Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [muirhead-inequality]

Inequality proof by using the Muirhead inequality.

1
vote
1answer
74 views

show this inequality with $xy+yz+zx=3$

let $x,y,z>0$ and such $xy+yz+zx=3$,show that $$\dfrac{x}{x^3+y^2+1}+\dfrac{y}{y^3+z^2+1}+\dfrac{z}{z^3+x^2+1}\le 1$$ To prove this inequality,I want use following Cauchy-Schwarz inequality $$(x^3+...
0
votes
3answers
92 views

How to find $f(X)$ such that $\sum\limits_{cyc}a^2-f(X)[abc-(1-a)(1-b)(1-c)]\geqq\frac{3}{4}X^2$ for $abc=(X- a)(X- b)(X- c),0\leqq a,\,b,\,c\leqq X$?

We have $\sum\limits_{cyc}\,a^{\,2}\geqq \frac{3}{4}\,X^{\,2}\tag{HaiDangel29}$ with $abc= (\,X- a\,)(\,X- b\,)(\,X- c\,),\,0\leqq a,\,b,\,c\leqq X$. Here is a hint to get you started from above. For $...
0
votes
2answers
50 views

A three variable inequality doubt , can I consider the three variables into just one variable , and show the inequality.

I was trying to prove the inequality : for a,b,c positive real numbers where $abc=1$ prove $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}}\leq 1 . $$ It is easy ...
-2
votes
1answer
109 views

Inequality with $a+b+c=1$, a,b,c positive numbers

If a,b,c are real positive numbers, such as $ a+b+c=1$ prove that $ \left( ab+bc+ca\right) \left( 1+3abc\right) \geq 10abc $. Thank you! First try:$$\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)(...
0
votes
2answers
81 views

Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

Let $a, b, c>0$ s.t. $abc (a+b+c)=3$. Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+ \frac{1}{c^2+a^2+1}\leq 1$. I have no idea how to start.
-1
votes
1answer
109 views

Show that $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq 256$. [duplicate]

Let $a, b, c, d\geq 0$ s.t. $a+b+c+d=4$. Show that $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq 256$. I don't know how can I deconditioned the inequality.
0
votes
0answers
51 views

Inequality using lengths of the edges of a triangle

If $a,b,c $ are the lengths of the edges of a triangle, show that: $$\frac {6 (a^2+b^2+c^2)}{a+b+c}\geq \frac {(a+b)^2}{b+c}+\frac {(b+c)^2}{a+c}+\frac {(c+a)^2}{a+b} $$ I have no idea how to start.
7
votes
2answers
156 views

Proving $\sum_{\text{cyc}} \frac{a}{b^2+c^2+d^2} \geq \frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$

Prove that $$\frac{a}{b^2+c^2+d^2}+\frac{b}{a^2+c^2+d^2}+\frac{c}{a^2+b^2+d^2}+\frac{d}{a^2+b^2+c^2}≥\frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$$ What I tried, was to say that $a^2+b^2+c^...
1
vote
2answers
165 views

Show that $\frac {1}{3+x^2+y^2} + \frac {1}{3+y^2+z^2} +\frac{1} {3+x^2+z^2}\leq \frac {3}{5} . $

Let $x, y, z>0$ s.t. $x+y+z=3$. Show that $$\frac {1}{3+x^2+y^2} + \frac {1}{3+y^2+z^2} +\frac{1 } {3+x^2+z^2}\leq \frac {3}{5}\ . $$ My idea: $$3 + x^2 + y^2 \geq 1 + 2x+ 2y=7-2z $$ I notice ...
1
vote
1answer
127 views

Prove: $x^4 + y^4 + z^4 - 4(x^3 + y^3 + z^3) + 5(x^2 + y^2 + z^2) \leqslant 4 $

Prove that $$x^4 + y^4 + z^4 - 4(x^3 + y^3 + z^3) + 5(x^2 + y^2 + z^2) \leqslant 4$$ for all $x, y, z \geqslant 0$ satisfying $x + y + z = 2$. When does equality occur? Here is my aproach: For each ...
0
votes
2answers
64 views

If $a$, $b$ and $c$ are positive then $\sum\limits_{cyc} \frac{a^{2}}{b^{2}+c^{2}+bc}\geq 1$

If $a$, $b$ and $c$ are positive then $\sum\limits_{cyc} \frac{a^{2}}{b^{2}+c^{2}+bc}\geq 1$. I tried to solve this problem by C-S. But I can't sovle it. Things I have done so far: $\sum\limits_{cyc}...
1
vote
1answer
105 views

Inequality from AMM problems section

This is Problem 12024 of AMM. It asks to show that if $x,y,z$ are positive reals, and $xyz=1$, then $(x^{10}+y^{10}+z^{10})^{2}\geq 3(x^{13}+y^{13}+z^{13})$. I could show it for the particular case ...
0
votes
1answer
74 views

Prove $\sum\limits_{cyc}\frac{1}{x^{2017}+ x^{2015}+ 1} \geq 1$ with $x,\,y,\,z>0,\,xyz= 1$ [closed]

Prove $$\sum\limits_{cyc}\frac{1}{x^{2017}+ x^{2015}+ 1} \geq 1$$ with $x,\,y,\,z>0,\,xyz= 1$ I try to use Jensen inequality, then: $$f\left ( x \right )+ f\left ( y \right )+ f\left ( z \right )\...
0
votes
1answer
62 views

An inequality with a condition

If $x,y,z>0$ and $\sum_{cyc}^{} x^2 (1-x) \ge 0 $, I have to prove that: $\sum_{cyc}^{} x^2 y^2 (1-xy) \ge 0 $ Then, if $\sum_{cyc}^{} x=3,$ then I have to prove $\sqrt [8] \frac {\sum_{cyc}^{} ...
-2
votes
1answer
90 views

Proof of Muirhead's inequality

I have seen Muirhead's inequality being stated and used in many, many places, but I haven't ever seen a proof. Does someone have a reference or simple proof for this result?
1
vote
1answer
30 views

Muirhead's inequality — equality occur

An equality occurs in muirhead's inequality when all variables are the same. But is it possible for the equality to occur when not all variables are the same? (also all variables are non-zero) If so, ...
2
votes
1answer
76 views

Inequality with $abc=1$,

Given 3 positive numbers $a, b, c$ satisfying $abc= 1$ Prove: $$\frac{1}{a^{5}+ b^{5}+ c^{2}}+ \frac{1}{b^{5}+ c^{5}+ a^{2}}+ \frac{1}{c^{5}+ a^{5}+ b^{2}}\leq 1$$ My opinion: Let: $x= \frac{a}{b}, ...
0
votes
1answer
94 views

Inequality involving cyclic sums (Muirhead? Schur? Something else?)

Let's define: $$M[a, b, c]=\sum_{cyc}x^a y^b z^c $$ I need to prove that for all positive and real $x$, $y$ and $z$: $$M[6, 3, 0] + M[3, 3, 3] \ge M[5, 2, 2] + M[4, 4, 1]$$ From Muirhead's ...
1
vote
2answers
102 views

Prove this Hard inequality to CS or AM-GM?

Let $x,y,z\ge 0$ such $x+y+z=1$, show that $$\sqrt{\dfrac{x(x+1)}{1-x}}+\sqrt{\dfrac{y(y+1)}{1-y}}+\sqrt{\dfrac{z(z+1)}{1-z}}\ge\sqrt{6}$$ This inequality is creat by wangyongxi,and when $x=y=z=\...
0
votes
2answers
75 views

Inequality with powers of 2

If $m,n$ and $p $ are positive integers show that : $$\frac{2^{mn}}{p}+\frac{2^{np}}{m}+\frac{2^{pm}}{n}\geq 2(m+n+p).$$ I tried using Bernoulli inequality and then Hölder's, but I cannot prove this ...
1
vote
1answer
68 views

for $x,y,z\ge 0$, $x+y+z=2$, prove $\frac{x}{1+y^2}+\frac{y}{1+z^2}+\frac{z}{1+x^2}\ge\frac{18}{13}$

$x,y,z\ge 0$, $x+y+z=2$, prove that $$ \frac{x}{1+y^2}+\frac{y}{1+z^2}+\frac{z}{1+x^2}\ge\frac{18}{13} $$ I had no idea besides brute-force: multiply everything, dehomogenize and use Muirhead. But ...
1
vote
1answer
53 views

Find the largest real $k$ such that: $(a+b+c)^2(ab+bc+ca)\geq k(a^2b^2+b^2c^2+c^2a^2)$

Find the largest real $k$ such that for every non negative real numbers $a,b,c$ : $$(a+b+c)^2(ab+bc+ca)\geq k(a^2b^2+b^2c^2+c^2a^2)$$ I expanded the LHS but the problem got more complicated and no ...
-4
votes
2answers
50 views

from cyclic to symetric sums

Can you explain to me how to pass from a cyclic to a symmetrical sum through weighed AM-GM (so I can use Muirhead's inequality) ? In particular by applying this to this inequality $$\sum_ {cyc} ^ {} a ...
2
votes
2answers
156 views

If $ab+bc+ca+abc=4$, then $\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq 3\leq a+b+c$

Let $a,b,c$ be positive reals such that $ab+bc+ca+abc=4$. Then prove $\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq 3\leq a+b+c $ So high guys im a high schooler trying to solve this inequality. I did a few ...
2
votes
2answers
413 views

$(xy+yz+zx)(1+6xyz) \geq 11xyz$

I've came across this inequality: Let $x>$, $y>0$ and $z > 0$ with $x+y+z=1$. Prove that $$(xy+yz+zx)(1+6xyz) \geq 11xyz.$$ I don't know where to take it from, I've tried means ...
2
votes
1answer
70 views

An inequality in four variables [closed]

Let $a$, $b$, $c$ and $d$ be positive real numbers. Prove that: $$\frac{ab+bc+ca}{a^3+b^3+c^3}+\frac{ab+bd+da}{a^3+b^3+d^3}+\frac{ac+cd+da}{a^3+c^3+d^3}+\frac{bc+cd+db}{b^3+c^3+d^3}\le\min\left [\...
0
votes
1answer
89 views

Inequality : $\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}$

Let $\{a, b, c\} \subset \mathbb{R}^+$. Prove that $$\displaystyle\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\displaystyle\sum_{cyc}\frac{a^3+b^3}{c^2+ab}$$ My work : WLOG, let $\;...
1
vote
1answer
57 views

is there a generalization of Muirhead theorem for negatives reals?

Is there a generalization of Muirhead theorem for negatives reals? Because the original theorem is for only non-negative real numbers.
1
vote
3answers
133 views

Prove the inequality $\sum_{cyc}\frac{a}{1+\left(b+c\right)^2}\le \frac{3\left(a^2+b^2+c^2\right)}{a^2+b^2+c^2+12abc}$

Let $a>0$, $b>0$ and $c>0$ such that $a+b+c=3$. Prove the inequality $$\frac{a}{1+\left(b+c\right)^2}+\frac{b}{1+\left(c+a\right)^2}+\frac{c}{1+\left(a+b\right)^2}\le \frac{3\left(a^2+b^2+...
0
votes
1answer
95 views

Wild inequality

I'm trying to solve this elementary inequality, but so far no clue. Can anybody solve it? I tried am/gm inequality on both side, but yield the same result hence no inequality. Also I tried other ...
5
votes
1answer
192 views

Proving $a^{-2}bcd+b^{-2}cda+c^{-2}dab+d^{-2}abc\geqslant a+b+c+d$

If $a,b,c,d$ are positive real numbers, then prove that $$\frac{bcd}{a^2}+\frac{cda}{b^2}+\frac{dab}{c^2}+\frac{abc}{d^2}\geqslant a+b+c+d$$ Attempt: $$\frac{bcd}{a^2}+\frac{cda}{b^2}+\frac{dab}{c^2}+...
0
votes
2answers
32 views

$a_1, … . a_{n+1} > 0$, then $(\prod_{i=1}^{n+1}a_i)(\sum_{i=1}^{n+1} 1/a_i^n) \ge \sum_{i=1}^{n+1}a_i$?

If $a_1, ... . a_{n+1} > 0$, then how to prove that $(\prod_{i=1}^{n+1}a_i)(\sum_{i=1}^{n+1} 1/a_i^n) \ge \sum_{i=1}^{n+1}a_i$ ? I couldn't get anywhere, I don't know where to start . Please help ....
0
votes
3answers
185 views

The isoperimetric inequality for triangles

If a triangle has perimeter $p$ and area $T$, how to prove that $p^2 \ge 12\sqrt{3} T$ and $T \le \frac{\sqrt{3}}{4}\cdot (abc)^{\frac{2}{3}}$? I try some AM-GM inequality but I am stuck on it.
5
votes
2answers
134 views

Prove that $\frac1{a+b+1}+\frac1{b+c+1}+\frac1{c+a+1}\le1$

If $abc=1$ then $$\frac1{a+b+1}+\frac1{b+c+1}+\frac1{c+a+1}\le1$$ I have tried AM-GM and C-S and can't seem to find a solution. What is the best way to prove it?
0
votes
2answers
76 views

Non - symetric inequality in 3 variables

How can we prove the following inequality for $a,b,c>0$: $$\dfrac{ab}{a^2+b^2}+\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}\leq \dfrac{a^2}{a^2+b^2}+\dfrac{b^2}{b^2+c^2}+\dfrac{c^2}{c^2+a^2} \ ?$$
3
votes
1answer
63 views

How to prove this $\sum\limits_{cyc}a\sqrt{b^2+(n^2-1)c^2}\le (a+b+c)^2+(n-3)(ab+bc+ca)$

Let $a,b,c>0$ and $n$ be postive integer,show that $$a\sqrt{b^2+(n^2-1)c^2}+b\sqrt{c^2+(n^2-1)a^2}+c\sqrt{a^2+(n^2-1)b^2}\le (a+b+c)^2+(n-3)(ab+bc+ac)$$ For $n=1$ it suffices to show that $$ab+bc+...
3
votes
2answers
47 views

How can I prove the this inequality?

$$ a^6+b^6+c^6+3a^2 b^2 c^2 \geq 2(a^3 b^3 + b^3 c^3 +c^3 a^3)$$ $\forall a,b,c \in \mathbb{R}$ Can this be done with just weighted AM-GM?
7
votes
4answers
183 views

Prove this inequality: $\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3$

Let $a$,$b$,$c$ be positive real numbers such that $abc=1$. Prove that $$\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3$$ I tried various methods. ...
3
votes
2answers
81 views

Prove the inequality $\frac{a^2+b^2}{c^2+a+b}+\frac{b^2+c^2}{a^2+b+c}+\frac{c^2+a^2}{b^2+c+a} \ge 2$

For every $a,b,c>0$ such that $abc=1$ prove the inequality $$\frac{a^2+b^2}{c^2+a+b}+\frac{b^2+c^2}{a^2+b+c}+\frac{c^2+a^2}{b^2+c+a} \ge 2$$ My work so far: $abc=1 \Rightarrow \frac 13 (a+b+c)\...
13
votes
2answers
360 views

How to prove that $\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$?

Let $a,b,c>0: (a+b)(b+c)(c+a)=ab+bc+ca$. How to prove that $$\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$$
-4
votes
1answer
275 views

How to prove or disprove this algebraic inequality?

How to prove or disprove the inequality (from math folklore) $$\sqrt{22(a^2+b^2+c^2)+5(ab+ac+bc)}\geq\sqrt{4a^2+ab+4b^2}+\sqrt{4b^2+bc+4c^2}+\sqrt{4c^2+ca+4a^2}$$ for nonnegative $a, b,$ and $c$? ...
5
votes
1answer
175 views

Seeking concise proof: $\frac18(a^2+b^2)(b^2+c^2)(c^2+a^2)\ge\frac1{27}(ab+bc+ca)^3$, where $a$, $b$, $c$ are positive numbers

I was just encountered an inequality in AoPs, Here it is: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=4569&view=next, that is: If $a$, $b$ and $c$ are positive ...
3
votes
3answers
113 views

Proving Cauchy inequality involving four expression

Show that $$(a^2 + b^2 + c^2) (a^2b^2 +b^2c^2 +c^2a^2) \geq (a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)$$ i should prove this inequality by making it a Cauchy form inequality(as teacher stated). my ...
0
votes
2answers
108 views

Let $a,b,c>0$ such $ \frac{b+c}{a}+ \frac{c+a}{b}+ \frac{a+b}{c} = 2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac} \right)$

Let $a$, $b$ and $c$ be positive numbers such that $$ \dfrac{b+c}{a}+ \dfrac{c+a}{b}+ \dfrac{a+b}{c} = 2\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac} \right).$$ Prove that $$a^2+b^2+c^2+3\ge 2(ab+...
5
votes
1answer
167 views

Proving AM-GM via $n \cdot (a_1^n + a_2^n + \dots + a_n^n) \ge (a_1^{n-1} + a_2^{n-1} + \dots + a_n^{n-1}) \cdot (a_1 + a_2 + \dots + a_n)$

I want to prove the arithmetic–geometric mean inequality. To prove that, I need the following inequality: Suppose that $n$ is an integer which is greater than or equal to $1$ and $a_1, a_2, \dots, ...
1
vote
5answers
3k views

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

This is supposed to be an application of AM-GM inequality. if $abc=1$, then the following holds true: $a^2+b^2+c^2\ge a+b+c$ First of all, $a^2+b^2+c^2\ge 3$ by a direct application of AM-GM....