inequality

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Vimath

396 posts

Vimath

#1 h Aug 8, 2016, 5:24 PM • 1 Y

Y by Adventure10

1)If $a,b,c$ are positive real numbers $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge 2(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})$
2)If $a,b,c,d$ are positive real numbers then prove that $(1+a^4)(1+b^4)(1+c^4)(1+d^4)\ge (1+(abcd)^4)$
3) If $n$ is a positive integer,prove that $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}>1$
4)If $a,b,c$ are three positive real numbers in harmonic progression .Prove that $\frac{a+b}{2a-b}+\frac{c+b}{2c-b}>4$

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bzhang512

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bzhang512

#2 h Aug 8, 2016, 5:56 PM • 1 Y

Y by Adventure10

1) By Cauchy-Schwarz, $\frac{1}{a}+\frac{1}{b} \geq \frac{4}{a+b}$ , $\frac{1}{b}+\frac{1}{c} \geq \frac{4}{b+c}$ , and $\frac{1}{c}+\frac{1}{a} \geq \frac{4}{c+a}$ . Add these inequalities to get $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge 2(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})$ .

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ohmcfifth

69 posts

ohmcfifth

#3 h Aug 8, 2016, 6:01 PM • 2 Y

Y by Adventure10, Mango247

S2

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ThisIsASentence

221 posts

ThisIsASentence

#4 h Aug 8, 2016, 6:10 PM • 2 Y

Y by Adventure10, Mango247

Solutions

This post has been edited 5 times. Last edited by ThisIsASentence, Aug 8, 2016, 6:44 PM
Reason: added note

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sqing

41157 posts

sqing

#5 h Aug 9, 2016, 12:38 AM • 2 Y

Y by Adventure10, Mango247

Vimath wrote:

3) If $n$ is a positive integer,prove that $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}>1$

Strengthening
If $n$ is a positive integer .Prove that $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}\ge\frac{13}{12}.$

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niraekjs

1861 posts

niraekjs

#6 h Aug 9, 2016, 1:15 AM • 1 Y

Y by Adventure10

1)

WLOG, let $a, b, c, \in \mathbb{R^+}$ s.t. $a \le b \le c$ . Then, $a = u, b = u+v, c = u+v+w$ , for some $u, v, w \ge 0$ . Then, if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge 2(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})$ we have
$\begin{align*} \frac{1}{a}+\frac{1}{b}+\frac{1}{c} &\ge 2(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}) \\ \implies 1/u+1/(u+v)+1/(u+v+w) &\ge 2/(2 u+2 v+w)+2/(2u+v+w)+2/(2 u+v) \\ \implies (3 u^2+4 u v+2 u w+v^2+v w)/(u (u+v) (u+v+w)) & \ge (2 (12 u^2+16 u v+8 u w+5 v^2+5 v w+w^2))/((2 u+v) (2 u+v+w) (2 u+2 v+w)) \\ \implies (24 u^5+80 u^4 v+40 u^4 w+102 u^3 v^2+102 u^3 v w+22 u^3 w^2+62 u^2 v^3+93 u^2 v^2 w+39 u^2 v w^2+4 u^2 w^3+18 u v^4+36 u v^3 w+22 u v^2 w^2+4 u v w^3+2 v^5+5 v^4 w+4 v^3 w^2+v^2 w^3) & \ge 24 u^5+80 u^4 v+40 u^4 w+98 u^3 v^2+98 u^3 v w+18 u^3 w^2+52 u^2 v^3+78 u^2 v^2 w+30 u^2 v w^2+2 u^2 w^3+10 u v^4+20 u v^3 w+12 u v^2 w^2+2 u v w^3 \\ \implies 4 u^3 v^2+4 u^3 v w+4u^3 w^2+10u^2 v^3+15 u^2 v^2 w+9 u^2 v w^2+2 u^2 w^3+8 u v^4+16 u v^3 w+10 u v^2 w^2+2 u v w^3+2 v^5+5 v^4 w+4 v^3 w^2+v^2 w^3) & \ge 0 \\ \end{align*}$

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mathaboy

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mathaboy

#7 h Aug 24, 2016, 1:21 PM • 2 Y

Y by Adventure10, Mango247

2)holder

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arqady

30150 posts

arqady

#8 h Aug 24, 2016, 1:24 PM • 3 Y

Y by uwu_, Adventure10, Mango247

mathaboy wrote:

2)holder

I think 2) is obvious without Holder.

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sqing

41157 posts

sqing

#9 h Oct 25, 2022, 3:27 PM

Y by

Vimath wrote:

1)If $a,b,c$ are positive real numbers $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge 2(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})$
2)If $a,b,c,d$ are positive real numbers then prove that $(1+a^4)(1+b^4)(1+c^4)(1+d^4)\ge (1+(abcd)^4)$
3) If $n$ is a positive integer,prove that $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}>1$
4)If $a,b,c$ are three positive real numbers in harmonic progression .Prove that $\frac{a+b}{2a-b}+\frac{c+b}{2c-b}>4$

https://artofproblemsolving.com/community/c6h1286362p6783919
https://artofproblemsolving.com/community/c6h2943652p26350136

This post has been edited 1 time. Last edited by sqing, Oct 26, 2022, 2:41 AM

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sqing

41157 posts

sqing

#10 h Apr 4, 2023, 2:14 PM

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Let $a,b,c>0.$ Prove that
$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \ge 3\left( \frac{1}{a +2 b} + \frac{1}{b + 2c} + \frac{1}{c + 2a}\right)$ $\frac{1}{a} + \frac{2}{b} + \frac{1}{c} \ge \frac{64}{25}\left( \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a}\right)$ $\frac{1}{a} + \frac{3}{b} + \frac{1}{c} \ge \frac{2(19+4\sqrt{21})}{25}\left( \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a}\right)$ Let $a,b>0.$ Prove that
$\frac{1}{a}+\frac{1}{b}\ge \frac{3+2\sqrt 2}{a+2b}$ $\frac{1}{a}+\frac{3}{b}\ge \frac{2(2+\sqrt 3)}{a+b}$

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youthdoo

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youthdoo

#11 h Apr 5, 2023, 4:25 AM • 1 Y

Y by uwu_

arqady wrote:

I think 2) is obvious without Holder.

C-S twice. :trampoline:

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DAVROS

1630 posts

DAVROS

#12 h Apr 5, 2023, 7:45 AM

Y by

sqing wrote:

Let $a,b,c>0.$ Prove that
$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \ge 3\left( \frac{1}{a +2 b} + \frac{1}{b + 2c} + \frac{1}{c + 2a}\right)$ $\frac{1}{a} + \frac{3}{b} + \frac{1}{c} \ge \frac{2(19+4\sqrt{21})}{25}\left( \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a}\right)$

solutions

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sqing

41157 posts

sqing

#13 h Apr 5, 2023, 1:19 PM

Y by

Very very nice.Thank DAVROS.

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sqing

41157 posts

sqing

#14 h Aug 20, 2023, 9:38 AM

Y by

Vimath wrote:

1)If $a,b,c$ are positive real numbers $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge 2(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})$

Let $a, b, c$ are sides of $ABC$ triangle such that $a+b+c=1$ . Prove that $4\left ( \frac{1}{a+b}+\frac{1}{b+c}+ \frac{1}{c+a}\right )\leq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+9$

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