ž / ž /
is decreasing with m.
wi xi
wi
n
wi q
Ý
F
ismq1
n
wi q
Ý ismq1
F
ž
xi
Ý ismq1
Ž n y m . x 1, n x mq1, n wi
/
wi ,
?
x 1, n wi
0
wi
0FmFn
728
ABRAMOVICH, MOND, AND PECARIC ˇ ´
The special case F Ž t . s t 1r p , p ) 1, n
B Ž n. s
Ý is1
F
xi
ž / wi
n
wi F
Ý is1
F
x 1, n
ž / wi
wi
was dealt with in w2x for positive x i , i s 1, . . . , n. REFERENCES 1. S. Abramovich, B. Mond, and J. E. Pecaric, inequality, J. Math. Anal. ˘ ´ Sharpening Holder’s ¨ Appl. 196 Ž1995., 1131]1134. 2. D. C. Barnes, Supplements of Holder’s inequality, Canad. J. Math. 34, No. 3 Ž1984., ¨ 405]420. 3. P. S. Bullen, D. S. Mitrinovic, ´ and P. M. Vasic, ´ ‘‘Means and Their Inequalities,’’ Reidel, DordechtrBostonrLancasterrTokyo, 1988. 4. J. W. Hovenier, Sharpening Cauchy’s inequality, J. Math. Anal. Appl. 186 Ž1994., 156]160. 5. D. S. Mitrinovic, ´ J. E. Pecaric, ˘ ´ and A. M. Fink, ‘‘Classical and New Inequalities in Analysis,’’ Kluwer Academic, DordrechtrBostonrLondon, 1993. 6. J. E. Pecaric ˘ ´ and S. Abramovich, On new majorization theorem, Rocky Mountain J. Math., in press.
214, 721]728 Ž1997.
AY975527
Sharpening Jensen’s Inequality and a Majorization Theorem S. Abramovich Department of Mathematics and Computer Science, Uni¨ ersity of Haifa, Haifa, Israel
B. Mond School of Mathematics, La Trobe Uni¨ ersity, Bundoora, Victoria, 3082, Australia
and J. E. Pecaric ˇ ´ Faculty of Textile Technology, Uni¨ ersity of Zagreb, Pierottije¨ a 6, 10000, Zagreb, Croatia Submitted by A. M. Fink Received October 10, 1995
1. INTRODUCTION In this paper we shall show that sharpening Holder’s Inequality can be ¨ generalized to sharpen Jensen’s inequality for concave and convex functions w1, 4x. We shall also sharpen a majorization theorem for concave functions and two sequences Žor functions. when only one of them is monotonic w6x.
2. SHARPENING JENSEN’S INEQUALITY Let us consider a concave function F of several variables defined on an open convex set u in R n w3, pp. 31, 32x. Let the real functions 0 - w Ž t ., and f i Ž t . w Ž t ., i s 1, . . . , n be integrable functions on the real interval 721 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
ABRAMOVICH, MOND, AND PECARIC ˇ ´
722
w a, b x. Then Jensen’s Inequality in its integral form w5, p. 10x is b
Ha
F Ž f 1 , . . . , f n . w Ž t . dt b
Ha
b
FF
w Ž t . dt
b
Ha
b
f 1w dt
Ha
Ha
,...,
b
w Ž t . dt
Ha
f n w dt
w Ž t . dt
0
. Ž 2.1.
Using this inequality we get the following Let Ž x 1 , . . . , x n . be a complex function in n complex ¨ ari-
THEOREM 1. ables and let
< F Ž x1 , . . . , x n . < F < F Ž < x1 < , . . . , < x n <. < .
Ž 2.2.
Let also < F Ž x 1 , . . . , x n .< be a conca¨ e function for x s Ž x 1 , . . . , x n . g R n. If f i Ž t ., i s 1, . . . , n, w Ž t . are complex functions of real ¨ ariables, and Ž f i t . w Ž t ., w Ž t . are integrable on w a, b x, then c
AŽ c . s
Ha w Ž t . F Ž f Ž t . , . . . , f Ž t . dt n
1
q
Hc
b
< w Ž t . < dt F
Hc
b
Hc
< w Ž t . f 1 < dt b
,...,
Hc
< w Ž t . < dt
b
Hc
< w Ž t . f n < dt b
< w Ž t . < dt
is a decreasing function in c, a F c F b. Analogous results for the discrete case are obvious. Proof. Let a F c - d F b, AŽ d . s
d
Ha
wF Ž f 1 , . . . , f n . dt b
b
Hd < w < dt F
q
F
Hd < wf < dt b
c n
q
b
Hd < w < dt F
Hc
Hd < wf < dt
d
b
0
< wF Ž < f 1 < , . . . , < f n < . < dt b
Hd < wf < dt n
1
Hd < w < dt
b
Hd < w < dt
b
q
Hd < wf < dt n
,...,
Hd < w < dt
Ha wF Ž f , . . . , f . dt 1
b
1
,...,
b
Hd < w < dt
0
0
Ž 2.3.
SHARPENING JENSEN’S INEQUALITY
F
c
Ha wF Ž f , . . . , f . dt n
1
q
Hc
d
< w < dt F
b
Hd < w < dt F
q
F
Hc
d
Hc
< wf 1 < dt
d
,..., < w < dt
b
b
Hc
d
< wf 1 < dt q
Hc
d
< w < dt q
q
< wf n < dt
d
< w < dt
Hd < wf < dt n
,...,
Hd < w < dt n
1
d
b
1
c
Hc
Hc
Hd < wf < dt
Ha wF Ž f , . . . , f . dt = F
s
723
b
Hd < w < dt
žH
d
c
< w < dt q
b
Hd < wf < dt 1
b
,...,
Hd < w < dt
0 0
b
Hd < w < dt
Hc
d
Hc
/ b
wf n < dt q
Hd < wf < dt
d
Hd < w < dt
< w < dt q
n
b
0
c
Ha wF Ž f , . . . , f . dt n
1
q
Hc
b
< w < dt F
Hc
b
Hc
< wf 1 < dt
b
,..., < w < dt
Hc
b
Hc
< wf n < dt
b
< w < dt
0
s AŽ c . .
The first inequality is a result of the given inequality Ž2.2.. The second inequality is Jensen’s inequality Ž2.1. applied on c F t F d to the given concave function < F Ž x 1 , . . . , x n .< on Ž x 1 , . . . , x n . g R n. The third inequality follows again from the concavity of < F Ž x 1 , . . . , x n .< on Ž x 1 , . . . , x n . g R n. The inequality AŽ d . F AŽ c . becomes equality only if the arguments of Hac wF Ž f 1 , . . . , f n . dt and of wF Ž f 1 , . . . , f n . on w c, d x are the same, < wf i <, i s 1, . . . , n, are constants on w c, d x, < F Ž f 1 , . . . , f n .< s < F Ž< f 1 <, . . . , < f n <.< on w c,d x, and if d - b, Hcd < f i w < dt s Hdb < f i w < dt, i s 1, . . . , n. EXAMPLE 1. Sharpening Holder’s Inequality. Let F Ž x 1 , . . . , x n . s ¨ n ai Ł is1 x i , x i g C, i s 1, . . . , n, where a i G 0, i s 1, . . . , n, Ý nis1 a i F 1, a0 s 1 y Ý nis1 a i .
ABRAMOVICH, MOND, AND PECARIC ˇ ´
724
Let f i Ž t ., i s 0, 1, . . . , n, be a complex integrable function on the real interval a F t F b. Then
AŽ c . s
s
n
c
fi Ž t .
ž /
Ha f Ž t . is1 Ł 0
c
ai
dt q
f0 Ž t .
n
n
Ha is0 Ł
Ž t . dt q Ł
f ia i
is0
žH c
b
Hc
b
n
< f 0 Ž t . < dt Ł is1
< f i Ž t . < dt
b
< f 0 Ž t . < dt
Hc Hc
ai
b
0
ai
< f i Ž t . < dt
/
n is a decreasing function in c, a F c F b, because F s Ł is1 x ia i , x i ) 0, n a i ) 0, i s 1, . . . , n, Ý is1 a i - 1, is a concave function. The case n s 1, a1 s 1r2 was considered in w4x, and the case n s 1, 0 - a1 - 1 was considered in w1x. Analogous results for the discrete case are obvious.
EXAMPLE 2. Sharpening Minkowski’s Inequality. Let F Ž x 1 , . . . , x n . s a. a Ž1 q Ý nis1 x 1r , a G 1. i n As F is a concave function for Ž x 1 , . . . , x n . g Rq , we get under the same conditions as in Example 1 on f i Ž t ., i s 1, . . . , n, and on w Ž t . s f 0 Ž t . that AŽ c . s
c
Hd f Ž t . 0
q
Hc
s
c
b
ž
n
1q
a
1ra
ž / / f0 Ž t .
n
Hc
b
dt 1ra
a
< f i Ž t . dt <
00
< f 0 Ž t . < dt 1 q
Ý
is1
n
ž
Ý is1
fi Ž t .
1ra
Hd Ý Ž f Ž t . . i
is0
a
Hc
b
< f 0 Ž t . dt <
n
dt q
ž Ý žH is0
c
b
a
1ra
< f i Ž t . < dt
/
/
is a decreasing function in c, d F c F b. Analogous results are obvious for the discrete case.
3. SHARPENING A MAJORIZATION THEOREM In w6x the following majorization theorem was proved. THEOREM A w6 Theorem 3x. Let f and grf be increasing functions on Ž0, 1.. Let w ) 0 and let g ) 0 or f ) 0. Let also fw and gw be integrable functions on w0, 1x and H01 fw dtrH01 gw dt G 0.
SHARPENING JENSEN’S INEQUALITY
725
Let F be a conca¨ e function. Then for e¨ ery k ) 0 1
1
H0
F Ž kg . w dt F
1
H0
H0
gw dt
0
F k
f w dt.
1
Ž 3.1.
fw dt
H0
Specifically, if g Ž t . is a conca¨ e function satisfying g Ž0. F 0 H01 g Ž t . w Ž t . dt G 0 we get the inequality 1
1
H0
F Ž g Ž t . . w Ž t . dt F
1
H0
F
H0
g Ž t . w Ž t . dt 1
H0
and
0
t w dt. tw Ž t . dt
In the following theorem we sharpen this Majorization Theorem as was done for Theorem 1. THEOREM 2. Let f ) 0 and grf be increasing functions on Ž a, b .. Let fw and gw be integrable on w a, b x. Also, let w ) 0 and Hab g Ž t . w Ž t . dt G 0. Let F Ž t . be a conca¨ e function and let
BŽ c. s
c
Ha
F Ž g . w Ž t . dt q
Hc
b
F
Hc
b
gw dt
0 0 Hc
b
f w Ž t . dt.
Ž 3.2.
fw dt
Then B Ž c . is a decreasing function of c, a F c F b, and
BŽ c. F
c
Ha F Ž g . w Ž t . dt q Hc
b
Hc
wŽ t. F
b
Hc
gw dt
b
dt.
Ž 3.3.
w dt
In the special case that g Ž t . is a con¨ ex function on w0, b x, g Ž0. F 0, and H0b g Ž t . w Ž t . dt G 0 we get that
BŽ c. s
c
H0
F Ž g . w Ž t . dt q
is a decreasing function of c, 0 F c F b.
Hc
b
F
Hc
b
gw dt
0 Hc
b
t w dt
tw dt
Ž 3.4.
ABRAMOVICH, MOND, AND PECARIC ˇ ´
726
Proof. Let a F c - d F b b
BŽ d. s
d
Ha
F Ž g . w Ž t . dt q
b
Hd F
Hd gw dt
0 b
f w dt
Hd fw dt
b
s
F
c
Ha F Ž g . w Ž t . dt q Hc c
Ha F Ž g . w Ž t . dt q Hc
d
b
F Ž g . w Ž t . dt q
F
Hc
b
b
Hd F
0 b
Hd
f w dt
fw dt
gw dt
0 Hc
Hd gw dt
b
f w dt s B Ž c . .
Ž 3.5.
fw dt
To prove Ž3.5., let us define
¡g Ž t . ,
cFt-d
~ Hd gw dt f, b fw dt H ¢d
d F t - b.
b
GŽ t . s
We shall see that GŽ t . plays the same role as g Ž t . in Theorem 4. In order to show that GŽ t .rf Ž t . is increasing on Ž c, b . it is enough to show that b
g Ž dq . f Ž dq .
F
Hd gw dt b
.
Hd fw dt
Let us assume that b
g Ž dq . f Ž dq .
)
Hd gw dt b
.
Hd fw dt
Because grf is increasing we therefore get g Ž t .Hdb fw dt ) f Ž t .Hdb gw dt, d F t F b which by integration on w d, b x leads to a contradiction. Also, Hcb GŽ t . w Ž t . dt s Hcb g Ž t . w Ž t . dt ) 0 and therefore we get the inequality in Ž3.5. by Theorem A. Thus the assertion that B Ž c . is decreasing is proved. By applying Jensen’s inequality to B Ž c . we get Ž3.3.. In case g Ž t . is a convex function for t G 0 satisfying g Ž0. F 0 we get that B Ž c . defined in Ž3.4. is decreasing because g Ž t .rt is increasing for t ) 0.
SHARPENING JENSEN’S INEQUALITY
727
By a similar proof as those of Theorems 1 and 2 we get THEOREM 3. Let F Ž t . be a complex function defined on C and let < F Ž t .< F < F Ž< t <.<. Let < f Ž t .<, < g Ž t .rf Ž t .< be increasing functions on the real inter¨ al a F t F b then c
Ha F Ž g Ž t . . w Ž t . dt
q
Hc
b
Hc
F
b
< gw < dt
b
< fw < dt
Hc
0
< f Ž t . < w Ž t . dt
is decreasing with c, a F c F b. Moreo¨ er, c
Ha
F Ž g Ž t . . w Ž t . dt q
c
F
Ha
Hc
b
F
F Ž g Ž t . . w Ž t . dt q
Hc
b
< gw < dt
b
< fw < dt
Hc Hc
b
0
< f Ž t . < w Ž t . dt
< w Ž t . dt < F
Hc
b
< gw < dt
0 Hc
b
.
< w < dt
EXAMPLE 3. In this example we use the obvious discrete version of Theorem 2. Let 0 - f i s d irwi and g irf i s x ird i , i s 1, . . . , n, be increasing sequences such that n
d i s x 1, n s
Ý
x irn,
wi ) 0, i s 1, . . . , n.
is1
Define n
x mq 2 , n s
xi
Ý ismq2
nymy1
.
Let F Ž t . be a concave function. Then, n m
B Ž m. s
Ý
F
is1 m
s
Ý is1
F
xi
ž / ž /
is decreasing with m.
wi xi
wi
n
wi q
Ý
F
ismq1
n
wi q
Ý ismq1
F
ž
xi
Ý ismq1
Ž n y m . x 1, n x mq1, n wi
/
wi ,
?
x 1, n wi
0
wi
0FmFn
728
ABRAMOVICH, MOND, AND PECARIC ˇ ´
The special case F Ž t . s t 1r p , p ) 1, n
B Ž n. s
Ý is1
F
xi
ž / wi
n
wi F
Ý is1
F
x 1, n
ž / wi
wi
was dealt with in w2x for positive x i , i s 1, . . . , n. REFERENCES 1. S. Abramovich, B. Mond, and J. E. Pecaric, inequality, J. Math. Anal. ˘ ´ Sharpening Holder’s ¨ Appl. 196 Ž1995., 1131]1134. 2. D. C. Barnes, Supplements of Holder’s inequality, Canad. J. Math. 34, No. 3 Ž1984., ¨ 405]420. 3. P. S. Bullen, D. S. Mitrinovic, ´ and P. M. Vasic, ´ ‘‘Means and Their Inequalities,’’ Reidel, DordechtrBostonrLancasterrTokyo, 1988. 4. J. W. Hovenier, Sharpening Cauchy’s inequality, J. Math. Anal. Appl. 186 Ž1994., 156]160. 5. D. S. Mitrinovic, ´ J. E. Pecaric, ˘ ´ and A. M. Fink, ‘‘Classical and New Inequalities in Analysis,’’ Kluwer Academic, DordrechtrBostonrLondon, 1993. 6. J. E. Pecaric ˘ ´ and S. Abramovich, On new majorization theorem, Rocky Mountain J. Math., in press.