Questions tagged [cauchy-schwarz-inequality]

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

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Equality characterization of the reverse Cauchy-Schwarz inequality in a Lorentzian manifold

Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone(ie, both future-directed or past-...
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I can't seem to understand the proof of the Cauchy-Schwarz inequality

The proof said to use the quadratic polynomial: $$P(t) = \sum_{i=1}^n(a_it - b_i)^2$$ by which we notice that $P(t) \ge 0$ , but then this is the part which I didn't understand where we conclude that ...
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A modified version of log-sum inequality.

Suppose $a_i>0, b_i>0, c_i>0 \; \forall i = 1, 2, \dots, n$, and $$\sum_{i} a_i b_i \ln(\frac{c_i}{b_i}) \geq 0, $$ where $a_i, b_i, c_i$ are not constant over $i$. Moreover, $$\sum_{i}a_ib_i\...
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A functional inequality over integers

Given positive integers $N$ and $n$, we define a function $f(x)$ as follows: $$f(x):=\frac{x^2-3x}{2}~\text{ when } x\leq N,$$ and $$f(x):=\frac{1}{2n}x^2+\frac{n-4}{2}x~\text{ when } x>N.$$ ...
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Demonstrate that: $\frac{1-a^2+c^2}{5c(3a+2b\sqrt{2})}+\frac{1-b^2+a^2}{5a(3b+2c\sqrt{2})}+\frac{1-c^2+b^2}{5b(3c+2a\sqrt{2})} \geq \frac{6}{5}$

The question Let $a,b,c \in (0,\infty)$ with $a^2+b^2+c^2=\frac{1}{4}$. Demonstrate that: $$\frac{1-a^2+c^2}{5c(3a+2b\sqrt{2})}+\frac{1-b^2+a^2}{5a(3b+2c\sqrt{2})}+\frac{1-c^2+b^2}{5b(3c+2a\sqrt{2})} \...
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challenging inequality with complex numbers

The statement of the problem: Let $n \in \mathbb N $ \ {0} and $z_1,z_2, ... z_n \in \mathbb C $. Prove that $$\sum_{i=1}^n |z_i||z-z_i| \ge \sum_{i=1}^n |z_i|^2$$ holds for any $z \in \mathbb C $ $\...
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Vertex and Edges [closed]

In a community every two acquaintances have no common acquaintances. Furthermore, every two people who do not know each other have exactly two common acquaintances. Show that everybody in this ...
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Prove the inequality knowing $x,y,z \ge 0$ and $xyz=1$ $\frac{x^5}{x^2+1}+\frac{y^5}{y^2+1}+\frac{z^5}{z^2+1}\ge\frac{3}{2}$

Prove the inequality knowing $x,y,z \ge 0$ and $xyz=1$ $$\frac{x^5}{x^2+1}+\frac{y^5}{y^2+1}+\frac{z^5}{z^2+1}\ge\frac{3}{2}$$ Starting from the condition set that $xyz =1$ I did this : $$x^3+y^3+z^3\...
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Positive matrices from positive linear functionals.

If $C=\sum\limits_{r=1}^K A_r \otimes B_r$, for elements $\{ A_r\}_r^K $ and $\{B_r\}_r^K$ of $C^*$ algebras $\mathscr{A}$ and $\mathscr{B}$, respectively; then if $\sigma$ is a positive linear ...
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Prove an inequality from a scalar product

This exercice is a proof of the convergence of the gradient descent towards a stationary point. Let $\mathbf{v}_1, \dots, \mathbf{v}_T$ be a sequence of gradients. We have $\mathbf{w}^{(1)} = \mathbf{...
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Enhanced Nesbitt's Inequality with Geometric Mean Term.

If a, b, c are positive real numbers such that $a + b + c = 1$, then how do you prove that $$\frac{a} {b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{3}{2} \sqrt[3]{abc} \ge {2}.$$ Here is my proof attempt: $...
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Why the Lipschitz constant of a matrix A is its largest singular value

The Lipschitz continuous is by $||Ax-Ay|| \leq L||x-y||$, where A is a matrix. Here by cauchy-schwarz inequality, it is easy to have $||Ax-Ay||^2 \leq ||A|| \times ||x-y||$. But the problem is, what ...
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Alternate proof of Cauchy Schwarz

My teacher gave me the following simple proof of CS inequality: Consider 2 vector $\mathbf A$ and $\mathbf B$ in an N dimensional space $\mathbf A = \sum_{i=0}^n{a_i \mathbf x_i}$ And $\mathbf B = \...
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An inequality involving real numbers....... [closed]

Given real numbers $a_1,a_2, \ldots , a_n$, let $S_1 = \sum_{i=1}^na_i$ and $S_2 = \sum_{i=1}^na_i^2$. Prove that $$S_1\sqrt{nS_2}\le S_1^2+\frac 12 \left \lfloor \frac{n^2}4 \right \rfloor \left(\...
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Product of dual norms in $\mathbb R^n$ inequality

Suppose $x\in \mathbb{R}^n$, do we have that $\|x\|_1 \|x\|_\infty \leq \|x\|_2^2$? I came across this inequality as an intermediate step in proving something else, and it appears to me that it might ...
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A strengthening of a known inequality; looking for a neater solution

If $a\ge b\ge c > 0$ then $$ \begin{split} \frac{(a-b)^2}{(a+b)} + \frac{(b-c)^2}{(b+c)} & \ge \sqrt{3(a^2+b^2+c^2)}- (a+b+c) \\&\ge \frac{(a-b)^2}{\frac{1+\sqrt{3}}{2}a+\frac{5-\sqrt{3}}{...
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Prove the inequality $\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right)$

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=2$. Prove the inequality: $\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right)$ I ...
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What does that mean by being less than infinity?

One of the most immediate qualitative inferences from Cauchy's inequality is the simple fact that $\sum_{k=1}^\infty a_k^2 < \infty. and. \sum_{k=1}^\infty b_k^2 < \infty$ imply that $\sum_{k=...
Mike Dong's user avatar
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How to prove this inequality $\frac{(\sum_i w_i x_i)}{(\sum_i w_i y_i^2)} \geq \sum_i w_i \frac{x_i}{y_i^2} - C$

This is what I am trying to show $\frac{(\sum_i w_i x_i)}{(\sum_i w_i y_i^2)} \geq \sum_i w_i \frac{x_i}{y_i^2} - C$, where $w_i \geq 1$, $y_i>0$, and $x_i \in \mathbb{R}$. We can assume $x_i>0$ ...
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Usage of Cauchy-Schwarz in lower bound argument

I am trying to understand the following step in a paper which the authors claim follows by Cauchy-Schwarz: $$ \left ( \int_D \left( \int_D k(x,y) dx\right)^2 dy \right)^{1/2} \ge \frac{1}{\sqrt{\text{...
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Cauchy-Schwarz-like inequality involving 2-by-2 minors and symmetric matrices

Let $x,y,z \in \mathbb{R}^n$ with $n \geq 3$ be orthonormal vectors and let also $A,B$ be $n$-by-$n$ real symmetric matrices. From numerical experiments it seems that the following inequality is true: ...
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$\frac{a}{\sqrt{2a^2+3bc}}+\frac{b}{\sqrt{2b^2+3ca}}+\frac{c}{\sqrt{2c^2+3ab}} \le \sqrt{ab+bc+ca}$

Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3$.Prove that $$\frac{a}{\sqrt{2a^2+3bc}}+\frac{b}{\sqrt{2b^2+3ca}}+\frac{c}{\sqrt{2c^2+3ab}} \le \sqrt{ab+bc+ca}.$$ By C-S we need proof $$4(...
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How to prove this inequality $\frac{(\sum_i w_i x_i)}{(\sum_i w_i y_i^2)} \leq \sum_i w_i \frac{x_i}{y_i^2}$

So this is what I am trying to prove $\frac{(\sum_i w_i x_i)}{(\sum_i w_i y_i^2)} \leq \sum_i w_i \frac{x_i}{y_i^2}$, where $w_i \geq 1$, $x_i>0$, and $y_i>0$. I have no idea where to begin with,...
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Proving $m_4m_2\geq m_3^2+m_2^2$

I've been trying to prove: $m_4m_2\geq m_3^2+m_2^2$ How I proceeded: From the Cauchy-Schwarz inequality: $m_3=E[(x-\bar{x})^3]$ $m_2=E[(x-\bar{x})^2]$ $m_4=E[(x-\bar{x})^4]$ $[E(XY)]^2 \leq E[X^2] \...
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Why is this proof in diff eq by George Simmons for Schwarz's inequality seemingly arbitrary?

In the problem section, there is a problem asking the reader to prove $|(f,g)|\le ||f||\cdot||g||$ where $|(f,g)|$ is the inner product and $||f||$ is the norm by defining the function $F(\alpha) = ||...
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How is this an application of Cauchy-Schwarz?

I want to deduce the following inequality: $$ (a_1 + ... + a_n)^2 \leq C_n \sum_{i = 1}^n a_i^2 $$ Here each $a_i$ is a real number and $C_n$ is a positive constant depends only on $n$. My first ...
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Upper bounding by 1 a sum of certain probability fractions

Introduction Let $(X,Y)$ be a discrete joint probability distribution, letting $X$ model the rows of a matrix and $Y$ the columns. Overlaying this matrix sits a (discrete or continuous) probability ...
Joseph Johnston's user avatar
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Generalization of an inequaliy related to Cauchy-Schwarz

A classic Cauchy-Schwarz problem is to show that if $p_1, p_2 \cdots p_k$ are positive real numbers with $p_1 + p_2 \cdots +p_k=1$ then $\sum_{k=1}^n (\frac{1}{p_k} +p_k)^2 \geq n^3 +2n + \frac{1}{n}$ ...
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Prove $\sum\limits_{\mathrm{cyc}}\frac{a}{a^2+bc+4} \leq \frac{1}{2}$ for $a,b,c>0$ with $a+b+c=6$

Prove that for $a, b, c > 0$ where $a + b + c = 6$, the following inequality holds: $$ \frac{a}{a^2+bc+4} + \frac{b}{b^2+ca+4} + \frac{c}{c^2+ab+4} \leq \frac{1}{2}. $$ From the AM-GM inequality, ...
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To what extent does the "Cauchy-Schwarz inequality" hold for a normed vector space not inner product space?

It is known to all that for inner product space $H$, Cauchy-Schwarz inequality hold: $| \langle x,y \rangle | \leq \| x \| \cdot \| y \| \ \forall x,y\in H$, and for a normed space $X$ without inner ...
anyon's user avatar
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A nice 3 variable inhomogeneous asymmetric inequality by TATA box.

This inequality is proposed by TATA box. Prove that for ${\forall}a,b,c \geq 0$ such that $ab+bc+ca=2$, prove the following inequality. $$\sum_{cyc}a^2 + abc \geq \frac{3}{8}\sum_{cyc}a^3 b +2$$
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Maximize $P=\frac{a^2+7ab+b^2}{a+b+ab}+\frac{b^2+7bc+c^2}{b+c+cb}+\frac{c^2+7ac+a^2}{a+c+ac}$ if $a+b+c=3$.

Let $a,b,c>0$ such that $a+b+c=3.$ Find the maximum $$P=\frac{a^2+7ab+b^2}{a+b+ab}+\frac{b^2+7bc+c^2}{b+c+cb}+\frac{c^2+7ac+a^2}{a+c+ac}.$$ I thought Max= 9 at $a=b=c=1.$ My attempt is using ...
Anonymous's user avatar
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Question regarding the Lipschitz continuity of the determinant function on the set of all $2\times 2$ real matrices.

$\mathbf{The \ Problem \ is}:$ Let $A$ and $B$ be two $2\times 2$ real matrices in $(\operatorname{M}_2(\mathbb{R}),\|\cdot\|)$ where $\|\cdot\|$ is the Frobenius norm in $\operatorname{M}_2(\mathbb{R}...
Rabi Kumar Chakraborty's user avatar
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2 answers
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Help to prove $\frac{a}{(b^2+3)^2}+\frac{b}{(c^2+3)^2}+\frac{c}{(a^2+3)^2}\ge\frac{3}{16}$ for $a+b+c=3.$

If $a,b,c\ge 0: a+b+c=3$ then prove$$\frac{a}{(b^2+3)^2}+\frac{b}{(c^2+3)^2}+\frac{c}{(a^2+3)^2}\ge\frac{3}{16}.$$ I've tried to use Holder inequality as$$\sum_{cyc}a\cdot\sum_{cyc}\frac{a}{(b^2+3)^2}...
Anonymous's user avatar
2 votes
1 answer
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Prove that $a^2(b^2+4)+b^2(c^2+4)+c^2(a^2+4) \geq 15$

Let $$a, b, c$$ be real numbers with the property that $$ a+b+c=3 $$. Prove that $a^2(b^2+4)+b^2(c^2+4)+c^2(a^2+4) \geq 15$ Initially, I thought to use Cauchy-Schwarz Inequality and simplify. $(a^2+b^...
Mogovan Jonathan's user avatar
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Reversing AM-GM Inequality When We Have Bounded Variable

Suppose $x_1,x_2,\dots,x_n$ are positive numbers. Define $A = \frac{1}{n}\sum_{i=1}^n x_i$, $G = (\Pi_{i=1}^n x_i)^{1/n}$. The well-known AM-GM inequality tells us that $$ A \geq G $$ Now suppose we ...
EggTart's user avatar
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Show for all positive real $a, b, c$ that $\sum_{\text{cyc}} \frac{a^3}{b^2 - bc + c^2} \ge a + b + c$

Show for all positive reals $a, b, c$ that: $$\sum_{\text{cyc}}\frac{a^3}{b^2-bc+c^2} \ge a + b + c$$ I tried the following AM-GM: $$bc \le \frac{b^2+c^2}{2} \implies \sum_{\text{cyc}}\frac{a^3}{b^2-...
Martin Westin's user avatar
3 votes
3 answers
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Proving $5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.$

For all $a,b,c\ge 0$ prove that$$\color{black}{5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.}$$ I've tried to use AM-GM without success. Indeed, $$3\cdot RHS=2\sum_{cyc}3a\sqrt{5a^...
Sickness's user avatar
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1 answer
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Find proof for sum of cyclic square root inequality where $a+b+c=3.$

While solving one inequality which posted by @arqady-AOPS. (See unsolved inequality.) $$\color{black}{\sqrt{24a^2b+25}+\sqrt{24b^2c+25}+\sqrt{24c^2a+25}\le 21,}$$I arrived at a much simpler, but still ...
Anonymous's user avatar
2 votes
2 answers
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Proving $\sum_{cyc}\frac{a}{\sqrt{8a+bc}}\le \frac{\sqrt{a+b+c+abc}}{2}$ if $ab+bc+ca=3.$

Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=3.$ Prove that$$\color{black}{\frac{a}{\sqrt{8a+bc}}+\frac{b}{\sqrt{8b+ca}}+\frac{c}{\sqrt{8c+ab}}\le \frac{\sqrt{a+b+c+abc}}{2}.}$...
TATA box's user avatar
1 vote
2 answers
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Inequality about $\sum_{cyc}\dfrac{a^3}{(bc)^2+1}$ and $a+b+c=ab+bc+ca.$

Context. My friend sent me problem which hard for me to think of good approach. It's Let $a,b,c>0:a+b+c=ab+bc+ca.$ Prove that$$\frac{a^3}{(bc)^2+1}+\frac{b^3}{(ca)^2+1}+\frac{c^3}{(ab)^2+1}+3\ge \...
Dragon boy's user avatar
2 votes
1 answer
81 views

Proving $\sqrt{31a+b+c} +\sqrt{31b+a+c} +\sqrt{31c+b+a}\le 3\sqrt{3}\cdot\sqrt{a+b+c+8}$? when $a+b+c+abc=4.$

I came up with the inequality accidentally so there is no original proof so far. It would be great if you can give some useful help to prove it. Problem. Given non-negative real numbers $a,b,c$ ...
TATA box's user avatar
1 vote
2 answers
126 views

Find a nice proof for an elegant problem.

Question. For any $a,b,c>0$ then prove that$$\frac{\sqrt{5a^2+4bc}}{bc}+\frac{\sqrt{5b^2+4ca}}{ca}+\frac{\sqrt{5c^2+4ab}}{ab}\le \frac{9}{4}\left(\frac{a^2+b^2+c^2}{abc}+\frac{a+b+c}{ab+bc+ca}\...
Dragon boy's user avatar
6 votes
1 answer
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Fourier Uncertainty Principle

This is about the Fourier uncertainty principle, which is closely related to, but not the same as the Heisenberg uncertainty principle in physics. So no $\hbar$. This may be expressed with scaling ...
robert bristow-johnson's user avatar
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1 answer
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Inequality sum of square root $3a^3+bc$ when $a^2+b^2+c^2=3$.

Remark. I really like inequality problems, but I've found that I have severe difficulties with ones that are very elegant form. I can prove something simple like this $$\color{black}{\sqrt{3a+bc}+\...
Sickness's user avatar
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1 answer
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How to solve $\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}$? [closed]

Let $a,b,c\ge 0: ab+bc+ca=3.$ Prove that $$\color{black}{\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}.}$$ Equality holds at $a=b=c=1.$ ...
Dragon boy's user avatar
1 vote
1 answer
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Conditional Cauchy-Schwarz Inequality with Product Moment Matrices

Let $X, Y, Z$ denote r.v.s. A possible proof for the unconditional Cauchy-Schwarz Inequality $$E(XY)^2 \leq E(X^2)E(Y^2)$$ uses the product-moment matrix $U := E(\vec{X}\vec{X}^T)$, with $\vec{X} = \...
V. Elizabeth's user avatar
1 vote
0 answers
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Conditional Cauchy-Schwarz Inequality

Let $X, Y, Z$ be r.v.s. I wish to show that $E(XY|Z)^2 \leq E(X^2|Z)E(Y^2|Z)$ almost surely. I can easily adapt the proof of the unconditional Cauchy-Schwarz inequality, shown in this answer, and so ...
V. Elizabeth's user avatar
0 votes
1 answer
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Prove $\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}$

For all $a,b,c\ge 0: ab+bc+ca>0$ then prove $$\color{black}{\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}....
Dragon boy's user avatar
0 votes
3 answers
188 views

Prove that $\sum\limits_{\mathrm{cyc}}\sqrt{\frac{xy}{z}}+6\sqrt{xyz}\ge \sum\limits_{\mathrm{cyc}} \sqrt{3x^3+6xyz}$

Context. I saw a problem on facebook.It's Let positive real numbers $x,y,z$ such that $x+y+z=x^2+y^2+z^2$. Prove that$$\sqrt{\frac{xy}{z}}+\sqrt{\frac{yz}{x}}+\sqrt{\frac{zx}{y}}+6\sqrt{xyz}\ge \sqrt{...
Dragon boy's user avatar

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