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Questions tagged [a.m.-g.m.-inequality]

For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.

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I want help knowing if my solution is correct

If $a, b, x, y$ are positive rational numbers such that $\frac 1x + \frac 1y = 1$ then prove that $\frac {a^{x}}{x}+ \frac {b^{y}}{y}$ $\ge ab$ This question is from Problems Plus in IIT Mathematics. ...
Phoenix's user avatar
  • 15
3 votes
2 answers
60 views

Given $A, B, C, D$ in $Oxyz$ space, find $M \in CD$ such that $MA + MB$ is smallest. Why can't I use AM-GM to solve this?

In the $Oxyz$ space, consider four points $A(-1, 1, 6),$ $B(-3,-2,-4),$ $C(1,2,-1),$ $D(2,-2,0).$ Find $M \in CD$ such that $△MAB$ has the smallest perimeter. As $AB$ is constant, the task is ...
ten_to_tenth's user avatar
  • 1,318
-2 votes
0 answers
49 views

Inequality $\forall a,b\in\mathbb{R}_{*}^{+}~~\text{then}~~ \dfrac{1}{a}+\dfrac{1}{b}+ab\geq 3$ [closed]

$\forall a,b\in\mathbb{R}_{*}^{+}\text{ then }\frac{1}{a}+\frac 1b+ab\geq 3$. Now this inequality: $a+\frac {1}{a}\geq 2$; $a+b\geq 2\sqrt{ab}$. But I can't use it!
Ellen Ellen's user avatar
  • 2,335
3 votes
5 answers
75 views

Minimizing $\left(\frac{c}{a} + \frac{c}{b}\right)^2$, where $c$ is the hypotenuse of a right triangle with legs $a$ and $b$

This question is regarding the following problem Given that $a, b, c$ are the sides of the $\triangle ABC$ which is right angled at $C$, then what is the minimum value of the following expression? $$\...
koiboi's user avatar
  • 766
1 vote
2 answers
102 views

Prove $(x_1 +x_2 +...+x_i +...+x_n)^2 \ge n(x_1x_2 +x_2x_3 +...+x_ix_{i+1} +...+x_nx_1)$

Determine all positive integers $n \ge 2$ such that for all POSITIVE real number $x_1, x_2,..., x_n$ the following inequality holds: $$(x_1 +x_2 +...+x_i +...+x_n)^2 \ge n(x_1x_2 +x_2x_3 +...+x_ix_{i+...
Kokos's user avatar
  • 418
4 votes
1 answer
104 views

prove that $\left( \sum_{i=1}^{n} a_i \right) \left( \sum_{i=1}^{n} a_i^{n-1} \right) \leq n \prod_{i=1}^{n} a_i + (n-1) \sum_{i=1}^{n} a_i^n.$

question:Let $a_1, a_2, \ldots, a_n$ be nonnegative real numbers. Prove that $$\left( \sum_{i=1}^{n} a_i \right) \left( \sum_{i=1}^{n} a_i^{n-1} \right) \leq n \prod_{i=1}^{n} a_i + (n-1) \sum_{i=1}^{...
user avatar
1 vote
1 answer
49 views

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 ...
FabDust's user avatar
  • 195
1 vote
2 answers
105 views

Knowing $a,b,c>0$ and $abc\le1$, prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge1+\frac{6}{a+b+c}$

Knowing $a,b,c>0$ and $abc\le1$, prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge1+\frac{6}{a+b+c}$ I tried to AM-GM the inequality, which gave this: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3\...
FabDust's user avatar
  • 195
2 votes
1 answer
95 views

A symmetric inequality involving product of three variables

Let $a, b,c \ge 0, ab + bc + ca + abc = 4$. Find the minimum of $S = \sqrt{a}+\sqrt{b}+\sqrt{c}$. My guess is that $S$ attends its minimum at $b = c = 2, a = 0$ and the other permutation of $(2, 2, 0)$...
anonimo's user avatar
  • 499
0 votes
1 answer
43 views

Hard Inequalities between harmonic and geometric means

Let's have $0 < \lambda_1 \leq \lambda_2 \leq ... \leq \lambda_n$ $n$ positive numbers How to show that : $$\left( \frac{n}{\sum_{k = 1}^n \frac{1}{\lambda_1 + \lambda_k}} - \lambda_1 \right)^n \...
badinmaths's user avatar
7 votes
2 answers
170 views

Minimizing $\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$ for real $a,b,c$ such that $a+b+c=-1$ and $abc\le -3$

Let $a,b,c$ be real number such that $a+b+c=-1$ and $abc\le -3$. Find the minimum value of $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ So, earlier this day I had a competition (it is ...
Kokos's user avatar
  • 418
1 vote
0 answers
80 views

prove that $\frac{x^x}{|x-y|} + \frac{y^y}{|y-z|} + \frac{z^z}{|z-x|} \geqslant \frac{7}{2}.$ where $x \neq y \neq z$

Question Statement: I came across the following intriguing inequality problem involving positive real numbers $x$, $y$, and $z$, where $x \neq y \neq z$: $$\frac{x^x}{|x-y|} + \frac{y^y}{|y-z|} + \...
Martin.s's user avatar
2 votes
2 answers
362 views

If $x, y, z$ are real numbers with $xy+yz+zx\geq0,$ then $\left(x^5 + y^5 + z^5\right)^2 \geq3xyz\left(x^7 +y^7 + z^7\right).$

Question: If $x, y, z$ are real numbers with $xy+yz+zx\geq0,$ then $$\left(x^5 + y^5 + z^5\right)^2 \geq3xyz\left(x^7 +y^7 + z^7\right).$$ Attempt: Let's consider the expression $$\left(x^5 + y^5 + z^...
Martin.s's user avatar
3 votes
2 answers
147 views

$a, b, c \in \mathbb{R}^+: abc = 1.$ Show that $\left(\dfrac{1}{a^3+1}+\dfrac{1}{b^3+1}+\dfrac{1}{c^3+1}\right)\le\dfrac{(a+b+c)^3}{18}$

$a, b, c \in \mathbb{R}^+: abc = 1.$ Show that $\left(\dfrac{1}{a^3+1}+\dfrac{1}{b^3+1}+\dfrac{1}{c^3+1}\right)\le\dfrac{(a+b+c)^3}{18}$ I ran into this inequality problem online and have been ...
ten_to_tenth's user avatar
  • 1,318
2 votes
1 answer
57 views

Prove that $2(a+b+c)+\frac{a^3}{bc}+\frac{b^3}{ac}+\frac{c^3}{ab}\le abc$ if $a,b,c\gt0$ and $a^2+b^2+c^2=abc$

I started solving the problem by using the AM-GM inequality to show that $\frac{a^3}{bc}+\frac{b^3}{ac}+\frac{c^3}{ab}\ge a+b+c$ Multiplied both sides by 2 and added $\frac{a^3}{bc}$, $\frac{b^3}{ac}$,...
Anna Németh's user avatar
-1 votes
1 answer
64 views

Prove that for $x,y,z$ positive integers the form of $\frac{x^2+y^2+z^2}{xy+yz+zx}$ can't be equal to 3. [duplicate]

Because it is positive integers so I can multiply both side by $(xy+yz+zx)$ I have tried to use completing square, like below $(3x-3y)^2+(3y-3z)^2+(3z-3x)^2=12x^2+12y^2+12z^2$ But in above form, it ...
Lim Zhao Sen's user avatar
1 vote
1 answer
148 views

Minimize $\sqrt{9x²+144}-x$ without calculus [closed]

How to find mimumum value of $\sqrt{9x²+144}-x$ for $x>0$ without calculus. I am spesificly looking for solution with QM-AM or CS inequality (If it is solvable with these, I tried to apply them but ...
Briston's user avatar
  • 192
3 votes
0 answers
128 views

Show that among the numbers $x_{1}, x_{2} ,...,x_n $there are two whose product is at most equal to $- 1 / n$. [duplicate]

the question Let $n \in \mathbb{N}, n \geq 3,$ and $x_1, x_2,\dots,x_n \in \mathbb{R}$ such that $x_1+x_2+\dots+x_n = 0$ and $x_1^2+x_2^2+\dots+x_n^2 = 1$. Show that among the numbers $x_1, x_2,\dots,...
IONELA BUCIU's user avatar
  • 1,125
3 votes
1 answer
141 views

Proving an inequality: $\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)} \ge \frac{1+abc}{2}$

The question is as follows: Let $a, b, c \ge 0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)} \ge \frac{1+abc}{2}$$ I started as follows: (1) Proving for one variable- $a$ $$\frac{1+a^2}{...
xoxo's user avatar
  • 119
5 votes
2 answers
120 views

Given $x_1^3+x_2^3+...+x_9^3=0$. Find the maximum value of $S=x_1+x_2+...+ x_9$.

Given 9 real numbers $x_1, x_2, ... , x_9\in [-1,1]$ such that $x_1^3+x_2^3+...+x_9^3=0$. Find the maximum value of $S=x_1+x_2+...+ x_9$. I have tried ordering the numbers from smallest to largest and ...
Nguyen Huy's user avatar
2 votes
2 answers
68 views

Finding Maximum Value using AM-GM Inequality

Let us have a set of natural numbers $S=\{x_1,x_2,...,x_n\}$ where $n≥4$, $n$ is even, such that all $(x_i\in S)≥0$ and $\sum_{i=1}^nx_i=1$.Find the maximum value of $\sum_{i=1}^{n-1}(x_i*x_{i+1})$.My ...
20DPCO190 Amanul Haque's user avatar
0 votes
0 answers
55 views

Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$ What's the minimum value of $(a^2+b^2)$? [duplicate]

Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$ What's the minimum value of $(a^2+b^2)$? Ok so here I tried to divide by $x^2$ and get this : $$x^2+\frac{1}{x^2}+a\left(...
FabDust's user avatar
  • 195
-2 votes
2 answers
60 views

Show that for every numbers a,b,c real positive we have $\sum a\frac{b^2+c^2}{b+c}\geq ab+bc+ca$ [closed]

Show that for every numbers $a,b,c$ real positive we have $$\sum a\frac{b^2+c^2}{b+c}\geq ab+bc+ca$$ That $a$ in front is really annoying so I tried: $abc$ and we get that $$\sum \frac{b^2+c^2}{bc(b+c)...
IONELA BUCIU's user avatar
  • 1,125
0 votes
1 answer
46 views

$\log_a 10 + \log_b 10 +\ log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$

Prove the following inequality for $a,b,c, \in (1,\infty)$ , such that $abc = 10$ $$\log_a 10 + \log_b 10 + \log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$$ I will transform logarithms to ...
Unknowduck's user avatar
1 vote
1 answer
93 views

$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$

Prove the following inequality $$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$$ where $x,y,z \ge 0$ and $x+y+z=3$ I don't really know how to approach such an ...
Unknowduck's user avatar
0 votes
2 answers
82 views

Prove $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$, with $x, y, a, b \in \mathbb{R}$ and $a, b > 0$

Prove $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$, with $x, y, a, b \in \mathbb{R}$ and $a, b > 0$ Proof: $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$ $\iff (bx^2 + ...
ten_to_tenth's user avatar
  • 1,318
3 votes
2 answers
79 views

If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$.

If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$. My Attempt I tried by putting $t=x+y$ $\Rightarrow 4x(t-x)=2^t$. On differentiation we have $4t-8x=\frac{dt}{dx}(...
Maverick's user avatar
  • 9,481
2 votes
2 answers
63 views

Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$

the question Let $a,b,c$ be positive numbers such that $a+b+c=1$. Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$. the idea After i put some values to $a$, $b$, ...
IONELA BUCIU's user avatar
  • 1,125
4 votes
7 answers
230 views

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\...
FabDust's user avatar
  • 195
2 votes
3 answers
107 views

Regarding the specifity of the A.M.-G.M. inequality in finding maximum and minimum in Number Theory

Today, my math teacher solved a problem which asked to find the maximum value of the expression $x^2y^3$ when $x$ and $y$ are related as $3x+4y=5$. It was solved using the classic A.M.-G.M. inequality ...
BlackKnight23's user avatar
3 votes
1 answer
155 views

Variant Proof of the AM-GM Inequality

The question at hand is: Let $(x_i)_{1}^{n}$ be a finite sequence of positive numbers whose mean is $m = \frac{1}{n}\sum_{i=1}^{n} x_i$ Use the fact that for each positive $t$, we have $\log t \leq t −...
altayir1's user avatar
3 votes
1 answer
108 views

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: $...
user1285922's user avatar
3 votes
1 answer
284 views

Finding all $x, y$ natural numbers

Find all natural numbers $x$ and $y$ such that $$ \frac{\sqrt{x} + \sqrt{y}}{\sqrt[3]{x^2 - y^2}} \in \mathbb{N}. $$ My approach: $x>y$ because otherwise the radical would be negative and we can ...
math.enthusiast9's user avatar
7 votes
1 answer
346 views

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}}{...
orangeskid's user avatar
  • 54.5k
0 votes
1 answer
171 views

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 ...
Kokos's user avatar
  • 418
5 votes
1 answer
200 views

Showing $\frac{1}{5}<(f(k,l,m,n))^3\leq\frac{64}{11}$, where $f(k,l,m,n)=\sum_{cyc}\frac{k^2}{\sqrt[3]{5k^6+6l^3m^3}}$ for positive $k$, $l$, $m$, $n$

For $k,l,m,n \in \mathbb{R}^+$ denote by $f(k,l,m,n)$ the following function: $$f(k,l,m,n) = \frac{k^2}{\sqrt[3]{5k^6+6l^3m^3}}+\frac{l^2}{\sqrt[3]{5l^6+6m^3n^3}}+\frac{m^2}{\sqrt[3]{5m^6+6n^3k^3}}+\...
Anonymous's user avatar
  • 4,253
2 votes
0 answers
89 views

$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}=2 $ then $ \sum_{cyc}\sqrt{ab} \geq \frac{3}{2}$

Prove that if $$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}=2 $$ then $$\sum_{cyc} \sqrt{ab} \geq \frac{3}{2}$$ I tried to use AM-GM inequality and Titu's lemma and was able to show that $a+b+c\...
roro roro's user avatar
1 vote
1 answer
102 views

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,...
coolname11's user avatar
0 votes
0 answers
75 views

Prove that $\prod_{i=1}^n(1-x_i)\ge 1/2$.

Let $x_1,x_2,\ldots, x_n\in \Bbb R^+\cup \{0\}$ such that $x_1+\cdots +x_n\le 1/2$. Prove that $\displaystyle \prod_{i=1}^n(1-x_i)\ge 1/2$. It is easy to prove it by induction but I have a doubt since ...
James A.'s user avatar
  • 824
2 votes
1 answer
90 views

Prove the inequality $(1+2^{x-1}+2^y)(1+2^{y-1}+2^z)(1+2^{z-1}+2^x)\leq 2^{x^2+y^2+z^2+3}$

question Let $x,y,z$ natural nonzero numbers. Prove that: $(1+2^{x-1}+2^y)(1+2^{y-1}+2^z)(1+2^{z-1}+2^x)\leq 2^{x^2+y^2+z^2+3}$ my idea I don't know why but the first thing I thought of was to break ...
IONELA BUCIU's user avatar
  • 1,125
0 votes
1 answer
94 views

Prove that $\sum\limits_{\mathrm{cyc}} \frac{a^5 + b^5}{(b^2 + c^2)(c^2 + a^2)} \geq \frac{3}{2} (abc)^{\frac{2}{3}}$ [closed]

$(a+b)(b+c)(a+c) = 8a^2b^2c^2$ Prove that $\frac{a^5 + b^5}{(b^2 + c^2)(c^2 + a^2)} + \frac{b^5 + c^5}{(c^2 + a^2)(a^2 + b^2)} + \frac{c^5 + a^5}{(a^2 + b^2)(b^2 + c^2)} \geq \frac{3}{2} \cdot (abc)^{\...
New31415's user avatar
3 votes
3 answers
85 views

How to check that the arithmetic mean of any subset $S\ne\{1\}$ of $A={1,2,...,n}$ is at least $\frac 32$?

How to check that the arithmetic mean of any subset $S\ne\{1\}$ of $A=\{1,2,...,n\}$ is at least $\frac 32$? I don't have any idea where to start this. It is clear that this is right and I need to ...
IONELA BUCIU's user avatar
  • 1,125
2 votes
1 answer
164 views

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, ...
math.enthusiast9's user avatar
0 votes
1 answer
36 views

Proving Inequality Involving Sums and Products of Variables in a Probabilistic Analysis

I'm exploring a probabilistic analysis problem where I have variables (or, probabilities) $x_1,\ldots, x_n \in (0,1)$ in the range $(0,1)$ satisfying $\sum^n_{i=1} x_i \in (0,1)$. I aim to prove the ...
John's user avatar
  • 193
-1 votes
1 answer
67 views

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. $$\sum_{cyc}a^2 + abc \geq \frac{3}{8}\sum_{cyc}a^3 b +2$$
d8g3n1v9's user avatar
  • 453
2 votes
1 answer
70 views

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
0 votes
0 answers
65 views

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
  • 385
0 votes
2 answers
93 views

Help to prove $\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$.

I deal with a problem belonged RMM magazine without success. It's Let $a,b,c>0: abc=1$. Prove that: $$\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$$ My idea is ...
Sickness's user avatar
1 vote
2 answers
49 views

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
80 views

Can anyone help prove or disprove that $\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{N+1}$, where $x_i>0$

I was hoping the generalize this result: How to prove $x^{x/(1+x)}/(1+x)\geq1/2$ I believe that the following inequality holds: $\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{...
Vance M's user avatar
  • 112

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