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Questions tagged [cauchy-schwarz-inequality]

Problems with using C-S (Cauchy-Schwarz inequality)

1
vote
7answers
293 views

Proof that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$

If $x_1,...,x_n$ are positive real numbers and if $y_k=1/x_k$, prove that $$\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2.$$ I've been learning induction, and I've come across this ...
9
votes
5answers
548 views

Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$

I am currently attempting to prove the following inequality $\dfrac{a}{b+c}+\dfrac{b}{a+c}+\dfrac{c}{a+b} \geq \dfrac{3}{2}$ for all $ a,b,c>0$ My instinctive plan of attack is to use the AM/GM ...
10
votes
4answers
2k views

Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
8
votes
3answers
722 views

A version of Hardy's inequality involving reciprocals.

How can one prove for any sequence of positive numbers $a_n, n\ge1,$ we have $$\sum_{n=1}^\infty \frac{n}{a_1+a_2+a_3+\cdots+a_n}\le 2\sum_{n=1}^\infty \frac{1}{a_n}$$ Added later: Apparently, this ...
2
votes
4answers
178 views

Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$

Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$. I computed the whole product ;If $(a+b+c)=x\implies (1+x)(1+\frac{bc+ca+ab}{abc})=16$. ...
20
votes
4answers
903 views

How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

let $a,b,c>0$,and such $a+b+c=3$, show that $$\dfrac{2}{(a+b)(4-ab)}+\dfrac{2}{(b+c)(4-bc)}+\dfrac{2}{(a+c)(4-ac)}\ge 1$$ I think this inequality use this $$ab\le\dfrac{(a+b)^2}{4}$$
10
votes
4answers
274 views

Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$

Given $a,b,c,d$ such that $a + b + c + d = 4$ show that $$(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) \geq 256$$ What I have tried so far is using CBS: $(a^2 + 3)(b^2 + 3) \geq (a\sqrt{3} + b\sqrt{3})^2 = ...
2
votes
1answer
230 views

Find the maximum of the expression

Let $a,b,c$ be real positive numbers so that $abc=1$. Find the maximum value that the following expression can attain: $$\frac{a}{a^8+1}+\frac{b}{b^8+1}+\frac{c}{c^8+1}$$ My try: I first though on ...
5
votes
4answers
292 views

Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$

How to prove this for positive real numbers? $$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$ I tried AM-GM, CS inequality but all failed.
26
votes
4answers
1k views

Inequality with five variables

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$ Easy to show ...
19
votes
12answers
2k views

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds. I spent a lot of time trying to solve this and, having consulted some books,...
0
votes
4answers
809 views

Given positive real numbers $a, b, c$ with $a<b+c$, show that $a/(1+a)<b/(1+b)+c/(1+c)$ [closed]

I am trying to prove the following: $$\frac{a}{1+a} < \frac{b}{1+b} + \frac{c}{1+c}$$ given that $a, b, c > 0$ and $a < b+c$. I tried various rearrangements but can't seem to get anywhere ...
11
votes
2answers
278 views

If $\frac {1}{2+a} + \frac {1}{2+b} + \frac {1}{2+c} = 1$, prove $\sqrt{ab} + \sqrt{ac} + \sqrt{bc} \leq 3$

Let $a,b,c$ be non-negative numbers such that $$\frac {1}{2+a} + \frac {1}{2+b} + \frac {1}{2+c} = 1.$$ Prove that $ \sqrt{ab} + \sqrt{ac} + \sqrt{bc} \leq 3 $.
2
votes
3answers
353 views

What are the most popular techniques of proving inequalities?

I know some of it, but I would like to learn other ones. I think this question is useful, as it will be helpful for other beginners. Such as: uvw, AM GM, Cauchy–Schwarz inequality, Jensen inequality. ...
12
votes
5answers
522 views

proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

Let $a,b,c>0$ how to prove that : $$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$$ I find that $$\ \frac{ab}{a^{2}+3b^{2}}=\frac{1}{\frac{a^{2}+3b^{2}}{ab}}=\frac{1}{\...
8
votes
1answer
654 views

Show inequality of integrals (cauchy-schwarz??)

$f:[0,1]\to\mathbb{C}$ continuous and differentiable and $f(0)=f(1)=0$. Show that $$ \left |\int_{0}^{1}f(x)dx \right |^2\leq\frac{1}{12}\int_{0}^{1} \left |f'(x)\right|^2dx $$ Well I know that $$ \...
7
votes
3answers
1k views

A inequality proposed at Zhautykov Olympiad 2008

A inequality proposed at Zhautykov Olympiad 2008 Let be $a,b,c >0$ with $abc=1$. Prove that: $$\sum_{cyc}{\frac{1}{(a+b)b}} \geq \frac{3}{2}.$$ $a=\frac{x}{y}$, $b=\frac{y}{z}$, $c=\frac{z}{x}$....
4
votes
3answers
174 views

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

If $a,b,c$ are sides of triangle Find Minimum value of $$S=\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$$ My Try: Let $$P=\sqrt{a}+\sqrt{b}+\sqrt{c}$$ we have $$S=\sum \frac{1}{\frac{\sqrt{b}}{...
8
votes
4answers
419 views

Proving :$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$

Let $a,b,c>0$ be real numbers such that $a+b+c=3$,how to prove that? : $$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$$
6
votes
3answers
793 views

Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$

I started to study inequalities - I try to solve a lot of inequlites and read interesting .solutions . I have a good pdf, you can view from here . The inequality which I tried to solve and I didn't ...
6
votes
5answers
393 views

How to find the minimum value of this function?

How to find the minimum value of $$\frac{x}{3y^2+3z^2+3yz+1}+\frac{y}{3x^2+3z^2+3xz+1}+\frac{z}{3x^2+3y^2+3xy+1}$$,where $x,y,z\geq 0$ and $x+y+z=1$. It seems to be hard if we use calculus methods. ...
1
vote
2answers
113 views

Minimum value of $\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2$

Given $a,b,c \in \mathbb{R^+}$ such that $a+b+c=12$ Find Minimum value of $$S=\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2$$ My Try: By Cauchy Schwarz ...
3
votes
3answers
361 views

Inequality $\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{3n+1}>1$ [duplicate]

Show that $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}>1,\:\forall n\in\mathbb{N}$$ This is a 9th grade problem. I was trying to take the greatest numerator, which is the last numerator of the ...
3
votes
2answers
867 views

Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$

In the pdf which you can download here I found the following inequality which I can't solve it. Exercise 2.1.11 Let $a,b,c \gt 0$. Prove that $$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{...
2
votes
1answer
91 views

inequality with three variables and condition

If $a$,$b$ and $c$ positive real numbers such that $a+b+c=1$, prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$. I have tried several methods to solve this,but ...
2
votes
5answers
81 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
1
vote
1answer
100 views

Bordered Hessian matrix to find a minimum of the function

I was trying to find the global minimum for the function $$(a + b) z + (a + c) y + (b + c) x $$ subject to the following constraint: $$(xy + xz + yz)(ab + bc + ac)=1.$$ By Lagrange multipliers I found ...
1
vote
2answers
97 views

Show the Trace$(B)^{2} \leq$ nTrace$(B^{T}B)$

The following definition is needed for my actual question: For $A, B \in \mathcal{M}_{n \times n}$ define $$ \langle A, B \rangle = \text{Trace}(B^{T}A) = \sum_{j=1}^{n}\sum_{i=1}^{n}b_{ij} \, a_{ij}...
0
votes
3answers
212 views

$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$

$a, b,c $ are positive real numbers such that $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$ Any ideas ?
0
votes
4answers
67 views

Employing Cauchy-Schwartz inequality question

Let $d_i\ge 0$ and $$\sum_{i=1}^n d_i=c$$ and $$\sum_{i=1}^nd_i^2\le C$$ Show that $$\frac{c^2}{n}\le C$$ The text says Using Cauchy-Schwartz inequality and the first equation, show the last ...
0
votes
4answers
86 views

Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$

Prove: that $$(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$$ for all real numbers $a$ and $b$. My attempt: $(a^2+b^2)(a^4+b^4)=a^6+a^2 b^4+b^2 a^4+b^6\geq 4\cdot(a^6\cdot b^6\cdot a^2 b^4\cdot a^4 b^2)^{\frac{...
11
votes
2answers
450 views

cyclic three variable inequality

Let $a,b,c$ be nonnegative real numbers and $a+b+c=3$. Prove the inequality $$ \sqrt{24a^2b+25}+\sqrt{24b^2c+25}+\sqrt{24c^2a+25}\le 21 $$ I have tried to find the solution using classical ...
-1
votes
3answers
9k views

If $a+b+c=6$ and $a,b,c$ belongs to positive reals $\mathbb{R}^+$; then find the minimum value of $\frac{1}{a}+\frac{4}{b}+\frac{9}{c}$ .

If $a+b+c=6$ and $a,b,c$ belongs to positive reals, then find the minimum value of $$\frac{1}{a}+\frac{4}{b}+\frac{9}{c}$$ using AM $\ge HM$ $\frac{a+b+c}{3}\ge\frac{3}{\frac{1}{a}+\frac{1}{b}+\...
9
votes
1answer
153 views

What kind of “geometric” regularity $f'^2$ gives on $f$

When solving real-analysis' problems I like to represent the functions involved and think geometrically what is going on. Today I got the following exercise : Let $f \in \mathcal{C}^1(\mathbb{R},\...
3
votes
2answers
453 views

Is there a proof of the AM-GM Inequality via Cauchy's Inequality? Or vice-versa?

The title says it all. There is already a list exhausting proofs of the AM-GM inequality here; however, none via Cauchy's Inequality. Why do I care? Recreation. But more than that, proving ...
5
votes
2answers
252 views

If $abc=1$ so $\sum\limits_{cyc}\frac{a}{\sqrt{a+b^2}}\geq\frac{3}{\sqrt2}$

Let $a$, $b$ and $c$ be positive numbers such that $abc=1$. Prove that: $$\frac{a}{\sqrt{a+b^2}}+\frac{b}{\sqrt{b+c^2}}+\frac{c}{\sqrt{c+a^2}}\geq\frac{3}{\sqrt2}$$ After substitution $a=\frac{y}{x}$.....
4
votes
2answers
78 views

Let $f:[0,1]\to[1,3]$ be continuous. Prove $1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}$

Let $f:[0,1]\to[1,3]$ be continuous. Prove $$1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}.$$ The left is just Cauchy's inequality with integral form, but ...
4
votes
1answer
121 views

$\frac {x_1^3}{y_1}+\frac {x_2^3}{y_3}+\ldots+\frac {x_n^3}{y_n}\leq \frac {a^4+b^4}{ab (a^2+b^2)}(x_1^2+x_2^2+\ldots+x_n^2) $

Let $b>a>0$ and $x_1, x_2,\ldots,x_n,y_1, y_2,\ldots,y_n\in [a,b]$. If $$x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2\,,$$ then $$\frac {x_1^3}{y_1}+\frac {x_2^3}{y_3}+\ldots+\frac {...
9
votes
2answers
1k views

Hard Olympiad Inequality

Let x,y,z be positive real numbers such that $xy+xz+yz=1$. Prove that $$\sqrt{x^3+x}+ \sqrt{y^3+y}+ \sqrt{z^3+z} \geq 2 \cdot \sqrt{x+y+z}$$. I tried to square expand homogenize then majorize. But I ...
3
votes
2answers
217 views

General Cauchy-Schwarz for adjoint positive operators

I'm trying to prove the next inequality, like Cauchy-Schwarz standard inequality: $$|\langle Tx,y\rangle |\leq\langle Tx,x\rangle ^{1/2}\langle Ty,y\rangle ^{1/2}\space\forall x,y\in\mathcal{H},$$ ...
4
votes
2answers
260 views

Prove that $(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$ [duplicate]

For the non-negative real numbers $a, b, c$ prove that $$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$$ What I did is applying Holder's inequality in LHS:$$(a^2+(\sqrt{2})^2)(b^2+(\sqrt{2})^2)(c^2+(\sqrt{2})^...
4
votes
8answers
133 views

Let $(x,y) \in \Bbb R^+$ Prove that $\Bigr(1+\frac{1}{x}\Bigl)\Bigr(1+\frac{1}{y}\Bigl)\ge \Bigr(1+\frac{2}{x+y}\Bigl)^2$

Let $(x,y) \in \Bbb R^+$ Prove that $$\Bigr(1+\frac{1}{x}\Bigl)\Bigr(1+\frac{1}{y}\Bigl)\ge \Bigr(1+\frac{2}{x+y}\Bigl)^2$$ My try Well, i didn't see a way to factorize this, so i put it in ...
2
votes
1answer
183 views

Inequality : $\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$

Let $a, b, c$ be positive real number such that $a+b+c = 3$. Prove that $$\displaystyle\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \displaystyle\sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$$ My attempt : ...
2
votes
1answer
212 views

Contest Inequality - Is it AM GM?

To prove: $$(a^2+2)(b^2+2)(c^2+2)\ge9(ab+bc+ca)$$ where $a,b,c$ are positive real numbers. When does equality hold? I thought equality would hold when $a = b = c$, but that doesn't seem to fit the ...
1
vote
3answers
80 views

For a given condition is it true that $a+b+c=3$. [duplicate]

Suppose a, b, c are positive real numbers such that $$(1+a+b+c)\left(1+\frac 1a+\frac 1b+\frac 1c\right)=16$$ Then is it true that we must have $a+b+c=3$ ? Please help me to solve this. Thanks ...
1
vote
2answers
342 views

Inequality $\frac{a}{\sqrt{bc}}\cdot\frac1{a+1}+\frac{b}{\sqrt{ca}}\cdot\frac1{b+1}+\frac{c}{\sqrt{ab}}\cdot\frac1{c+1}\leqslant\sqrt2.$

Let $a,b,c>0$ and $\frac1{a+1}+\frac1{b+1}+\frac1{c+1}=1.\qquad $ Prove $$\frac{a}{\sqrt{bc}}\cdot\frac1{a+1}+\frac{b}{\sqrt{ca}}\cdot\frac1{b+1}+\frac{c}{\sqrt{ab}}\cdot\frac1{c+1}\leqslant\sqrt2.$...
1
vote
2answers
164 views

Find the minimum value of $P=\sum _{cyc}\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}$

For $x>0$, $y>0$, $z>0$ and $x+y+z=3$ find the minimize value of $$P=\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}+\frac{\left(y+1\right)^2\left(z+1\right)^2}{x^2+1}+\frac{\left(z+1\...
0
votes
3answers
404 views

If $a$ and $b$ are positive real numbers such that $a+b=1$, then prove that $\big(a+\frac{1}{a}\big)^2 +\big(b+\frac{1}{b}\big)^2 \ge\frac{25}{2}$ [duplicate]

If $a$ and $b$ are positive real numbers such that $a+b=1$, then prove that $$\bigg(a+ \dfrac {1}{a}\bigg)^2 +\bigg(b+ \dfrac {1}{b}\bigg)^2 \ge \dfrac {25}{2}$$ My tries: I am really unable to see ...
-2
votes
2answers
115 views

If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Prove that for any set of three positive real $x, y, z$ such that $xy+yz+zx=3$ $x^2(y+z)+y^2(x+z)+z^2(x+y)+2\sqrt {xyz}\left(\sqrt{x^3+3x}+\sqrt{y^3+3x}+\sqrt{z^3+3x}\right)\ge$ $\ge 2xyz(x^2+y^2+z^2+...
6
votes
3answers
232 views

How to prove $\frac 1{2+a}+\frac 1{2+b}+\frac 1{2+c}\le 1$?

How to prove this inequality ? $$\frac 1{2+a}+\frac 1{2+b}+\frac 1{2+c}\le 1$$ for $a,b,c>0 $ and $a+b+c=\frac 1a+\frac 1b+\frac 1c$. I do not know where to start. I need some idea and advice on ...