MATH1050 Cauchy-Schwarz Inequality and Triangle Inequality

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1 MATH050 Cuchy-Schwrz Inequlity nd Tringle Inequlity 0 Refer to the Hndout Qudrtic polynomils Definition (Asolute extrem for rel-vlued functions of one rel vrile) Let I e n intervl, nd h : D R e rel-vlued function of one rel vrile with domin D which contins I entirely Let p e point in I () h is sid to ttin solute mximum t c on I if for ny x I, the inequlity h(x) h(c) The numer h(c) is clled the solute mximum vlue of h on I () h is sid to ttin solute minimum t c on I if for ny x I, the inequlity h(x) h(c) The numer h(c) is clled the solute minimum vlue of h on I Theorem () (Asolute extrem for qudrtic functions) Let,,c R, with 0 Let f : R R e the qudrtic function given y f(x) = x + x + c for ny x R Denote the discriminnt of f(x) y f () Suppose > 0 Then f ttins solute minimum t onr, with solute minimum vlue f () Suppose < 0 Then f ttins solute mximum t onr, with solute mximum vlue f Proof of () Let,,c R, with > 0 Let f : R R e the qudrtic function given y f(x) = x + x + c for ny x R Denote the discriminnt of f(x) y f ( For ny x R, we hve f(x) = x+ ) f 4 f Also note tht f( 4 ) = f Hence f ttins solute minimum t onr, with solute minimum vlue f 3 Theorem (), s Corollry to Theorem () Let,,c R Suppose > 0, f = 4c, nd f : R R is the qudrtic polynomil function defined y f(x) = x +x+c for ny x R Then the sttements ( ), ( ) re logiclly equivlent: ( ) f(x) 0 for ny x R ( ) f 0 Equlity in ( ) holds iff is repeted rel root of the polynomil f(x) Remrk This result will ply key role in the proof of the Cuchy-Schwrz Inequlity Proof of Theorem () Let,,c R Suppose > 0, f = 4c, nd f : R R is the qudrtic polynomil function defined y f(x) = x +x+c for ny x R By Theorem (), f ttins solute minimum vlue t, with f( ) = f [( ) = ( )?] Suppose f(x) 0 for ny x R Note tht R Then, y ssumption, we hve 0 f( ) = f ( Since > 0, we hve 4 < 0 Then f = 4 ) f 0 4 [( ) = ( )?] Suppose f 0 Then, since > 0, we hve f 4 0 Pick ny x R We hve f(x) f( ) = f 4 0 ( f = 0 iff f(x) = x+ ) s polynomils This hppens iff is repeted rel root of the polynomil f(x)

2 4 Theorem (3) (Cuchy-Schwrz Inequlity for rel vectors ) Let x,x,,x n,y,y,,y n R Suppose x,x,,x n re not ll zero nd y,y,,y n re not ll zero Then the sttements elow hold: () The inequlity j= j= j= () The sttements ( ), ( ) re logiclly equivlent: ( ) = j= j= j= ( ) There exist some p,q R\{0} such tht px +qy = 0, px +qy = 0,, nd px n +qy n = 0 Remrks () In the context of the sttement of Theorem (3), if (x = x = = x n = 0 or y = y = = y n = 0), then the inequlity in () trivilly reduces to the equlity in ( ) of () () We my re-formulte Theorem (3) in the lnguge of liner lger, nd cover the trivil cses mentioned ove: Let x,x,,x n,y,y,,y n R Suppose x, y re vectors inr n defined y x = (x,x,,x n ), y = (y,y,,y n ) Then the sttements elow hold: () x,y x y () Equlity holds iff x, y re linerly dependent over R 5 Theorem (4) (Tringle Inequlity for rel vectors ) Let x,x,,x n,y,y,,y n R Suppose x,x,,x n re not ll zero nd y,y,,y n re not ll zero Then the sttements elow hold: () The inequlity ( + ) j= j= j= j= () The sttements ( ), ( ) re logiclly equivlent: ( ) ( + ) = j= j= ( ) There exist s > 0,t > 0 such tht sx = ty, sx = ty,, nd sx n = ty n Remrks () In the context of the sttement of Theorem (4), if (x = x = = x n = 0 or y = y = = y n = 0), then the inequlity in () trivilly reduces to the equlity in ( ) of () () We my re-formulte Theorem (4) in the lnguge of liner lger, nd cover the trivil cses descried ove: Let x,x,,x n,y,y,,y n R Suppose x, y re vectors inr n defined y x = (x,x,,x n ), y = (y,y,,y n ) Then the sttements elow hold: () x+y x + y () Equlity holds iff one of x, y is non-negtive sclr multiple of the other

3 6 Proof of Theorem (3) Let x,x,,x n,y,y,,y n R Suppose x,x,,x n re not ll zero nd y,y,,y n re not ll zero () Define the function F :R R y F(t) = By definition, for ny t R, we hve F(t) = ( t+ ) for ny t R j= ( t+ ) 0 [We re going to identify F s qudrtic polynomil function] Define A =, B =, C = y j, nd = B 4AC j= j= For ny t R, we hve F(t) = j= j= j= t + t+ j= y j = At +Bt+C Since t lest one of x,x,,x n is non-zero, we hve A > 0 Then F is qudrtic polynomil function with rel coefficients Recll tht F(t) 0 for ny t R Then y Theorem (), 0 Therefore = B 4 AC = j= () i [( ) = ( )?] Suppose = j= j= j= j= Then = B 4AC = 0 By Theorem (), the qudrtic polynomil F(t) hs repeted rel root It is t 0 = B/A Then, for the sme t 0, we hve F(t 0 ) = ( t 0 + ) = 0 j= Therefore for ech j =,,,n, we hve ( t 0 + ) = 0 Tke p = t 0, q = For ech j =,,,n, we hve p +q = t 0 + = 0 Note tht p 0; otherwise it would hppen tht = 0 for ech j =,,,n (Why?) j= ii [( ) = ( )?] Suppose there exist some p,q R\{0} such tht for ech k =,,,n, the equlity px k +qy k = 0 Define t 0 = p q Then for ech k =,,,n, we hve x kt 0 +y k = 0 Therefore F(t 0 ) = 0 Now the qudrtic polynomil F(t) hs rel root, nmely t 0 Then y Theorem (), 0 Also recll 0 Then = 0 Hence = j= j= j= j= 3

4 7 Proof of Theorem (4) Let x,x,,x n,y,y,,y n R Suppose x,x,,x n re not ll zero nd y,y,,y n re not ll zero () j= j= = x j + y j + j= j= j= j= x j + j= j= y j + j= (y Cuchy-Schwrz Inequlity) x j + y j + = (x j + +y j ) = ( + ) j= j= j= j= j= Hence ( + ) j= j= j= () i [( ) = ( )?] Suppose ( + ) j= j= Then = j= = j= j= j=, nd = j= By Theorem (3), there exist some p,q R\{0} such tht for ech k =,,,n, the equlity px k +qy k = 0 Since =, we hve q x j = p q p x j = q p = q x j p j= j= j= Then p,q re of opposite signs Without loss of generlity, suppose q < 0 Then p > 0 Tke s = p, t = q Then s > 0 nd t > 0 Moreover, for ech k =,,,n, we hve sx k = px k = qy k = ty k ii [( ) = ( )?] Suppose there exist some s > 0,t > 0 such tht for ech k =,,,n, the equlity sx k = ty k Then for ech k =,,,n, we hve sx k ty k = 0 Therefore j= = j= j= Moreover, = t s = t s = t x t j = s s = j= j= j= j= j= j= j= j= j= j= 4

5 8 Appendix Cuchy-Schwrz Inequlity for squre-summle infinite sequences in R With the help of the Bounded-Monotone Theorem (which you hve lernt in your clculus of one vrile course) nd sic result (Theorem (A)) on solutely convergent infinite series (which you will lern in your nlysis course), oth stted elow, we cn extend the Cuchy-Schwrz Inequlity nd Tringle Inequlity to nlogous results for squre-summle infinite sequences in R (Theorem (5), Theorem (6) respectively) Bounded-Monotone Theorem (for incresing infinite sequences which re ounded ove) Let {u n } n=0 e n infinite sequence of rel numers Suppose {u n } n=0 is incresing nd is ounded ove inr, sy, y β Then lim u n exists inr, nd the inequlity lim u n β Theorem (A) Let {v n } n=0 e n infinite sequence of rel numers Suppose the limit lim lim v n exists, nd the inequlity lim v n lim v n v n exists Then the limit Theorem (5) (Cuchy-Schwrz Inequlity for squre-summle infinite sequences in R ) Let { }, {} e infinite sequences of rel numers Suppose the limits lim Then the limits lim () The inequlity lim x j, lim, lim lim y j exist exist Moreover, the sttements elow hold: () The sttements ( ), ( ) re logiclly equivlent: ( ) lim = lim lim lim ( ) There exist some p,q R, not oth zero, such tht p + q = 0 for n N (The infinite sequences { }, {} re linerly dependent overr ) Theorem (6) (Tringle Inequlity for squre-summle infinite sequences in R ) Let { }, {} e infinite sequences of rel numers Suppose the limits lim Then the limit lim () The inequlity lim x j, lim y j exist ( + ) exists Moreover, the sttements elow hold: ( + ) lim () The sttements ( ), ( ) re logiclly equivlent: ( ) lim ( + ) = lim lim lim ( ) There exist non-negtive rel numers s,t, not oth zero, such tht s = t for n N (One of the infinite sequences { }, {} is non-negtive sclr multiple of the other) 5

6 9 Appendix Cuchy-Schwrz Inequlity for definite integrls With the help of sic result (Theorem (B)) for definite integrls of continuous rel-vlued functions on closed nd ounded intervls (which you hve lernt in your clculus of one vrile course nd will prove in your nlysis course), we cn deduce the Cuchy-Schwrz Inequlity nd Tringle Inequlity for continuous rel-vlued functions on closed nd ounded intervls (Theorem (7), Theorem (8) respectively) The rgument is lmost exctly the sme s tht for Theorem (3) nd Theorem (4), with summtion replced y definite integrtion Theorem (B) Let, e rel numers, with <, nd h : [,] R e function Suppose h is continuous on [,] nd h(u) 0 for ny u [,] Then the inequlity (h(u) = 0 for ny u [,]) Theorem (7) (Cuchy-Schwrz Inequlity for definite integrls) h(u)du 0 Moreover, equlity holds iff Let, e rel numers, with <, nd f,g : [,] Re functions Suppose f,g re continuous on [,] Then the sttements elow hold: () The inequlity [ ] [ f(u)g(u)du (f(u)) du () The sttements ( ), ( ) re logiclly equivlent: [ ] [ ( ) f(u)g(u)du = (f(u)) du (g(u)) du ] (g(u)) du ] ( ) There exist some p,q R, not oth zero, such tht pf(u)+qg(u) = 0 for ny u [,] (The functions f,g re linerly dependent over R ) Theorem (8) (Tringle Inequlity for squre-summle infinite sequences in R ) Let, e rel numers, with <, nd f,g : [,] Re functions Suppose f,g re continuous on [,] Then the sttements elow hold: [ () The inequlity (f(u)+g(u)) du ] [ (f(u)) du () The sttements ( ), ( ) re logiclly equivlent: [ ] [ ] [ ( ) (f(u)+g(u)) du = (f(u)) du + (g(u)) du ] [ + (g(u)) du ] ] ( ) There exist non-negtive rel numers s,t, not oth zero, such tht sf(u) = tg(u) (One of the functions f,g is non-negtive sclr multiple of the other) 0 Appendix 3 () There re complex nlogues for the rel versions of Cuchy-Schwrz Inequlities nd Tringle Inequlities stted here () The Cuchy-Schwrz Inequlity for rel vectors cn e seen s specil cse of Hölder s Inequlity for rel vectors The Tringle Inequlity for rel vectors cn e seen s specil cse of Minkowski s Inequlity for rel vectors You will encounter these inequlities in dvnced courses in mthemticl nlysis (Nevertheless, find wht these re nd try to prove them) 6