... (a1 a2 · · · ak ) 1/ k = k =1 (a1 c1 a2 c2 · · · ak ck )1/ k (c1 c2 · · · ck )1/ k k =1 ∞ ≤ a1 c1 + a2 c2 + · · · + ak ck k(c1 c2 · · · ck )1/ k k =1 ∞ = ∞ ak ck k =1 j=k , j(c1 c2 · · · cj )1/ j (2 .17 ) ... e(ak /A) 1 = 1, (2 .13 ) unless A A and we always have ak ≤ e(ak /A) 1 , A so we see that one also has a1 A p1 a2 A p2 ··· an A n pn < exp pk k =1 ak A 1 = 1, (2 .14 ) unless ak = A for all k = 1, 2, ... c1 c2 · · · cj 1 = j j 1 and c1 c2 · · · cj = (j + 1) j , so division gives us the explicit formula cj = (j + 1) j =j 1+ j j 1 j j From this formula and our original bound (2 .17 ) we find ∞ ∞ (a1...
... to d d (uj + vj )2 ≤ j =1 d u2 + j d 1/ 2 u2 j j =1 j =1 d 1/ 2 vj j =1 j =1 and this in turn may be simplified to the equivalent bound d d u j vj ≤ j =1 1/2 d u2 j j =1 1/2 vj j =1 vj , + On Geometry and ... satisfy the On Geometry and Sums of Squares 71 triangular system of linear relations x1 = x1 , e1 e1 x2 = x2 , e1 e1 + x2 , e2 e2 ··· = ··· xn = xn , e1 e1 + xn , e2 e2 + · · · + xn , en en Exercise ... upper bound for a product of two linear forms: n n vj xj ≤ uj xj j =1 j =1 n n j =1 n u2 j u j vj + j =1 1/2 n vj j =1 1/2 x2 (4.8) j j =1 The charm of this inequality is that it leverages the presence...
... nonnegative real numbers a1 , a2 , b1 , and b2 satisfy max{a1 , a2 } ≥ max{b1 , b2 } and a1 + a2 = b1 + b2 , then for nonnegative x and y, one has xb1 y b2 + xb2 y b1 ≤ xa1 y a2 + xa2 y a1 (5.22) Prove ... inequality (5 .12 ) 80 Consequences of Order aj an 1 an ··· bσ(n 1) bσ(n) ··· bτ (n 1) bτ (n) ak a1 a2 bσ (1) bσ(2) ··· bσ(j) ··· bσ(k) bτ (1) bτ (2) ··· bτ (j) ··· bτ (k) Fig 5 .1 An interchange ... function n n aθ+x bθ−x j j fθ (x) = j =1 aθ−x bθ+x , j j x ∈ R j =1 If we set θ = 1, we see that f1 (0 )1/ 2 gives us the left side of Cauchy’s inequality while f1 (1) 1/2 gives us the right side Show...
... value V = (1+ r1 ) (1+ r2 ) · · · (1+ rn ) of our investments after n years must satisfy the bounds n (1 + rG )n ≤ (1 + rk ) ≤ (1 + rA )n , (6.23) k =1 where rG = (r1 r2 · · · rn )1/ n and rA = (r1 + r2 ... before in Exercise 2 .11 that for nonnegative aj and bj , j = 1, 2, , n one has superadditivity of the geometric mean: (a1 a2 · · · an )1/ n +(b1 b2 · · · bn )1/ n ≤ {(a1 + b1 )(a2 + b2 ) · · · ... n n pk f (xk ) − f k =1 n pk xk k =1 = µ j =1 n pj pk (xj − xk )2 (6 .16 ) k =1 Context and a Plan This result is from the same famous 18 85 paper of Otto Ludwig H¨lder o (18 59 -19 37) in which one finds...
... + 1) I = xα+β +1 |f (x)| dx Thus, if we then apply Schwarz’s inequality to the splitting xα+β +1 |f (x)| = {x(2α +1) /2 |f (x) |1/ 2 } {x(2β +1) /2 |f (x) |1/ 2 } we find the nice intermediate bound ∞ (1 ... 1) 2 = dx t t ≤ f (x) dx 11 dx, f (x) (7 .14 ) so, by our hypothesis we find t c 1 t−2 (t − 1) 2 ≤ 1 dx, f (x) and when we let t → ∞ we find the bound ∞ dx c 1 ≤ f (x) (7 .15 ) Since we were challenged ... for p1 + p2 + · · · + pn = the bound (7. 21) reduces to exactly to the Integral Intermezzo 11 5 classic AM-GM bound, n n apk ≤ k k =1 pk ak (7.22) k =1 Incidentally, the integral analog (7. 21) of...
... has n k =1 n 1/ s pk xs k ≤ 1/ t pk xt k (8 .10 ) k =1 Finally, show that then one has equality in the bound (8 .10 ) if and only if x1 = x2 = · · · = xn 12 4 The Ladder of Power Means Fig 8 .1 If x > ... the bound of Case I gives us n pk x−t k 1/ t n pk x−s k ≤ k =1 1/ s k =1 Now, when we take reciprocals we find n pk x−s k n 1/ s pk x−t k ≤ k =1 1/t , k =1 1 so when we substitute xk = yk , we get ... weighted sums of reciprocals: · · · xpn n ≤ p1 x1 + p2 x2 + · · · + pn xn ≤ xp1 xp2 p1 p2 + + ··· + x1 x2 p1 p2 + + ··· + x1 x2 pn , xn pn xn (8 .15 ) (8 .16 ) Going to Extremes The last of the power...
... formula (1/ s, 1/ t) = θ (1/ s1 , 1/ t1 ) + (1 − θ) (1/ s0 , 1/ t0 ) the bound (9. 41) automatically recaptures the inequality (9.24) from Challenge Problem 9.6; one only needs to set t1 = 1, s1 = 1, M1 = ... relation 1/ p + 1/ q = By splitting and by H¨lder’s inequality we find o m m n n q (cjk xp )1/ p (cjk yj )1/ q k cjk xk yj = j =1 k =1 j =1 k =1 m n 1/ p m n cjk xp k ≤ 1/ q q cjk yj j =1 k =1 , (9.26) j =1 k =1 and ... n 1 + = p q r n 1/ r ⇒ j =1 n 1/ p ap j ≤ (aj bj )r 1/ q bq j j =1 j =1 (b) Given p, q, and r are bigger than 1, show that if 1 + + = 1, p q r then one has the triple produce inequality n n 1/ p j=1...
... we see that as → we have ∞ a2 ( ) m ∞ b2 ( ) n m =1 ∞ = n =1 n =1 ∞ ∼ n1+2 dx = (10 .11 ) x1+2 Closing the Loop To complete the solution of Problem 10 .2, we only need to show that the corresponding ... the Plan When we apply Cauchy’s inequality (10 .2) to the pair (10 .4), we find ∞ ∞ am bn m+n m =1 n =1 ∞ ∞ ≤ a2 m m m+n n m =1 n =1 2λ ∞ ∞ b2 n n m+n m n =1 m =1 2λ , so, when we consider the first factor ... explicit: ∞ 1 π dy = 2λ (1 + y) y sin 2πλ for < λ < 1/ 2 (10 .8) Now, since sin 2πλ is maximized when λ = 1/ 4, we see that the smallest value attained by Cλ with < λ < 1/ 2 is equal to ∞ C = C1/4 = 1 √...
... (x)) dx = ak (11 .17 ) n 1 and, since ϕ(x)/x is nondecreasing, we also have the bound n 1/ n ak = exp k =1 −ϕ(n) n n ≤ exp n 1 −ϕ(x) x dx (11 .18 ) When we sum the relations (11 .17 ) and (11 .18 ), we then ... bound in our summation by parts inequality (11 .11 ), we find N n =1 (a1 +a2 +· · ·+an ) n N ≤4 n =1 (a1 +a2 +· · ·+an ) an , (11 .13 ) n and this is almost the inequality (11 .10 ) that we wanted to prove ... k=n ≤ k2 ∞ k=n ∞ k(k − 1) = k=n 1 − k 1 k = ≤ , n 1 n and, since s1 = + s2 ≤ + 1/ (2 − 1) = 2, we see that ∞ k=n ≤ k2 n for all n ≥ (11 .12 ) Hardy’s Inequality and the Flop 17 1 Now, when we use this...
... chapter Problem 13 .5 Given x, y, z ∈ (0, 1) such that max(x, y, z) ≤ (x + y + z)/2 < 1, (13 .19 ) show that one has the bound 1+ x 1 x 1+ y 1 y 1+ z 1 z ≤ + (x + y + z) − (x + y + z) 2 (13 .20) If this ... sum n f (x1 , x2 , , xn ) = φ(xk ) (13 .17 ) k =1 is Schur convex Thus, for α, β ∈ (a, b)n with α ≺ β one has n n φ(αk ) ≤ k =1 φ(βk ) (13 .18 ) k =1 Orientation To familiarize the bound (13 .18 ), one ... 19 2 Majorization and Schur Convexity corresponding ordered values, {α[j] } and {β[j] }, we could just as well write the chain (13 .1) as (1, 1, 1, 1) ≺ (0, 1, 1, 2) ≺ (1, 3, 0, 0)...
... cn−h n =1 h=0 N +H H H = cn−j n =1 j=0 N +H cn−k ¯ H k=0 H 1 H |cn−s | + Re = n =1 s=0 cn−s cn−t ¯ s=0 t=s +1 N H 1 H N +H |cn | + Re = (H + 1) cn−s cn−t ¯ n =1 s=0 t=s +1 n =1 N H 1 H N +H ≤ (H + 1) |cn ... j =1 k =1 yj , (14 .39) j =1 for any n real numbers y1 , y2 , , yn , then we also have k max 1 k≤n n cj ψj (x) dx ≤ C log2 (4n) j =1 c2 k (14 .40) k =1 for all real c1 , c2 , , cn Exercise 14 .8 ... one has the bound M N | um , | m =1 n =1 M M ≤ 1/ 2 | um , uµ | m =1 µ =1 N N 1/ 2 | v n , vν | n =1 ν =1 Exercise 14 .12 (Selberg’s Inequality) Prove that if x and y1 , y2 , , yn are elements of...
... SDT: 08 360 311 43_ Số 1/ 100 hẻm 56 Lữ Gia, Quận 11 SDT: 08 36020443 • Do you like (to go to/going to) parties? (Why?/Why not?) • Do you often go to parties? • Do you usually have these parties at ... Minh: Số 42 Nguyễn Phi Khanh, Quận SDT: 08 360 311 43_ Số 1/ 100 hẻm 56 Lữ Gia, Quận 11 SDT: 08 36020443 • (For university students) What are your favourite classes/courses/subjects at university? • ... 360 311 43_ Số 1/ 100 hẻm 56 Lữ Gia, Quận 11 SDT: 08 36020443 Sport/Exercise In the questions below, the word "sport" might be replaced by the word, "exercise" for some questions • Do you like any particular...