Inequality 2 - Assignment

Pure Mathematics – Inequalities 2

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Inequalities Assignment 2 1. Let a, b and c be positive real numbers such that a2 + b2 + c2 = 3. (a) Using Cauchy-Schwarz’s Inequality, prove that (i) a + b + c ≤ 3 . (ii) a 3 + b 3 + c 3 ≥ 3 . (b) For every n = 2, 3, 4, …, let P(n) be the statement that a n + b n + c n ≥ 3 . Prove that for any integer k ≥ 2 , (i) If P(k) is true, then P(2k) is true; (ii) If P(k) is true, then P(2k – 1) is true. 2.

(a) Let m and k be positive integers and k ≤ m . m(m − 1)L (m − k + 1) (m + 1)m L (m − k + 2) < for k > 1. Prove that mk (m + 1)k

Prove that the above inequality does not hold when k = 1. (a) Let m be a positive integer. Using (a) or otherwise, prove that m

1⎞ 1 ⎞ ⎛ ⎛ ⎜1 + ⎟ < ⎜1 + ⎟ ⎝ m⎠ ⎝ m + 1⎠ 3.

m +1

.

Let n be a positive integer. (a) Suppose 0 < p < 1. (i) By considering the function f(x) = xp – px on (0, ∞), or otherwise, prove that x p ≤ px + 1 − p for all x > 0. (ii) Using (a)(i), or otherwise, prove that a p b1− p ≤ pa + (1 − p )b for all a, b > 0. (iii) Let a1, a2, a3, …, an and b1, b2, b3, …, bn be positive real numbers. Using (a)(ii), or otherwise, prove that p

n

∑a i =1

p i

bi

1− p

⎛ n ⎞ ⎛ n ⎞ ≤ ⎜ ∑ ai ⎟ ⎜ ∑ bi ⎟ ⎝ i =1 ⎠ ⎝ i =1 ⎠

1− p

.

(b) Suppose 0 < s < 2. Let x1, x2, x3, …, xn and y1, y2, y3, …, yn be positive real numbers. Prove that (i)

n

∑x y i =1

i

i

1 2

1 2

⎛ 2− s ⎞ ⎛ 2− s s s ⎞ ≤ ⎜ ∑ xi y i ⎟ ⎜ ∑ x i y i ⎟ , ⎝ i =1 ⎠ ⎝ i =1 ⎠ n

n

⎛ n s 2− s ⎞⎛ n 2− s s ⎞ ⎛ n 2 ⎞⎛ n 2 ⎞ (ii) ⎜ ∑ xi y i ⎟⎜ ∑ xi y i ⎟ ≤ ⎜ ∑ xi ⎟⎜ ∑ y i ⎟ ⎝ i =1 ⎠⎝ i =1 ⎠ ⎝ i =1 ⎠⎝ i =1 ⎠

Pure Mathematics – Inequalities 2

4.

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(a) Let a1, a2, …, an be real numbers and b1, b2, …, bn be non-zero real numbers. n

By considering

∑ (a x + b ) i =1

2

i

i

, or otherwise, prove Schwarz’s inequality

2

⎛ n ⎞ ⎛ n 2 ⎞⎛ n 2 ⎞ ⎜ ∑ aibi ⎟ ≤ ⎜ ∑ ai ⎟⎜ ∑ bi ⎟ , and that the equality holds if and only if ⎝ i =1 ⎠ ⎝ i =1 ⎠⎝ i =1 ⎠

a1 a2 a = =L n b1 b2 bn 2

(b) (i)

⎛ n ⎞ ⎜ ∑ xi ⎟ Prove that ⎜ i =1 ⎟ ≤ ⎜ n ⎟ ⎜ ⎟ ⎝ ⎠

n

∑x i =1

2

i

n

, where x1, x2, …, xn are real numbers.

2

⎛ n ⎞ ⎛ n ⎞⎛ n 2⎞ (ii) Prove that ⎜ ∑ λi xi ⎟ ≤ ⎜ ∑ λi ⎟⎜ ∑ λi xi ⎟ , where x1, x2, …, xn are real ⎝ i =1 ⎠ ⎝ i =1 ⎠⎝ i =1 ⎠

numbers and λ1, λ2, …, λn are positive numbers. Find a necessary and sufficient condition for the equality to hold. (iii) Using (b)(ii) or otherwise, prove that 2

2

2

2

y ⎞ y y y ⎛ y1 y 2 ⎜ + 2 + L + nn ⎟ < 1 + 22 + L + nn , where y1, y2, …, yn are t t t ⎠ t t ⎝ t

real numbers, not all zero, and t ≥ 2 .