Show-that-for-all-a-b-c-d-R-with-a-b-c-d-0-1-ab-cd-1-4-a-2-b-2-c-2-d-2-2-abcd-1-4-1-4-a-b-c-d-Help-

Question Number 193116 by Mastermind last updated on 04/Jun/23

Show that for all a, b, c, d \in R with

a, b, c, d ⩾ 0

1) \sqrt{ab} \sqrt{cd} ⩽ \frac{1}{4} (a^{2} + b^{2} + c^{2} + d^{2})

2) {(abcd)}^{\frac{1}{4}} ⩽ \frac{1}{4} (a + b + c + d)

Help!

Answered by Subhi last updated on 04/Jun/23

1) 4 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ {(a + b + c + d)}^{2}

a + b + c + d ⩾ 4^{4} \sqrt{a b c d} (A M - G M)

{(a + b + c + d)}^{2} ⩾ 16 \sqrt{a b c d}

4 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ 16 \sqrt{a b c d}

\frac{1}{4} (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ \sqrt{a b} . \sqrt{c d}

Commented by Subhi last updated on 04/Jun/23

4 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ {(a + b + c + d)}^{2}

{(a + b + c + d)}^{2} = a^{2} + b^{2} + c^{2} + d^{2} + 2 c d + 2 a b + 2 a c + 2 a d + 2 b c + 2 b d

= a^{2} + b^{2} + c^{2} + d^{2} + 2 (a b + a c + a d + b c + b d + c d) (i)

a^{2} + b^{2} + c^{2} + d^{2}

a^{2} + b^{2} ⩾ 2 \sqrt{a^{2} . b^{2}} = 2 a b (A M - G M)

b^{2} + c^{2} ⩾ 2 b c ⇛ c^{2} + d^{2} ⩾ 2 c d

a^{2} + d^{2} ⩾ 2 a d ⇛ b^{2} + d^{2} ⩾ 2 b d

a^{2} + c^{2} ⩾ 2 a c

s u m t h e 5 e q u a t i o n s

3 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ 2 (a b + b c + a c + a d + b d + c d) (i i)

f r o m (i), (i i)

{(a + b + c + d)}^{2} ⩽ 4 (a^{2} + b^{2} + c^{2} + d^{2})

Commented by Mastermind last updated on 04/Jun/23

Thank you so much

Commented by Mastermind last updated on 04/Jun/23

Thank you my boss

Commented by Mastermind last updated on 04/Jun/23

{What}^{'} s AM - GM ?

Commented by Subhi last updated on 04/Jun/23

A r i t h m e t i c g e o m e t r i c m e a n i n e q u a l i t y

i t s p r o o f i s b e l o w ↓

Answered by Subhi last updated on 04/Jun/23

p r o o f f o r A M - G M

{(\sqrt{a} - \sqrt{b})}^{2} ⩾ 0

a + b - 2 \sqrt{a b} ⩾ 0

\frac{a + b}{2} ⩾ \sqrt{a b}

\frac{a_{1} + a_{2} + \dots \dots \dots a_{n}}{n} ⩾^{n} \sqrt{a_{1} . a_{2} \dots \dots a_{n}}

\frac{a + b + c + d}{4} ⩾^{4} \sqrt{a b c d}

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