Mathematics

Questions tagged [schur-inequality]

For questions related to Schur's inequality. Schur's inequality is a classical inequality that relates three non-negative real numbers.

In mathematics, Schur's inequality, named after Issai Schur, establishes that $$\displaystyle \sum _{cyc}x^{t}(x-y)(x-z)=x^{t}(x-y)(x-z)+y^{t}(y-z)(y-x)+z^{t}(z-x)(z-y)\geq 0 \ \forall \ x,y,z,t\in\Bbb R^{0+}$$ with equality iff $x = y = z$ or two of them are equal and the other is $0$. When $t$ is an even positive integer, the inequality holds for all $x,y,z\in\Bbb R$.

When $t=1$, the following well-known special case can be derived:

$$\displaystyle x^{3}+y^{3}+z^{3}+3xyz\geq xy(x+y)+xz(x+z)+yz(y+z)$$

For more on this check this link.

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Prove that $\frac{m}{\sqrt{7m+2}} + \frac{n}{\sqrt{7n+2}} + \frac{p}{\sqrt{7p+2}} \ge 1.$

Let $m,n,p \ge 0$ such that $m+n+p = mn+np+pm > 0$ . Prove that: $$\frac{m}{\sqrt{7m+2}} + \frac{n}{\sqrt{7n+2}} + \frac{p}{\sqrt{7p+2}} \ge 1.$$ The equality occurs when $(m,n,p)=(1,1,1)$ or $(m,n,p)=(2,2,0)$ I tried to use Holder inequality:…
ttp266
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Prove the inequality $\sum_{cyc}\frac{a^2}{\sqrt{(b+c)(b^3+c^3)}}\geq\frac 32$ where $a,b,c$ are positive reals.

Let $a,b,c$ be positive real numbers, prove that $$\sum_{cyc}\frac{a^2}{\sqrt{(b+c)(b^3+c^3)}}\geq\frac 32.$$ The problem is from an inequality handout. Here is my attempt to solve the problem: I first rewrote the inequality as…
Oshawott
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How to prove $2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$?

Question. Let $a,b,c>0: abc=a+b+c+2.$ Prove that $$2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$$ I am looking for a nice proof by hand, for which there is a…
user1071608
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If $a,b,c\ge0$ and $a^2+b^2+c^2=2$, show that $\frac{a^2}{a^2+2bc+1}+\frac{b^2}{b^2+2ca+1}+\frac{c^2}{c^2+2ab+1}\le1$.

This is a high school contest exercise question. If $a,b,c\ge0$ and $a^2+b^2+c^2=2$ , show that $\frac{a^2}{a^2+2bc+1}+\frac{b^2}{b^2+2ca+1}+\frac{c^2}{c^2+2ab+1}\le1$ . It is straightforward to show that the quantity is $\ge \frac{2}{3}$ , where…
Zack Fisher
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Proving non-negativity of Schur's interpolating function provided it holds for $t=1$ and $t\to∞$?

I learnt that the function $x^{2t}(x-y)(x-z)+y^{2t}(y-z)(y-x)+z^{2t}(z-x)(z-y)$ is non-negative for all $x,y,z\in\mathbb{R}$ . I wonder what we can say about the case when the power is odd. Let…
hbghlyj
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How to prove $ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6$?

Question. Prove $$ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6,$$ when $a,b,c\ge 0: ab+bc+ca+abc=4.$ My idea: I've tried to use AM-GM as $$\bullet \sum \sqrt{ab}\ge 2\sum \frac{ab}{a+b}=2(ab+bc+ca)\sum…
user979185
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For positive $a,b,c$ with $abc=1$, how to prove $2(a^3+b^3+c^3)+3\ge 3(ab^2+bc^2+ca^2)$?

This is a question derived from another exercise problem for high school math contests. I hope this new question is also interesting. If $a>0$ , $b>0$ , and $c>0$ with $abc=1$ , prove that $$2(a^3+b^3+c^3)+3\ge 3(ab^2+bc^2+ca^2).$$ However, I only…
Zack Fisher
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How prove this $a+b+c+3\sqrt[3]{abc}\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})$

let $a,b,c>0$, show that $$a+b+c+3\sqrt[3]{abc}\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})$$ I know this $$a+b+c\ge 3\sqrt[3]{abc}$$ so $$\Longleftrightarrow 6\sqrt[3]{abc}\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})$$ But this maybe not true?
math110
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Schur's inequality for even exponent and $x,y,z\in\mathbb{R}$

In Schur's inequality When $t=0,2,4\dots$ is an even integer, Wlog set $z=1$ , $$x^t(x-y)(x-1)+y^t(y-1)(y-x)+(1-x)(1-y)\ge0$$ how to prove the above inequality is true for all $x,y\in\mathbb{R}$ ? (The proof in the link only applies to non-negative…
hbghlyj
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Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.

I am not sure if the inequality is true. My first attempt was to try AM-GM inequality in clever ways. I also tried Schur's inequality which gives $$ x^6+y^6+z^6 + 6x^2y^2z^2 \geq (x^2+y^2+z^2)(x^2y^2+y^2z^2+z^2x^2) $$ but it is not true that $$…
Razor56
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If $x + y + z = 1$, what is $\min(xyz)$ where $x$, $y$ and $z$ are positive numbers?

$x,y,z$ are positive numbers and given $x + y + z = 1$ . I was initially required to prove that $xy+yz+xz - 3xyz \leq \frac{1}{4}$ . I then manipulated such that I was required to prove that $xyz \geq \frac{1}{36}$ (using $x^3 +y^3 +z^2 -3xyz =…
maffhard
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Prove $\sum_{cyc} \sqrt{\frac{a}{2bc+a+6}}\ge 1, \forall a,b,c \ge 0:a+b+c=ab+bc+ca>0.$

I'm looking for some ideas to solve the following inequality. Problem. For any non-negative real numbers $a,b,c$ with $a+b+c=ab+bc+ca>0,$ prove that $$\sqrt{\frac{a}{2bc+a+6}}+\sqrt{\frac{b}{2ca+b+6}}+\sqrt{\frac{c}{2ab+c+6}}\ge 1.$$ Source: Tran…
30 Anh Ti 711
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Proving the inequality $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \leq \sqrt[3]{3(3 + a + b + c + ab + bc + ca)}$

Let $a,b,c$ be positive real numbers such that $abc = 1$ . Prove that $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \leq \sqrt[3]{3(3 + a + b + c + ab + bc + ca)}$ Attempt : We are given that $a,b,c$ are positive real numbers such that $abc = 1$ , and we…
Amlan Gayan
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Knowing $x,y,z\ge0$ prove $x^2+xy^2+xyz^2\ge4xyz-4$

Knowing $x,y,z\ge0$ prove $x^2+xy^2+xyz^2\ge4xyz-4$ I thought that I should rearrange this inequality to be somewhat of the form of Schur's Inequality and WLOG I assumed $x\ge y\ge z$ . Trying this way it did bring me out to nowhere so I tried to…
user1291864
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Trigonometric version of Schur's inequality

Let $\alpha+\beta+\gamma=\frac{\pi}{2}$ and $t>0$ , prove that $$\cos^t{(\alpha)}\sin{(\alpha-\beta)}\sin{(\alpha-\gamma)}+\cos^t{(\beta)}\sin{(\beta-\alpha)}\sin{(\beta-\gamma)}+\cos^t{(\gamma)}\sin{(\gamma-\alpha)}\sin{(\gamma-\beta)}\ge 0.$$ This…
Emmanuel José García
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