GENERAL ⎜ ARTICLE
The Logarithmic Mean
Rajendra Bhatia
T h e in e q u a lity b e tw e e n th e a r ith m e tic m e a n (A M )
a n d g e o m e tr ic m e a n (G M ) o f tw o p o sitiv e n u m b e r s is w e ll k n o w n . T h is a r tic le in tr o d u c e s th e
lo g a r ith m ic m e a n , sh o w s h o w it le a d s to r e ¯ n e m e n ts o f th e A M { G M in e q u a lity . S o m e a p p lic a tio n s a n d p r o p e r tie s o f th is m e a n a r e sh o w n .
S o m e o th e r m e a n s a n d r e la te d in e q u a litie s a re
d isc u sse d .
O n e o f th e b est k n ow n a n d m o st u sed in eq u a lities in
m a th em a tics is th e in eq u a lity b etw een th e h a rm o n ic,
g eo m etric, a n d a rith m etic m ea n s. If a a n d b a re p o sitiv e n u m b ers, th ese m ea n s a re d e¯ n ed , resp ectiv ely, a s
µ ¡1
¶¡1
a + b¡ 1
H (a ;b) =
;
2
p
a+ b
G (a ;b) =
a b; A (a ;b) =
;
(1 )
2
a n d th e in eq u a lity say s th a t
H (a ;b) · G (a ;b) · A (a ;b):
Rajendra Bhatia is a
Professor at the Indian
Statistical Institute, New
Delhi. Some of his recent
work is on means of
matrices. But he is not a
mean theorist.
(2 )
M ea n s o th er th a n th e th ree `cla ssica l' o n es d e¯ n ed in
(1 ) a re u sed in d i® eren t p ro b lem s. F o r ex a m p le, th e
root m ean squ are
µ 2
¶1 = 2
a + b2
;
(3 )
B 2 (a ;b) =
2
is o ften u sed in va rio u s co n tex ts. F o llow in g th e m a th em a ticia n 's p en ch a n t fo r g en era lisa tio n , th e fo u r m ea n s
m en tio n ed a b ov e ca n b e su b su m ed in th e fa m ily
µ p
¶
a + bp 1 = p
B p (a ;b) =
; ¡1 < p < 1 ;
(4 )
2
RESONANCE ⎜ June 2008
Keywords
Inequalities, logarithmic mean,
power mean, Holder mean, arithmetic-geometric mean (AGM),
arithmetic-mean geometricmean inequality (AM-GM inequality), Heinz mean.
583
GENERAL ⎜ ARTICLE
A substantial part of
the mathematics
classic Inequalities
by Hardy, Littlewood
va rio u sly k n ow n a s bin om ial m ean s, pow er m ean s, o r
H oÄ lder m ean s. W h en p = ¡ 1;1; a n d 2 ; resp ectiv ely,
B p (a ;b) is th e h a rm o n ic m ea n , th e a rith m etic m ea n , a n d
th e ro o t m ea n sq u a re. If w e u n d ersta n d B 0 (a ;b) to m ea n
B 0 (a ;b) = lim B p (a ;b);
and Pölya is devoted
to the study of these
means and their
applications.
p! 0
th en
B 0 (a ;b) = G (a ;b):
(5 )
In a sim ila r v ein w e ca n see th a t
µ p
¶1 = p
a + bp
B 1 (a ;b) := lim
= m a x (a ;b);
p! 1
2
µ p
¶
a + bp 1 = p
B ¡1 (a ;b) :=
lim
= m in (a ;b):
p! ¡1
2
A little ca lcu la tio n sh ow s th a t
B p (a ;b) · B q (a ;b)
if p · q :
(6 )
T h is is a stro n g g en era liza tio n o f th e in eq u a lity (2 ). W e
m ay say th a t fo r ¡ 1 · p · 1 , th e fa m ily B p in terpolates
b etw een th e th ree m ea n s in (1 ) a s d o es th e in eq u a lity
(6 ) w ith resp ect to (2 ).
A su b sta n tia l p a rt o f th e m a th em a tics cla ssic In equ alities b y G H a rd y, J E L ittlew o o d a n d G P oÄ ly a is d ev o ted
to th e stu d y o f th ese m ea n s a n d th eir a p p lica tio n s. T h e
b o o k h a s h a d q u ite a few su ccesso rs, a n d y et n ew p ro p erties o f th ese m ea n s co n tin u e to b e d iscov ered .
T h e p u rp o se o f th is a rticle is to in tro d u ce th e rea d er to
th e logarithm ic m ean , so m e o f its a p p lica tio n s, a n d so m e
v ery p retty m a th em a tics a ro u n d it.
T h e lo g a rith m ic m ea n o f tw o p o sitiv e n u m b ers a a n d b
is th e n u m b er L (a ;b) d e¯ n ed a s
L (a ;b) =
584
a¡ b
lo g a ¡ lo g b
fo r a 6= b;
(7 )
RESONANCE ⎜ June 2008
GENERAL ⎜ ARTICLE
w ith th e u n d ersta n d in g th a t
The logarithmic
L (a ;a ) = lim L (a ;b) = a :
mean always falls
betwen the
T h ere a re o th er in terestin g rep resen ta tio n s fo r th is o b ject, a n d th e rea d er sh o u ld ch eck th e va lid ity o f th ese
fo rm u la s:
Z1
a tb 1 ¡t d t;
(8 )
L (a ;b) =
0
Z1
1
dt
=
;
(9 )
L (a ;b)
ta + (1 ¡ t)b
0
Z1
dt
1
=
:
(1 0 )
L (a ;b)
(t + a )(t + b)
0
geometric mean
and the arithmetic
b! a
mean.
T h e lo g a rith m ic m ea n a lw ay s fa lls b etw een th e g eo m etric a n d th e a rith m etic m ea n s; i.e.,
G (a ;b) · L (a ;b) · A (a ;b):
(1 1 )
W e in d ica te th ree d i® eren t p ro o fs o f th is a n d in v ite th e
rea d er to ¯ n d m o re.
W h en a = b; a ll th e th ree m ea n s in (1 1 ) a re eq u a l to a :
S u p p o se a > b; a n d p u t w = a = b: T h e ¯ rst in eq u a lity in
(1 1 ) is eq u iva len t to say in g
p
w ·
w ¡ 1
lo g w
fo r w > 1:
R ep la cin g w b y u 2 ; th is is th e sa m e a s say in g
2 lo g u ·
u2 ¡ 1
u
fo r u > 1 :
(1 2 )
T h e tw o fu n ctio n s f (u ) = 2 lo g u ; a n d g (u ) = (u 2 ¡ 1 )= u
a re eq u a l to 0 a t u = 1; a n d a sm a ll ca lcu la tio n sh o w s
th a t f 0(u ) < g 0(u ) fo r u > 1: T h is p rov es th e d esired
in eq u a lity (1 2 ), a n d w ith it th e ¯ rst in eq u a lity in (1 1 ).
In th e sa m e w ay, th e seco n d o f th e in eq u a lities (1 1 ) ca n
b e red u ced to
u ¡ 1
lo g u
·
fo r u ¸ 1 :
u + 1
2
RESONANCE ⎜ June 2008
585
GENERAL ⎜ ARTICLE
In this form we
recognize it as one
of the fundamental
inequalities of
analysis:
t ≤ sinh t for all
t ≥ 0.
an d p roved b y calcu latin g d erivatives.
A seco n d p ro o f go es a s fo llow s. T w o a p p licatio n s of th e
arith m etic-geom etric m ea n in eq u ality sh ow th at
p
t2 + 2 t a b + a b · t2 + t(a + b) + a b
µ
¶
a+ b 2
2
· t + t(a + b) +
2
for a ll t ¸ 0 : U sin g th is on e ¯ n d s th at
Z1
Z1
Z1
dt
dt
dt
p
·
:
·
(t + a )(t + b)
(t + a +2 b )2
(t + a b)2
0
0
0
E va lu a tio n o f th e in teg ra ls sh ow s th a t th is is th e sa m e
as th e assertion in (1 1).
S in ce a an d b are p o sitive, w e can ¯ n d rea l n u m b ers
x an d y su ch th a t a = e x an d b = e y : T h en th e ¯ rst
in eq u ality in (11 ) is eq u iva len t to th e sta tem en t
e (x +
y )= 2
·
ex ¡ ey
;
x ¡ y
or
1·
e (x ¡ y )= 2 ¡ e (y ¡ x )= 2
:
x ¡ y
T h is can b e ex p ressed also as
1·
sin h (x ¡ y )= 2
:
(x ¡ y )= 2
In th is form w e recogn ise it a s on e of th e fu n d am en tal
in eq u alities of an a ly sis: t · sin h t for all t ¸ 0 : V ery
sim ila r calcu lation s sh ow th a t th e seco n d in eq u a lity in
(11 ) can b e red u ced to th e fa m ilia r fa ct tan h t · t for
all t ¸ 0 :
E ach of ou r th ree p ro o fs sh ow s th a t if a 6= b; th en
G (a ;b) < L (a ;b) < A (a ;b): O n e o f th e rea son s for th e
586
RESONANCE ⎜ June 2008
GENERAL ⎜ ARTICLE
in terest in (1 1 ) is th a t it p rov id es a re¯ n em en t o f th e
fu n d a m en ta l in eq u a lity b etw een th e g eo m etric a n d th e
a rith m etic m ea n s.
T h e lo g a rith m ic m ea n p lay s a n im p o rta n t ro le in th e
stu d y o f co n d u ctio n o f h ea t in liq u id s ° ow in g in p ip es.
L et u s ex p la in th is b rie° y. T h e ° ow o f h ea t b y stea d y
u n id irectio n a l co n d u ctio n is g ov ern ed b y N ew ton's law
of coolin g : if q is th e ra te o f h ea t ° ow a lo n g th e x -a x is
a cro ss a n a rea A n o rm a l to th is a x is, th en
q= kA
dT
;
dx
The logarithmic
mean plays an
important role in
the study of
conduction of heat
in liquids flowing in
pipes.
(1 3 )
w h ere d T = d x is th e tem p era tu re g ra d ien t a lo n g th e x
d irectio n a n d k is a co n sta n t ca lled th e th erm a l co n d u ctiv ity o f th e m a teria l. (S ee, fo r ex a m p le, R B h a tia ,
F ou rier S eries, M a th em a tica l A sso cia tio n o f A m erica ,
p .2 , 2 0 0 4 .) T h e cro ss-sectio n a l a rea A m ay b e co n sta n t,
a s fo r ex a m p le in a cu b e. M o re o ften (a s in th e ca se o f
a ° u id trav ellin g in a p ip e) th e a rea A is a va ria b le. In
en g in eerin g ca lcu la tio n s, it is th en m o re co n v en ien t to
rep la ce (1 3 ) b y
q = kA
m
4 T
;
4 x
(1 4 )
w h ere 4 T is th e d i® eren ce o f tem p era tu res a t tw o p o in ts
a t d ista n ce 4 x a lo n g th e x -a x is, a n d A m is th e m ean
cross section o f th e b o d y b etw een th ese tw o p o in ts. F o r
ex a m p le, if th e b o d y h a s a u n ifo rm ly ta p erin g recta n g u la r cro ss sectio n , th en A m is th e a rith m etic m ea n o f th e
tw o b o u n d a ry a rea s A 1 a n d A 2 :
C o n sid er h ea t ° ow in a lo n g h o llow cy lin d er w h ere en d
e® ects a re n eg lig ib le. T h en th e h ea t ° ow ca n b e ta k en
to b e essen tia lly ra d ia l. (S ee, fo r ex a m p le, J C ra n k : T he
M athem atics of D i® u sion , C la ren d o n P ress, 1 9 7 5 ). T h e
cro ss-sectio n a l a rea in th is ca se is p ro p o rtio n a l to th e
d ista n ce fro m th e cen tre o f th e p ip e. If L is th e len g th
o f th e p ip e, th e a rea o f th e cy lin d rica l su rfa ce a t d ista n ce
RESONANCE ⎜ June 2008
587
GENERAL ⎜ ARTICLE
x fro m th e a x is is 2 ¼ x L : S o , th e to ta l h ea t ° ow q a cro ss
th e sectio n o f th e p ip e b o u n d ed b y tw o co a x ia l cy lin d ers
a t d ista n ce x 1 a n d x 2 fro m th e a x is, u sin g (1 3 ), is seen
to sa tisfy th e eq u a tio n
Z x2
dx
q
= k4 T;
(1 5 )
x 1 2¼ x L
o r,
q=
k 2¼ L 4 T
:
lo g x 2 ¡ lo g x 1
If w e w ish to w rite th is in th e fo rm (1 4 ) w ith x 2 ¡ x 1 =
4 x ; th en w e m u st h av e
A
m
= 2¼ L
x2 ¡ x1
2 ¼ L x 2 ¡ 2¼ L x 1
=
:
lo g x 2 ¡ lo g x 1
lo g 2 ¼ L x 2 ¡ lo g 2 ¼ L x 1
In o th er w o rd s,
A
m
=
A2¡ A1
;
lo g A 2 ¡ lo g A 1
th e lo g a rith m ic m ea n o f th e tw o a rea s b o u n d in g th e
cy lin d rica l sectio n u n d er co n sid era tio n . In th e en g in eerin g litera tu re th is is ca lled th e logarithm ic m ean area.
If in stea d o f tw o co a x ia l cy lin d ers w e co n sid er tw o co n cen tric sp h eres, th en th e cro ss-sectio n a l a rea is p ro p o rtio n a l to th e sq u a re o f th e d ista n ce fro m th e cen tre. In
th is ca se w e h av e, in stea d o f (1 5 ),
Z x2
dx
q
= k4 T :
2
x1 4 ¼ x
A sm a ll ca lcu la tio n sh ow s th a t in th is ca se
p
Am =
A 1A 2;
th e g eo m etric m ea n o f th e tw o a rea s b o u n d in g th e a n n u la r sectio n u n d er co n sid era tio n .
588
RESONANCE ⎜ June 2008
GENERAL ⎜ ARTICLE
T h u s th e g eo m etric a n d th e lo g a rith m ic m ea n s a re u sefu l
in ca lcu la tio n s rela ted to h ea t ° ow th ro u g h sp h erica l a n d
cy lin d rica l b o d ies, resp ectiv ely. T h e la tter rela tes to th e
m o re co m m o n p h en o m en o n o f ° ow th ro u g h p ip es.
L et u s retu rn to in eq u a lities rela ted to th e lo g a rith m ic
m ea n . L et t b e a n y n o n zero rea l n u m b er. In th e eq u a lity
(1 1 ) rep la ce a a n d b b y a t a n d b t ; resp ectiv ely. T h is g iv es
(a b)t= 2 ·
a t + bt
a t ¡ bt
·
;
t(lo g a ¡ lo g b)
2
fro m w h ich w e g et
t(a b)t= 2
a¡ b
a¡ b
a t + bt a ¡ b
·
t
·
:
a t ¡ bt
lo g a ¡ lo g b
2
a t ¡ bt
T h e m id d le term in th is eq u a lity is th e lo g a rith m ic m ea n .
L et G t a n d A t b e d e¯ n ed a s
G t (a ;b) =
A t (a ;b) =
a¡
at ¡
a t + bt a ¡
t
2 at ¡
t(a b)t= 2
b
;
bt
b
:
bt
W e h av e a ssu m ed in th ese d e¯ n itio n s th a t t 6= 0: If w e
d e¯ n e G 0 a n d A 0 a s th e lim its
G 0 (a ;b) =
lim G t (a ;b);
A 0 (a ;b) =
lim A t (a ;b);
t! 0
t! 0
th en
G 0 (a ;b) = A 0 (a ;b) = L (a ;b):
T h e rea d er ca n v erify th a t
p
G 1 (a ;b) =
G
¡ t(a ;b)
=
a b;
G t (a ;b);
RESONANCE ⎜ June 2008
a+ b
;
2
A ¡ t(a ;b) = A t (a ;b):
A 1 (a ;b) =
589
GENERAL ⎜ ARTICLE
We have an infinite
family of inequalities
that includes the
AM–GM inequality,
and other interesting
F o r ¯ x ed a a n d b; G t (a ;b) is a d ecrea sin g fu n ctio n o f
jtj; w h ile A t (a ;b) is a n in crea sin g fu n ctio n o f jtj: (O n e
p ro o f o f th is ca n b e o b ta in ed b y m a k in g th e su b stitu tio n
a = e x ; b = e y :) T h e la st in eq u a lity o b ta in ed a b ov e ca n
b e ex p ressed a s
G t (a ;b) · L (a ;b) · A t (a ;b);
inequalities.
(1 6 )
fo r a ll t: T h u s w e h av e a n in ¯ n ite fa m ily o f in eq u a lities
th a t in clu d es th e a rith m etic{ g eo m etric m ea n in eq u a lity,
a n d o th er in terestin g in eq u a lities. F o r ex a m p le, ch o o sin g t = 1 a n d 1 = 2; w e see fro m th e in fo rm a tio n o b ta in ed
a b ov e th a t
p
ab ·
a 3 = 4 b1 = 4 + a 1 = 4 b3 = 4
· L (a ;b)
2
µ
·
a 1 = 2 + b1 = 2
2
¶2
·
a+ b
:
2
(1 7 )
T h is is a re¯ n em en t o f th e fu n d a m en ta l in eq u a lity (1 1 ).
T h e seco n d term o n th e rig h t is th e b in o m ia l m ea n
B 1 = 2 (a ;b): T h e seco n d term o n th e left is o n e o f a n o th er
fa m ily o f m ea n s ca lled H ein z m ean s d e¯ n ed a s
H º (a ;b) =
a º b 1 ¡ º + a 1 ¡º b º
;
2
0 · º · 1:
(1 8 )
C lea rly
H 0 (a ;b) =
H 1 = 2 (a ;b) =
H 1 ¡ º (a ;b) =
H 1 (a ;b) =
p
a b;
H º (a ;b):
a+ b
;
2
T h u s th e fa m ily H º is y et a n o th er fa m ily th a t in terp o la tes b etw een th e a rith m etic a n d th e g eo m etric m ea n s.
T h e rea d er ca n ch eck th a t
H
590
1 = 2 (a ;b)
· H º (a ;b) · H 0 (a ;b);
(1 9 )
RESONANCE ⎜ June 2008
GENERAL ⎜ ARTICLE
fo r 0 · º · 1 : T h is is a n o th er re¯ n em en t o f th e a rith m etic{
g eo m etric m ea n in eq u a lity.
If w e ch o o se t = 2 ¡n ; fo r a n y n a tu ra l n u m b er n ; th en w e
g et fro m th e ¯ rst in eq u a lity in (1 6 )
2 ¡n (a b)2
¡ (n + 1 )
a 2¡
a¡ b
· L (a ;b):
n
¡ b2 ¡ n
U sin g th e id en tity
³ ¡n
´³ ¡ n
´³ ¡ n + 1
´
¡ n
¡ n
¡ n+ 1
a ¡ b = a 2 ¡ b2
a 2 + b2
a2
+ b2
¢¢¢
³ ¡1
´
¡ 1
a 2 + b2
;
w e g et fro m th e in eq u a lity a b ov e
2¡
(n + 1 )
(a b)
Yn a 2 ¡ m + b 2 ¡ m
· L (a ;b):
2
m = 1
(2 0 )
S im ila rly, fro m th e seco n d in eq u a lity in (1 6 ) w e g et
L (a ;b) ·
a2
¡ n
+ b2
2
¡ n
Yn a 2 ¡ m + b 2 ¡ m
:
2
m = 1
(2 1 )
If w e let n ! 1 in th e tw o fo rm u la s a b ov e, w e o b ta in a
b ea u tifu l p ro d u ct fo rm u la :
¡ m
¡ m
1
Y
a 2 + b2
:
L (a ;b) =
2
m = 1
(2 2 )
T h is a d d s to o u r list o f fo rm u la s (7 ){ (1 0 ) fo r th e lo g a rith m ic m ea n .
C h o o sin g b = 1 in (2 2 ) w e g et a fter a little m a n ip u la tio n
th e rep resen ta tio n fo r th e lo g a rith m fu n ctio n
lo g x = (x ¡ 1 )
1
Y
m = 1
2
;
1 + x 2¡ m
(2 3 )
fo r a ll x > 0 :
RESONANCE ⎜ June 2008
591
GENERAL ⎜ ARTICLE
There are more
analytical delights in
store; the logarithmic
mean even has a
connection with the
fabled Gauss
arithmetic-geometric
mean.
W e ca n tu rn th is a rg u m en t a ro u n d . F o r a ll x > 0 w e
h av e
¡
¢
lo g x = lim n x 1 = n ¡ 1 :
(2 4 )
n!1
R ep la cin g n b y 2 n ; a sm a ll ca lcu la tio n lea d s to (2 3 ) fro m
(2 4 ). F ro m th is w e ca n o b ta in (2 2 ) b y a n o th er little
ca lcu la tio n .
T h ere a re m o re a n a ly tica l d elig h ts in sto re; th e lo g a rith m ic m ea n ev en h a s a co n n ectio n w ith th e fa b led G a u ss
a rith m etic{ g eo m etric m ea n th a t a rises in a to ta lly d i® eren t co n tex t. G iv en p o sitiv e n u m b ers a a n d b; in d u ctiv ely
d e¯ n e tw o seq u en ces a s
a0 =
an+ 1 =
a;
a n + bn
;
2
b0 = b
bn + 1 =
p
a n bn :
T h en f a n g is a d ecrea sin g , a n d f b n g a n in crea sin g , seq u en ce. A ll a n a n d b n a re b etw een a a n d b: S o b o th
seq u en ces co n v erg e. W ith a little w o rk o n e ca n see
th a t a n + 1 ¡ b n + 1 · 12 (a n ¡ b n ) ; a n d h en ce th e seq u en ces
f a n g a n d f b n g co n v erg e to a co m m o n lim it. T h e lim it
A G (a ;b) is ca lled th e G a u ss a rith m etic{ g eo m etric m ea n .
G a u ss sh ow ed th a t
Z
1
dx
2 1
p
=
A G (a ;b)
¼ 0
(a 2 + x 2 )(b 2 + x 2 )
Z ¼ =2
2
d'
p
=
: (2 5 )
¼ 0
a 2 co s2 ' + b 2 sin 2 '
T h ese in teg ra ls ca lled `ellip tic in teg ra ls' a re d i± cu lt o n es
to eva lu a te, a n d th e fo rm u la a b ov e rela tes th em to th e
m ea n va lu e A G (a ;b): C lea rly
G (a ;b) · A G (a ;b) · A (a ;b):
(2 6 )
S o m ew h a t u n ex p ected ly, th e m ea n L (a ;b) ca n a lso b e
rea lised a s th e o u tco m e o f a n itera tio n clo sely rela ted to
592
RESONANCE ⎜ June 2008
GENERAL ⎜ ARTICLE
th e G a u ss iteration . L et A t a n d G t b e th e tw o fa m ilies
d e¯ n ed earlier. A sm all ca lcu lation , th at w e leav e to th e
read er, sh ow s th at
A t+ G
2
t
= A
p
t= 2 ;
A
t= 2 G t
= G
t= 2 :
(2 7)
F or n = 1 ;2 ;::: ; let t = 2 1 ¡n ; an d d e¯ n e tw o seq u en ces
a 0n a n d b 0n as a 0n = A t ; b 0n = G t; i.e.,
a 01
=
a 02 =
..
.
a 0n + 1 =
p
a+ b
A1 =
; b 01 = G 1 = a b;
2
p
a 01 + b 01 0
A 1=2 =
A
; b2 = G 1 = 2 =
2
1=2 G 1
=
p
Somewhat
unexpectedly, the
logarithmic mean
can also be
realized as the
outcome of an
iteration closely
related to the
Gauss iteration.
a 02 b 01 ;
q
a 0n + b 0n
a 0n + 1 b 0n :
; b 0n + 1 =
2
W e leav e it to th e read er to sh ow th at th e tw o seq u en ces
f a 0n g an d f b 0n g con verge to a com m on lim it, a n d th at
lim it is eq u a l to L (a ;b): T h is g iv es o n e m ore ch aracterisation of th e loga rith m ic m ea n . T h ese co n sid era tio n s
also b rin g h o m e a n oth er in terestin g in eq u a lity
L (a ;b) · A G (a ;b):
(2 8)
F in a lly, w e in d ica te yet an o th er u se th at h as recen tly
b een fou n d for th e in eq u a lity (11 ) in d i® eren tial geom etry. L et k T k 2 b e th e E u clid ean n orm on th e sp ace of
n £ n com p lex m atrices; i.e.,
kT
k 22
¤
= tr T T =
Xn
jtij j2 :
i;j = 1
A m a trix v ersion of th e in eq u ality (11) say s th a t for all
p o sitive d e¯ n ite m a trices A a n d B an d for all m atrices
X ; w e h av e
¯¯Z 1
¯¯
¯¯
¯¯
¯
¯
¯
¯
¯
¯
¯¯
¯
¯
¯¯ 1 = 2
A
X
+
X
B
¯¯ :
¯¯A X B 1 = 2 ¯¯ · ¯¯ A t X B 1 ¡t d t¯¯ · ¯¯
¯¯
¯¯
¯¯
¯¯
2
2
0
2
2
(2 9)
RESONANCE ⎜ June 2008
593
GENERAL ⎜ ARTICLE
T h e sp a ce H n o f a ll n £ n H erm itia n m a trices is a rea l
v ecto r sp a ce, a n d th e ex p o n en tia l fu n ctio n m a p s th is
o n to th e sp a ce Pn co n sistin g o f a ll p o sitiv e d e¯ n ite m a trices. T h e la tter is a R iem a n n ia n m a n ifo ld . L et ± 2 (A ;B )
b e th e n a tu ra l R iem a n n ia n m etric o n Pn : A v ery fu n d a m en ta l in eq u a lity ca lled th e expon en tial m etric in creasin g property say s th a t fo r a ll H erm itia n m a trices H a n d
K
¡
¢
± 2 e H ;e K ¸ jjH ¡ K jj2 :
(3 0 )
A sh o rt a n d sim p le p ro o f o f th is ca n b e b a sed o n th e ¯ rst
o f th e in eq u a lities in (2 9 ). T h e in eq u a lity (3 0 ) ca p tu res
th e im p o rta n t fa ct th a t th e m a n ifo ld Pn h a s n o n p o sitiv e
cu rva tu re. F o r m o re d eta ils see th e S u g g ested R ea d in g .
Suggested Reading
[1]
G Hardy, J E Littlewood and G Pölya, Inequalities, Cambridge University Press, Second edition, 1952. (This is a well-known classic. Chapters
II and III are devoted to ‘mean values’.)
[2]
P S Bullen, D S Mitrinovic and P M Vasic, Means and their Inequalities,
D Reidel, 1998. (A specialised monograph devoted exclusively to various
means.)
W H McAdams, Heat Transmission, Third edition, McGraw Hill, 1954.
(An engineering text in which the logarithmic mean is introduced in the
context of fluid flow.)
[3]
[4]
[5]
Address for Correspondence
Rajendra Bhatia
Theoretical Statistics and
Mathematics Unit
Indian Statistical Institute
Delhi Centre
[6]
7, SJS Sansanwal Marg
New Delhi 110 016, India.
[7]
Email: [email protected]
[8]
594
B C Carlson, The logarithmic mean, American Mathematical Monthly,
Vol.79, pp.615–618, 1972. (A very interesting article from which we have
taken some of the material presented here.)
R Bhatia, Positive Definite Matrices, Princeton Series in Applied Mathematics, 2007, and also TRIM 44, Hindustan Book Agency, 2007.
(Matrix versions of means, and inequalities for them, can be found here.
The role of the logarithmic mean in this context is especially emphasized
in Chapters 4-6.)
R Bhatia and J Holbrook, Noncommutative Geometric Means, Mathematical Intelligencer, Vol.28, pp.32–39, 2006. (A quick introduction to
some problems related to matrix means, and to the differential geometric
context in which they can be placed. )
Tung-Po Lin, The Power Mean and the Logarithmic Mean, American
Mathematical Monthly, Vol.81, pp.879–883, 1974.
S Chakraborty, A Short Note on the Versatile Power Mean, Resonance,
Vol. 12, No. 9, pp.76–79, September 2007.
RESONANCE ⎜ June 2008