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An “Error Term” for the Ky Fan Inequality
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
220, 774]777 Ž1998.
AY975834
NOTE An ‘‘Error Term’’ for the Ky Fan Inequality A. McD. Mercer Department of Mathematics and Statistics, Uni¨ ersity of Guelph, Guelph, Ontario, N1G 2W1, Canada Submitted by A. M. Fink Received August 20, 1997
For k s 1, 2, . . . , n let w k G 0 and let Ýw k s 1. Let F be a function defined and continuous on some interval J of the real axis. Take x s Ž x 1 , x 2 , . . . , x n . g J n and write L Ž x, F . s
Ý wk F Ž x k . y F Ž Ý wk x k . .
Jensen’s classical inequality asserts that if F is convex on J then L Ž x, F . G 0
for all x g J n .
If we write e k for the function defined by e k Ž x . ' x k Ž k s 0, 1, . . . . we see that L Ž x, e0 . s L Ž x, e1 . s 0 ; x g J n
but
L Ž x, e2 . / 0 in general.
This suggests that when F has a continuous second derivative we might expect Jensen’s inequality to possess an ‘‘error term’’ involving F Y . And indeed it is easy to prove the following lemma. LEMMA. When the function F has a continuous second deri¨ ati¨ e in J then L Ž x, F . s 12 L Ž x, e2 . F Y Ž j .
for some j g sp Ž x k . .
Ž Here we ha¨ e written spŽ x k . to denote the interior of the smallest closed inter¨ al containing all of the x k .. 774 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
775
NOTE
Proof. Denoting Ýw k x k by M we have F Ž x k . s F Ž M . q Ž x k y M . F X Ž M . q 12 Ž x k y M . F Y Ž j k . , 2
where k s 1, 2, . . . , n and j k lies between x k and M. Multiplying this by w k and summing over k gives L Ž x, F . s
1 2
Ý wk Ž x k y M . 2 F Y Ž j k . s 12 F Y Ž h . Ý wk Ž x k y M . 2 , h g sp Ž x k .
because of the continuity of F Y . When we expand the term in the last summation we get the result. Note that L Žx, e2 . s Ýw k Ž x k y M . 2 G 0. EXAMPLE. Let M r Žu. s wÝw k u kr x1r r Ž r / 0. be the generalized mean of the positive numbers u k and write x k s u ks . Now take F Ž x . s x r r s and apply the Lemma. We get
Ý wk x kr r s y Ý wk x k
rrs
s
1 2
L Ž x, e2 .
r Ž r y s. s2
j r r sy2 ,
j g sp Ž x k .
In terms of the u k this reads M rr Ž u . y M rs Ž u . s
1 2
L Ž x, e2 .
r Ž r y s. s2
h ry2 s ,
h g sp Ž u k .
which improves the classical result that M r Ž u. G M s Ž u.
whenever r ) s.
In general, this Lemma can be used to find an ‘‘error term’’ for any inequality derived from Jensen’s inequality provided the function F involved has a continuous second derivative, but our main objective here is to improve the following theorem which is known as the Žweighted. Ky Fan inequality. THEOREM A. Let a k g Ž0, 12 x Ž k s 1, 2, . . . , n.. If AŽ a. and GŽ a. are the weighted arithmetic and geometric means of the a k and if AŽ1 y a. and GŽ1 y a. denote the corresponding means of the numbers 1 y a k , then AŽ a. G Ž a.
G
AŽ 1 y a. G Ž 1 y a.
.
Ž In this we take the weights to be the numbers w k mentioned in the first paragraph..
776
NOTE
This inequality has been proved many times and we refer the reader to the references. Now the proof in w10x used Jensen’s inequality and so the above lemma can be applied to it. When this is done we get the following result. THEOREM 1.
Let a k g Ž0, 1. Ž k s 1, 2, . . . , n.. Then
AŽ a. G Ž 1 y a. y AŽ 1 y a. G Ž a. s 12 L Ž x, e2 . G Ž a . q G Ž 1 y a . a Ž 1 y 2 a . Ž 1 y a . , where a g spŽ a k . and x k s log a k y logŽ1 y a k .. This result implies Theorem A whenever the a k are in the interval Ž0, 12 x. Another consequence in this case is that A Ž a . G Ž 1 y a . y A Ž 1 y a . G Ž a . s g L Ž x, e2 . G Ž a . q G Ž 1 y a . , where 0FgF
'3 36
.
Proof. With a k g Ž0, 1. we write ak s
bk 1 q bk
and
e x k s bk
Ž 1.
so that bk g Ž0, q`. and x k g Žy`, q`.. Take F Ž x . s Ž e x y K .rŽ1 q e x . and apply the Lemma to this function. We get L Ž x, F . s 12 L Ž x, e2 . Ž 1 q K . e j Ž 1 y e j . Ž 1 q e j .
y3
Ž 2.
for some j g spŽ x k .. Here L Žx, F . is, by definition,
Ý wk
e xk y K 1 q e xk
y
eM y K 1 q eM
and, as before, M s Ýw k x k . We now use the latter of Ž1. to replace the x k by bk . We write g s e M s Ł bkw k and then put K s g. When this is done and b is written for e j , Ž2. reads bk
Ý wk 1 q b
y g Ý wk k
for some b g spŽ bk ..
1 1 q bk
s
1 2
L Ž x, e2 . Ž 1 q g .
b Ž1 y b .
Ž1 q b .
3
NOTE
777
Finally, since a k s bkrŽ1 q bk . and g s GŽ a.rGŽ1 y a. we can now express this in terms of the a k to get the required result, namely AŽ a. G Ž 1 y a. y AŽ 1 y a. G Ž a. s 12 L Ž x, e2 . G Ž a . q G Ž 1 y a . a Ž 1 y a . Ž 1 y 2 a . for some a s brŽ1 q b . g spŽ a k .. For the sake of brevity we have not replaced the x in the L functional but it is to be calculated as x k s log a k y logŽ1 y a k .. To conclude, we mention that the upper bound '3 r36 Žs 0.048112 . . . . for g in Theorem 1 is likely best-possible as one can find, already in case n s 4, values of a k namely 0.20648, 0.20852, 0.21343, 0.21840 which give g s 0.048096 . . . .
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
H. Alzer, On an inequality of Ky Fan, J. Math. Anal. Appl. 137 Ž1989., 168]172. H. Alzer, A short proof of Ky Fan’s inequality, Arch. Math. B 27 Ž1991., 199]200. E. F. Beckenbach and R. Bellman, ‘‘Inequalities,’’ Springer-Verlag, Berlin, 1961. G. H. Hardy, J. E. Littlewood, and G. Polya, ‘‘Inequalities,’’ Cambridge Univ. Press, Cambridge, UK, 1952. N. Levinson, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 Ž1964., 133]134. M. T. McGregor, On some inequalities of Ky Fan and Wang-Wang, J. Math. Anal. Appl. 180 Ž1993., 182]188. C. L. Wang, On a Ky Fan inequality of the complementary A-G type and its variants, J. Math. Anal. Appl. 73 Ž1980., 501]505. C. L. Wang, Functional equation approach to inequalities, II, J. Math. Anal. Appl. 78 Ž1980., 522]530. W. L. Wang and P. F. Wang, A class of inequalities for the symmetric functions, Acta Math. Sinica 27 Ž1984., 485]497. wIn Chinesex A. McD. Mercer, A short proof of Ky Fan’s arithmetic-geometric inequality, J. Math. Anal. Appl. 204 Ž1996., 940]943.
220, 774]777 Ž1998.
AY975834
NOTE An ‘‘Error Term’’ for the Ky Fan Inequality A. McD. Mercer Department of Mathematics and Statistics, Uni¨ ersity of Guelph, Guelph, Ontario, N1G 2W1, Canada Submitted by A. M. Fink Received August 20, 1997
For k s 1, 2, . . . , n let w k G 0 and let Ýw k s 1. Let F be a function defined and continuous on some interval J of the real axis. Take x s Ž x 1 , x 2 , . . . , x n . g J n and write L Ž x, F . s
Ý wk F Ž x k . y F Ž Ý wk x k . .
Jensen’s classical inequality asserts that if F is convex on J then L Ž x, F . G 0
for all x g J n .
If we write e k for the function defined by e k Ž x . ' x k Ž k s 0, 1, . . . . we see that L Ž x, e0 . s L Ž x, e1 . s 0 ; x g J n
but
L Ž x, e2 . / 0 in general.
This suggests that when F has a continuous second derivative we might expect Jensen’s inequality to possess an ‘‘error term’’ involving F Y . And indeed it is easy to prove the following lemma. LEMMA. When the function F has a continuous second deri¨ ati¨ e in J then L Ž x, F . s 12 L Ž x, e2 . F Y Ž j .
for some j g sp Ž x k . .
Ž Here we ha¨ e written spŽ x k . to denote the interior of the smallest closed inter¨ al containing all of the x k .. 774 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
775
NOTE
Proof. Denoting Ýw k x k by M we have F Ž x k . s F Ž M . q Ž x k y M . F X Ž M . q 12 Ž x k y M . F Y Ž j k . , 2
where k s 1, 2, . . . , n and j k lies between x k and M. Multiplying this by w k and summing over k gives L Ž x, F . s
1 2
Ý wk Ž x k y M . 2 F Y Ž j k . s 12 F Y Ž h . Ý wk Ž x k y M . 2 , h g sp Ž x k .
because of the continuity of F Y . When we expand the term in the last summation we get the result. Note that L Žx, e2 . s Ýw k Ž x k y M . 2 G 0. EXAMPLE. Let M r Žu. s wÝw k u kr x1r r Ž r / 0. be the generalized mean of the positive numbers u k and write x k s u ks . Now take F Ž x . s x r r s and apply the Lemma. We get
Ý wk x kr r s y Ý wk x k
rrs
s
1 2
L Ž x, e2 .
r Ž r y s. s2
j r r sy2 ,
j g sp Ž x k .
In terms of the u k this reads M rr Ž u . y M rs Ž u . s
1 2
L Ž x, e2 .
r Ž r y s. s2
h ry2 s ,
h g sp Ž u k .
which improves the classical result that M r Ž u. G M s Ž u.
whenever r ) s.
In general, this Lemma can be used to find an ‘‘error term’’ for any inequality derived from Jensen’s inequality provided the function F involved has a continuous second derivative, but our main objective here is to improve the following theorem which is known as the Žweighted. Ky Fan inequality. THEOREM A. Let a k g Ž0, 12 x Ž k s 1, 2, . . . , n.. If AŽ a. and GŽ a. are the weighted arithmetic and geometric means of the a k and if AŽ1 y a. and GŽ1 y a. denote the corresponding means of the numbers 1 y a k , then AŽ a. G Ž a.
G
AŽ 1 y a. G Ž 1 y a.
.
Ž In this we take the weights to be the numbers w k mentioned in the first paragraph..
776
NOTE
This inequality has been proved many times and we refer the reader to the references. Now the proof in w10x used Jensen’s inequality and so the above lemma can be applied to it. When this is done we get the following result. THEOREM 1.
Let a k g Ž0, 1. Ž k s 1, 2, . . . , n.. Then
AŽ a. G Ž 1 y a. y AŽ 1 y a. G Ž a. s 12 L Ž x, e2 . G Ž a . q G Ž 1 y a . a Ž 1 y 2 a . Ž 1 y a . , where a g spŽ a k . and x k s log a k y logŽ1 y a k .. This result implies Theorem A whenever the a k are in the interval Ž0, 12 x. Another consequence in this case is that A Ž a . G Ž 1 y a . y A Ž 1 y a . G Ž a . s g L Ž x, e2 . G Ž a . q G Ž 1 y a . , where 0FgF
'3 36
.
Proof. With a k g Ž0, 1. we write ak s
bk 1 q bk
and
e x k s bk
Ž 1.
so that bk g Ž0, q`. and x k g Žy`, q`.. Take F Ž x . s Ž e x y K .rŽ1 q e x . and apply the Lemma to this function. We get L Ž x, F . s 12 L Ž x, e2 . Ž 1 q K . e j Ž 1 y e j . Ž 1 q e j .
y3
Ž 2.
for some j g spŽ x k .. Here L Žx, F . is, by definition,
Ý wk
e xk y K 1 q e xk
y
eM y K 1 q eM
and, as before, M s Ýw k x k . We now use the latter of Ž1. to replace the x k by bk . We write g s e M s Ł bkw k and then put K s g. When this is done and b is written for e j , Ž2. reads bk
Ý wk 1 q b
y g Ý wk k
for some b g spŽ bk ..
1 1 q bk
s
1 2
L Ž x, e2 . Ž 1 q g .
b Ž1 y b .
Ž1 q b .
3
NOTE
777
Finally, since a k s bkrŽ1 q bk . and g s GŽ a.rGŽ1 y a. we can now express this in terms of the a k to get the required result, namely AŽ a. G Ž 1 y a. y AŽ 1 y a. G Ž a. s 12 L Ž x, e2 . G Ž a . q G Ž 1 y a . a Ž 1 y a . Ž 1 y 2 a . for some a s brŽ1 q b . g spŽ a k .. For the sake of brevity we have not replaced the x in the L functional but it is to be calculated as x k s log a k y logŽ1 y a k .. To conclude, we mention that the upper bound '3 r36 Žs 0.048112 . . . . for g in Theorem 1 is likely best-possible as one can find, already in case n s 4, values of a k namely 0.20648, 0.20852, 0.21343, 0.21840 which give g s 0.048096 . . . .
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
H. Alzer, On an inequality of Ky Fan, J. Math. Anal. Appl. 137 Ž1989., 168]172. H. Alzer, A short proof of Ky Fan’s inequality, Arch. Math. B 27 Ž1991., 199]200. E. F. Beckenbach and R. Bellman, ‘‘Inequalities,’’ Springer-Verlag, Berlin, 1961. G. H. Hardy, J. E. Littlewood, and G. Polya, ‘‘Inequalities,’’ Cambridge Univ. Press, Cambridge, UK, 1952. N. Levinson, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 Ž1964., 133]134. M. T. McGregor, On some inequalities of Ky Fan and Wang-Wang, J. Math. Anal. Appl. 180 Ž1993., 182]188. C. L. Wang, On a Ky Fan inequality of the complementary A-G type and its variants, J. Math. Anal. Appl. 73 Ž1980., 501]505. C. L. Wang, Functional equation approach to inequalities, II, J. Math. Anal. Appl. 78 Ž1980., 522]530. W. L. Wang and P. F. Wang, A class of inequalities for the symmetric functions, Acta Math. Sinica 27 Ž1984., 485]497. wIn Chinesex A. McD. Mercer, A short proof of Ky Fan’s arithmetic-geometric inequality, J. Math. Anal. Appl. 204 Ž1996., 940]943.