How many proofs of the Nebitt s Inequality?

Transcription

1 How mny roofs of the Neitt s Inequlity? Co Minh Qung. Introdution On Mrh, 90, Nesitt roosed the following rolem () The ove inequlity is fmous rolem. There re mny eole interested in nd solved (). In this er, I would send reders some roofs of (). And now, we egin.. Some roofs Proof. Usingthe well - known inequlity ( y z) y 9 for ; y; z > 0. z We get = = [( ) ( ) ( )] 9 Therefore, 9 = Proof. Setting = ; y = ; z = ; then = y z ; = z y ; = y z () n e written s y z z y y z y z () y y y z z z y 6 z This inequlity is lerly true y the well - known inequlity q q for ; q > 0. Proof. () is equivlent to [ ( ) ( ) ( ) ( ) ( ) ( )] ( ) ( ) ( ) () ( ) ( ) ( ) The inequlity follows ythe well - known inequlity y y ( y) () ( y) ( y) 0 for ; y 0. Proof. Using the AM - GM inequlity, we otin r = Similrly, we get ; nd Adding three ove inequlities, we hve

2 () ( ) ( ) ( ) ( ) () ( ) () Proof 5. Setting A = ; B = ; C = then B C = ; A B = ; nd A C = By the AM - rgm inequlity, we hve AB =, AC Thus, A B C 6 or A Proof 6. By the AM - GM inequlity, we hve 6 =, r = nd 6 = Adding two inequlities, we get ( ) Thus, Similrly, we hve ; Adding three ove inequlities, we otin = Proof 7. Using the well - known inequlity ( y) y; for ; y 0. We hve [ ( )] ( ) or ( ) ( ) 8 ( ) Hene, ( ) ( ) [8 ( )] =) 8 Similrly, we get 8 ; 8 Adding three inequlities, we hve ( ) 6 = Proof 8. (By Co Minh Qung) Using the AM - GM inequlity, we hve ( ) ( ) ( ) = q ( ) Thus, Similrly, we get q( )

3 nd q( ) q( ) Adding three inequlities, we hve q( ) We hve to show tht () q ( ) ( ) () We set = ; y = ; z = () n e written s y z y z By the Cuhy - Shwrz inequlity, we hve y z y z ( y z) () By the Cheyshev s inequlity, we hve y z ( y z) y z () Multilying () nd () yields, we otin y z y z Proof 9. () is equivlent to () () ( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) 0 The lst inequlity is lerly true. Proof 0. By the Cuhy - Shwrz inequlity, we hve ( ) ( ) ( ) ( ) = ( ) ( ) = ( ) ( ) = Proof. Without loss of generlity, we n ssume tht, then By the Cheyshev s inequlity nd the AM - GM inequlity, we hve ( ) = = 6 [( ) ( ) ( )] 9 6 =

4 Proof. Setting = ; y = ; z =, nd A = y z We need to rove tht A We hve y y z z = = Thus, = yz y yz z By the AM - GM inequlity, we get = yz y yz z A A =) (A ) (A ) 0 Sine A > 0, hene A Proof. Using the sme sustitution s the th roof, nd setting f (t) = t It is esy to show tht f (t) is inrese nd onve on (0; ) By the t Jensen s inequlity, we hve f = = y z [f () f (y) f (z)] f =) y z or Proof. (By Co Minh Qung) Firstly, we stte nd rove lemm. Lemm. If i ; y i ; (i = ; ; ) re ositive rel numers stisfy tht ; y y y ; then y y y y i y i y i ; (5) where (i ; i ; i ) is ermuttion of (; ; ) Proof (5). We set z = y i ; z = y i ; z = y i (5) eomes y y y z z z () (y z ) (y z ) (y z ) 0 It is esy to see tht y z ; y y z z nd y y y = z z z Therefore, (y z ) (y z ) (y z ) (y z ) (y z ) (y z ) = [(y y ) (z z )] (y z ) [(y y ) (z z )] (y z ) = [(y y y ) (z z z )] = 0 Let us now rove (). Without loss of generlity, we n ssume tht, then Using (5), we get nd Adding two inequlities, we otin = or Proof 5. Without loss of generlity, we n ssume tht We set = ; y = ; then y () eomes

5 y y y (6) Using the AM - GM inequlity, we hve y y =) y y It su es to rove tht y y () y y () y (y ) y ( ) ( y) (y ) [( ) ( y) (y )] () 0 ( y) ( ) (y ) y The lst inequlity i true sine y To rove (6), eside the ove roof, we lso hve n nother roof. Proof 6. We set m = y; n = y (6) eomes m n m m n m () m m m n (7m ) We note tht 7m > 0 nd m n It su es to rove tht m m m m (7m ) () (m ) (m ) 0 This inequlity is lerly true. Proof 7. (By Co Minh Qung) By setting = ; y = ; z = ; then yz = () n e written s z y y z zy () y y z z ( y z) (y yz z) (7) Using the AM - GM inequlity, we hve y y z z = [ y y z y z y z z z z y y ] y 5 z y z 5 z 5 y = y z Using the AM - GM inequlity gin, we hve y y z z = [ y z z y z y y z y z y z ] z y y z y z = z y yz Adding two inequlities, we get (7). Sine () in the homogeneous, we n ssume tht = We hve to rove We hve some roofs of ( ) Proof 8. For 0 < < ; we note tht ( ) (9 ) = ( ) ; where = (0 ) 5

6 Hene, ( ) (9 ) or 9 Therefore, = ( ) = = Proof 9. (By Co Minh Qung) For 0 < < ; we note tht ( ) 0 () 9 ( ) Hene 9 or 9 Therefore, y Cuhy - Shwrz Inequlity, we hve = ( ) ( ) = Proof 0. (By Co Minh Qung) For 0 < <, y the AM - GM inequlity, we get ( ) ( ) ( ) ( ) ( ) ( ) = 6 7 Therefore, 7 ( ) Using the well - known inequlity r y r z r r y z ; for ; y; z 0; r We get = 7 X ( ) = 7 X 5 ;; 7 ;; " 6 X X ;; ;; # = Proof. (By Co Minh r Qung) By the AM - GM inequlity, we get 9 ( ) ( ) 9 = 9 ( ) Similrly, we hve ; Adding three inequlities nd noting tht ( ) = ( ) ( ) We otin ( ) 9 9 ( ) ( ) = ( ) ( ) 9 ( ) 6

7 Proof. Setting f (t) = t It is esy to show tht f (t) is onve t funtion on (0; ). By the Jensen s inequlity, we otin f () f () f () f or Proof. Without loss of generlity, we n ssume tht We get ; ; whih follows tht (; ; ) ; ; Sine f (t) = t is onve funtion on (0; ), y the Krmt s inequlity, we otin t f () f () f () f f f or We re done. Proof. (By Co Minh Qung) To rove ( ), we need to rove lemm. Lemm. (Co Minh Qung nd Trn Tun Anh) If ; ; re ositive rel numers suh tht = ; then Proof. Using the well - known inequlity y z y yz z; for ; y; z 0 We get ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) = We now rove ( ). Using the well - known inequlity y y ( y) ; for ; y > 0 We otin [( ) ( ) ( )] = Lst, we use "Miing Vriles Theorem" to rove (). Proof 5. By setting E (; ; ) = v = It is esy to see tht E (; ; ) = ( ) ( ) Thus, E (; ; ) E (v; v; v) =. Some generl results of the Nesitt s Inequlity, t = ; nd t t t t = E (t; t; ) t 7

8 In reent yers, y some owerful tools to rove inequlities, eole found out some inequlities whih is "stronger" thn (). There re Prolem. [ Titu Vreesu, Mire Lsu ] Let ; ; ; e ositive rel numers suh tht = nd Prove tht Prolem. [ Trn Tun Anh ] Let ; ; ; k e ositive rel numers suh tht k Prove tht k k k k And rolem is "stronger" thn rolem is Prolem. [ Vsile Cirtoje ] Let ; ; ; r e ositive rel numers suh tht r ln Prove tht ln r r r Prolem. [ Trn Nm Dung ] Let ; ; e ositive rel numers Prove tht But rolem is weker thn rolem 5 Prolem 5. [ Vsile Cirtoje ] Let ; ; e ositive rel numers Prove tht 6 Prolem 6. [ Cezr Luu ] Let ; ; e ositive rel numers Prove tht ( ) ( ) ( ) ( ) ( ) ( ) Prolem 7. [ Co Minh Qung ] Let ; ; e ositive rel numers nd m; n e nonnegtive rel numers suh tht = ; 6m 5n Prove tht m n m n m n m n The story of "the Nesitt s Inequlity" is still. Hoing nyone will nd out n nother of () on reent dy. Address Co Minh Qung, Nguyen Binh Khiem high shool, Vinh Long town, Vinh Long, Vietnm. E-mil ktqung@yhoo.om Referenes []. Hojoo Lee, Tois in Inequlities - Theorems nd Tehniques, 006, unulished. []. Phm Kim Hung (in Vietnmese), Serets in Inequlities, 006. []. Titu Andressu, Vsile Cirtoje, Griel Posinesu, Mire Lsu, Old nd New Inequlities, Gil ulishing House, 00. []. Vsile Critoje, Algeri Inequlities, Old nd New Methods, Gil ulishing House, 00. [5]. Mthemtis nd Youth Mgzine, (in Vietnmese) (link 8