bat dang thuc karamata

TP CH KHOA HC V CNG NGH, I HC NNG - S 6(29).2008

77

V BT NG THC KARAMATA V NG DNG ON THE KARAMATAS INEQUALITY AND ITS APPLICATIONS

CAO VN NUI - NGUYN QUANG THI Trng i hc S phm, i hc Nng

TM TT nh l Karamata v cc tnh cht ca hm li l mt phn quan trng v kh ca cc bt ng thc. Da vo nh l Karamata, ngi ta chng minh c cc bt ng thc: T. Popoviciu, bt ng thc A. Lupas v bt ng thc Vasile Cirtoaje 2.1.[2]. Cc bt ng thc ny c nhng ng dng trong vic gii mt s bi ton kh. V chng ti thy rng: vic xy dng cc bt ng thc mi l rt cn thit. Trong bi bo ny, chng ti xy dng hai nh l mi v nhng ng dng ca chng. Bi bo trnh by hai bt ng thc mi m phng php chng minh ca n da vo nh l Karamata v cc tnh cht ca hm li.

ABSTRACT Karamata's theorem and properties of the convex function are important and difficult part of inequalities.Base on Karamatas theorem, it proved the inequalities: T. Popovicius inequality, A. Lupas inequalitys and Vasile Cirtoajes inequality 2.1.[2]. These inequalities have application in solving difficult problems. And we see that: building of inequalities are very necessary. In this paper, we built two new theorems and their applications. We present two new inequalities which are that their demonstration method based on the Karamata's theorem and properties of the convex function.

1. M u

( )I a, bTa k hiu l mt tp hp c mt trong 4 dng sau y: [ ]a,b ( )a,b, , [ )a,b v ( ]a,b .

nh ngha (Cc b tri). ( )1 2 nx , x , , x Cho v ( )1 2 ny , y , , y l hai b s thc. Ta ni rng dy ( )1 2 nx , x , , x tri hn ( )1 2 ny , y , , y hay dy ( )1 2 ny , y , , y c lm tri bi dy ( )1 2 nx , x , , x v ta vit ( ) ( )1 2 n 1 2 nx , x , , x y , y , , y , nu cc iu kin sau tho mn:

(1) 1 2 nx x x v 1 2 ny y y .

(2) 1 2 i 1 2 ix x x y y y+ + + + + + , i 1, n 1 = .

(3) 1 2 n 1 2 nx x x y y y+ + + = + + + .

Hm s

nh ngha (Hm li)

( )f x c gi l li trn tp ( )I a, b ( )I a, b nu n xc nh trn , vi mi


78

( )1 2x , x I a, b v vi mi cp s khng m , c tng 1 + = , ta u c

( ) ( ) ( )1 2 1 2f x x f x f x + + (1.1)

Nu ng thc trong (1.1) xy ra khi v ch khi 1 2x x= th f c gi l li tht s trn

( )I a, b

( )f x

.

Hm s c gi l lm trn tp ( )I a, b ( )I a, b nu n xc nh trn , vi mi ( )1 2x , x I a, b v vi mi cp s khng m , c tng 1 + = , ta u c

( ) ( ) ( )1 2 1 2f x x f x f x + + (1.2)

Nu ng thc trong (1.2) xy ra khi v ch khi 1 2x x= th f c gi l lm tht s

trn ( )I a, b .

nh l. f Nu kh vi bc hai trn ( )I a, b th hm ( )f x li trn ( )I a, b nu v ch nu ( )f '' x 0 vi mi ( )x I a,b .

nh l (Karamata). Cho hai b s thc ( )1 2 nx , x , , x v ( )1 2 ny , y , , y , ( )( )i ix , y I a, b tho mn iu kin ( ) ( )1 2 n 1 2 nx , x , , x y , y , , y . Khi , vi mi

hm f (x) li tht s trn ( )I a, b ( )( )f '' x 0> , ta lun c

( ) ( )n n

i ii 1 i 1

f x f y= =

ng thc xy ra khi v ch khi i ix y= , i 1;n = .

2. Cc kt qu

2.1. Chng minh nh l Karamata Trc ht, chng ti gii thiu cch chng minh nh l Karamata c trnh by trong 0.

Chng minh (nh l Karamata). Ta c ( ) ( ) ( ) ( )f x f y x y f ' y , ( )x, y I a, b vi mi hm ( )f x li tht s trn ( )I a, b . Tht vy, do ( )f '' x 0> nn ( )f ' x l hm s tng trn ( )I a, b . Xt 3 trng hp sau:

+ Nu x y= th bt ng thc hin nhin ng.

+ Nu x y> th ( ) ( ) ( ) ( )1f x f y

f ' c f ' yx y

= >

, ( )1c y, x .

+ Nu x y< th ( ) ( ) ( ) ( )2f y f x

f ' c f ' yy x

= . Nu ( )1 2 nx , x , , x v ( )1 2 ny , y , , y l hai b s thc tho mn cc iu kin sau:

[ ]

( ) ( )i i

1 2 n 1 2 n

x , y 0,a , i 1;n

x , x , , x y , y , , y

=

( ) ( ) ( ) ( )n n n

i i i ii 1 i 1 i 1

f x f y x f y n f0= = =

+ +

th

Chng minh.

1 1

1 2 1 2

1 2 n 1 1 2 n 1

1 2 n 1 2 n

x yx x y y

x x x y y yx x x y y y

+ + + + + + + +

+ + + = + + +

Ta c

( ) ( )( ) ( ) ( )( ) ( ) ( )

1 1

1 1 2 2

1 1 2 2 n 1 n 1

1 1 2 2 n n

0 y x0 y x y x

0 y x y x y x

0 y x y x y x

+ + + + = + + +

t i i it y x , i 1;n= = . Gi ( )* * *1 2 2nk , k , , k l b gm 2n s nhn c t cc b ( )1 2 nx , x , , x v ( )1 2 nt , t , , t bng cch sp xp cc s 1 2 nx , x , , x , 1 2 nt , t , , t theo th t gim dn. Theo tnh cht ca b tri, ta suy ra

( ) ( )* * *1 2 2n 1 2 nk , k , , k y , y , , y ,0, ,0 Tht vy, gi s tn ti t N v 1 t n 1 sao cho

* * * * * * *1 2 n n 1 n t n t 1 2nk k k k k 0 k k+ + + +

Hin nhin

* * *1 2 p 1 2 p 1 2 pk k k x x x y y y , p 1;n+ + + + + + + + + =


80

* * *1 2 n q 1 2 n 1 2 nq

k k k x x x y y y 0 0, q 1; t++ + + + + + + + + + + + =

s

v

* * * * * * * *1 2 n q 1 2 n q n q 1 2n

1 2 nq

k k k k k k k k

y y y 0 0, q t 1;n 1+ + + ++ + + + + + + + + =

= + + + + + + = +

s

T nh l Karamata, ta suy ra

( ) ( ) ( )

( ) ( ) ( ) ( )

2n n*i i

i 1 i 1n n n

i i i ii 1 i 1 i 1

f k f y nf 0

f x f y x f y n f0

= =

= = =

+

+ +

Kt lun: nh l c chng minh. ng thc xy ra khi v ch khi *i ik y= , i 1;n= v *ik 0= , i n 1;2n= + .

Sau y ta ng dng nh l 1 gii mt s bi ton.

Bi ton 1. Cho ABC tho mn A B C2 4 . Chng minh rng:

cos A cos B cos C cos A cos B cos C 3 22 4 4 + + + + +

Li gii. ( )f x cos x= Xt hm s trn ,2 2

. Ta c ( )f '' x cos x 0= ,

x ,2 2

. Vy, hm s ( )f x li trn ,2 2

.

Do A B C2 4 nn ta c:

A2

C A B2 4 4

A B C2 4 4

+ = = +

+ + = + +

( ), , A,B,C2 4 4

p dng nh l 1, ta c

cos cos cos cos A cos B cos C2 4 4 2 4 4cos A cos B cos C 3cos 0

Hay


81

cos A cos B cos C cos A cos B cos C 3 22 4 4 + + + + +

ng thc xy ra khi v ch khi ABC vung cn ti A . Bi ton 2. x, y, z Cho l cc s thc tho mn cc iu kin:

2 x y z 1x y z 4

+ + =

( ) ( ) ( )2 2 2

2 2 2

4 x 2 4 y 1 4 z 1 2 2 2 5 6

4 x 4 y 4 z

+ + + + + + +

+ + + + +

Chng minh rng:

Li gii.

x 2, y z 1= = =Nhn xt. ng thc xy ra nu v ch nu . T iu kin bi ton, d dng ta c

2 x2 1 4 z x y2 1 1 x y z

+ = + + + = + +

Khi , xt hm s ( ) 2f x 4 x= + trn [ ]2,2 .

Do ( )2 2

4f ''(x) 04 x 4 x

= >+ +

, [ ]x 2,2 nn hm s ( )f x li trn [ ]2, 2 . Khi

, p dng nh l 1, ta thu c

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )f 2 f 1 f 1 f x 2 f y 1 f z 1 f x f y f z 3f 0+ + + + + + + + Hay

( ) ( ) ( )2 2 2

2 2 2

2 2 2 5 4 x 2 4 y 1 4 z 1

4 x 4 y 4 z 6

+ + + + + + +

+ + + + + +

Kt lun: Bt ng thc c chng minh. Chng ti chng minh c nh l sau y:

nh l 2. ( )f x Nu hm s li trn ( )I a, b v ( )x, y ,z I a, b

( ) ( ) ( ) x y zf x f y f z f3

2 2x y 2y x 2y z 2z y 2x z 2z xf f f f f f3 3 3 3 3 3 3

+ + + + +

+ + + + + + + + + +

th

Chng minh. t


82

1 2

1 2 1 2

x y z 2x y 2y xd ,a ,a3 3 3

2x z 2z x 2y z 2z yb ,b ,c ,c3 3 3 3

+ + + += = =

+ + + += = = =

Khng mt tnh tng qut, gi s rng: x y z . Xt 2 trng hp sau:

2y x z +Trng hp 1: x x x y y y d d d z z z . D thy

v 1 1 2 2 1 1 1 1 2 2 2 2a a a a b b c c b b c c .

Ta s chng minh

( ) ( )1 1 2 2 1 1 1 1 2 2 2 2x, x, x, y ,y ,y ,d,d,d, z, z, z a ,a ,a ,a , b , b ,c ,c , b , b ,c ,c (1.3) Tht vy,

( ) ( )

( ) ( )

( )

1 1 1 2

1 2 1 2 1

1 2 1

1 2 1 1

1 2 1 1

2 x y 4 x yx yx a 0,2x 2a 0,3x 2a a 03 3 3

x z3x y 2a 2a x y 0,3x 2y 2a 2a b 03

2y x z y z3y x 2z3x 3y 2a 2a 2b 03 3

2 y z3x 3y d 2a 2a 2b c 0

3x y 2z3x 3y 2d 2a 2a 2b 2c 0

3

3 x3y 3d 2

= = =

+ = + =

+ + = =

+ + =

+ + + =

+ +

( )

( )

1 2 1 1 2

1 2 1 1 2

1 2 1 1 2 2

x 2y 3za 2a 2b 2c b 032 y z

3x 3y 3d z 2a 2a 2b 2c 2b 03

y z3x 3y 3d 2z 2a 2a 2b 2c 2b c 0

3

+ =

+ + + =

+ + + =

V

1 2 1 1 2 23x 3y 3d 3z 2a 2a 2b 2c 2b 2c 0+ + + =

Suy ra (1.3) ng. Theo nh l Karamata, ta c bt ng thc ng. 2y x z +Trng hp 2: x x x d d d y y y z z z . Tng t, ta c

v 1 1 1 1 2 2 2 2 1 1 2 2a a b b a a b b c c c c . D dng, ta c

( ) ( )1 1 1 1 2 2 2 2 1 1 2 2x, x, x,d,d,d, y ,y ,y ,z, z, z a ,a , b , b ,a ,a , b , b ,c ,c ,c ,c Theo nh l Karamata, ta c bt ng thc ng. nh l c chng minh. ng thc xy ra khi v ch khi x y z= = .

Chng ta c mt ng dng ca nh l 2 trong bi ton sau y:


83

Bi ton 3. a,b,c 0> Cho . Chng minh rng:

( )

( ) ( ) ( ) ( ) ( ) ( )( )

33 3 3

3 3 3 3 3 3

1a b c a b c27

2 2a b 2b a 2b c 2c b 2a c 2c a81

+ + + + +

+ + + + + + + + + + +

Li gii. ( ) 3f x x= Xt hm s trn ( )0,+ . Do ( )f '' x 6x 0= > nn ( )f x li trn ( )0,+ . Bt ng thc c vit li nh sau:

( ) ( ) ( ) a b cf a f b f c f

32 2a b 2b a 2b c 2c b 2a c 2c af f f f f f3 3 3 3 3 3 3

+ + + + +

+ + + + + + + + + + +

Khng mt tnh tng qut, ta gi s a b c . T , ta d dng p dng nh l 2, ta suy ra bt ng thc ng.

Chng ta c cch gii khc cho bi ton ny da vo bt ng thc Muirhead v bt ng thc Schur

Nhn xt.

0. Tht vy, bt ng thc cn chng minh tng ng vi:

( ) ( )

( )

3 3 3 2 3 3 3 2

sym sym

3 3 3 2

sym

3 2 3 3 3 2

sym sym sym

1 228 a b c 3 a b 6abc 18 a b c 18 a b27 3.27

16 a b c 6abc 9 a b

7 a a b 2 a b c 3ab c a b 0

+ + + + + + +

+ + +

+ + + +

T bt ng thc Muirhead v bt ng thc Schur, ta suy ra bt ng thc cui cng l ng.

TI LIU THAM KHO [1] Phm Kim Hng, Sng to bt ng thc, NXB Tri thc, 2006. [2] Nguyn Vn Mu, Bt ng thc: nh l v p dng, NXB Gio dc, 2006. [3] Mitrinovic D. S., Pecaric J. E., Fink A. M., Classical and New Inequalities in

Analysis, Kluwer Acadmemic Publishers, 1993. [4] Titu Andresscu, Vasile Cirtoaje, Gabriel Dospinescu, Mircea Lascu, Old and New

Inequalities, Gil Publishing House, 2004. [5] Pachpatte B.G., Mathematical Inequalities, vol. 67, Elsevier, 2005.

CAO VN NUI - NGUYN QUANG THIM uCc kt quChng minh nh l Karamata

2.2. V hai bt ng thcPhm Kim Hng, Sng to bt ng thc, NXB Tri thc, 2006.Nguyn Vn Mu, Bt ng thc: nh l v p dng, NXB Gio dc, 2006.Mitrinovic D. S., Pecaric J. E., Fink A. M., Classical and New Inequalities in Analysis, Kluwer Acadmemic Publishers, 1993.Titu Andresscu, Vasile Cirtoaje, Gabriel Dospinescu, Mircea Lascu, Old and New Inequalities, Gil Publishing House, 2004.Pachpatte B.G., Mathematical Inequalities, vol. 67, Elsevier, 2005.

bat dang thuc karamata

Documents