Questions on proving and manipulating inequalities.

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2
votes
2answers
97 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
2
votes
3answers
75 views

How to prove $ \frac{e^{x}+e^{-x}}{2} \le e^{\frac{x^2}{2}} $?

Let $x\in \mathbb{R}$, how to prove $$ \frac{e^{\large x}+e^{\large-x}}{2} \le e^{\large\frac{x^2}{2}} $$
2
votes
3answers
34 views

prove $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

$a,b,c$ are positive real numbers.prove:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$ Additional info:we cant use induction.we should mostly use Cauchy inequality.other ...
2
votes
0answers
40 views

$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $ [on hold]

Show that for every arbitrary complex number a,b and c we have $$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $$ Thanks.
1
vote
0answers
71 views

Practicing the arithmetic-geometric means inequality

I am struggling with learning the AM-GM Inequality that is considered vital to know for math olympiads, contests, etc. I just don't really know when to use it, when it is necessary to use, the purpose ...
2
votes
6answers
196 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
5
votes
2answers
115 views

prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$

If $a_i$ and $b_i$ are positive, and $b= \sum ^n_{i=1} b_i$,$a= \sum ^n_{i=1} a_i$ prove $$\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$$ Additional: we should just use Cauchy ...
-3
votes
0answers
12 views

Maximum absolute column sum of the matrix norm inequality [on hold]

$A=(a_{ij})\in \mathbb{C}^{n\times n}$, $\upsilon(A)=n \max_{ij} |a_{ij}|$ is matrix norm. Prove that $\|A\|_{1} \leq \upsilon(A) \leq n \|A\|_1$.
1
vote
1answer
29 views

Prove $\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$

If $\space x_1+x_2+\cdots+x_n=1$ and all $x_1,x_2,\cdots,x_n$ are positive and real numbers, prove:$$\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$$ ...
0
votes
4answers
43 views

Prove that $r(n+1−r) \ge n$ for any positive integer $n$ and $1 \le r \le n$. [on hold]

Prove that $r(n+1−r) \ge n$ for any positive integer $n$ and $1 \le r \le n$. Nothing I tried worked, because the coefficient of $r$ is negative so I only get maximums and not minumums.
0
votes
0answers
18 views

Deriving inequalities featuring bounded variables

I have a model which fits certain thermodynamic data, of the form $$y = \frac{x}{ 1 + (a - 1)x} + bx(1 - x) \quad a,b \in \mathbb{R} \quad 0 \leq x \leq 1$$ Thermodynamics dictate that ...
1
vote
1answer
44 views

Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
3
votes
3answers
108 views

Prove that $n^n \le (n!)^2$.

Prove that $n^n \le (n!)^2$. There is an elementary solution, which I haven't been able to find. So far I tried manipulating and pairing terms but nothing worked. I would appreciate any help!
1
vote
0answers
21 views

Entropy of the product of two random variables

Consider a random matrix $X$ and a random vector $Y$. Let the Shannon entropies $H(X) = H(Y) = n$. Is there a simple upper bound for entropy $H(XY)$? I believe $H(XY) \leq 2n$ as that is a simple ...
4
votes
2answers
65 views

How to prove that $ 1- \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1-\frac{x}{n})^n$

How would I prove this inequality (assuming its true, its from a textbook) $$1 - \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1+\frac{-x}{n})^n$$ if $n > |x|$, $x\in R$ and $n\in N$ I first ...
3
votes
0answers
44 views

How prove $\max\lbrace \cot{A}(\cot{Y}+\cot{Z}),\cot{B}(\cot{Z}+\cot{X}),\cot{C}(\cot{X}+\cot{Y})\rbrace\ge\frac{2}{3}$

let $\Delta ABC,\Delta XYZ$ are acute triangle show that $$\max\lbrace\cot{A}(\cot{Y}+\cot{Z}),\cot{B}(\cot{Z}+\cot{X}),\cot{C}(\cot{X}+\cot{Y})\rbrace\ge\dfrac{2}{3}$$ My idea: since ...
0
votes
0answers
18 views

Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
1
vote
1answer
36 views

Is the following sequence monotonic?

Suppose $a_i>0$ for all $i$ and p>1. Let $$y_n = \frac{(\sum_{i=1}^n a_i)^p}{n^{p-1}\sum_{i=1}^n(a_i^p)}.$$ Is $y_n$ monotonic? How can you prove or disprove this? Otherwise, can it be shown that ...
5
votes
3answers
134 views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that ...
0
votes
2answers
56 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
1
vote
2answers
26 views

Where did I go wrong with this inequality involving absolute value function?

Question: Find all $x \in \mathbb R$ such that the inequality $4<|x+2|+ |x-1|<5$ is satisfied. This is my attempt at solving the problem: Case (i): If $x+2 \geq 0 $ and $ x-1\geq0$, then ...
1
vote
4answers
107 views

Is the minimum of the product of two functions equal to the product of their minima?

I have stuck with following equality, For all $x$, assume function $a(x)$, $b(x)$ have nonzero, and non negative values. (i.e $a(x)>0$, $b(x)>0$, Is the following equality true? ...
11
votes
1answer
249 views
+50

How prove this $x_{1}+x_{2}+\cdots+x_{n}<\frac{5}{3}$

Question: Let $x_{1},x_{2},\cdots,x_{n}\ge 0$ with $$x_{i}x_{j}\le 4^{-|i-j|}$$ for all $i, j = 1, \dots, n.$ Show that $$x_{1}+x_{2}+\cdots+x_{n}<\dfrac{5}{3}.$$ This problem is from ...
10
votes
3answers
55 views

Solve inequality: $-5 < \frac{1}{x} < 0$

Solve inequality: $-5 < \frac{1}{x} < 0$ I thought about how I can solve this. If I multiply all sides by $x$ I'm afraid I'm removing the answer, cause $\frac{x}{x}=1$. And when $x$ 'leaves' ...
4
votes
2answers
102 views

How prove $x^3+y^3+z^3-3xyz\ge C|(x-y)(y-z)(z-x)|$

let $x,y,z\ge 0$,and such $$x^3+y^3+z^3-3xyz\ge C|(x-y)(y-z)(z-x)|$$ Find the maximum of the $C$ witout loss of we assume that $$x+y+z=1$$ I think ...
0
votes
0answers
29 views

A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
0
votes
0answers
39 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
1
vote
1answer
63 views

Is the following Eigenvalue inequality holds or not?

Can anyone help me with the following problem? Suppose $u=(u_1,u_2,...u_n)^T$, $e=(1,1,...1)^T$, and we have $u\geq e$. Now for any symmetric matrix $A\in S^n$ with $diag(A)=0$, can we claim the ...
0
votes
0answers
25 views

How to solve the inequality: $\prod_{k=1}^N\left(x^k-k^2\right)\gt0$

Given the inequality: $$\displaystyle\prod_{k=1}^N\left(x^k-k^2\right)\gt0$$ how can I solve it? I suppose there is a difference if $N=2n$ or $N=2n+1$ with $n\in\mathbb{N}$, but I'm unable to find a ...
1
vote
1answer
37 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
2
votes
2answers
37 views

minimum value of $y= \frac {x^n+a}{x^m}$

Question if $n>m$, $\frac {a}{x^m} > 0$ and $x^{n-m} > 0$,prove $y= \frac {x^n+a}{x^m}$ is minimum when $x= \sqrt[n]{\frac {am}{n-m}}$ and value of minimum is equal to $y= ...
1
vote
0answers
92 views
+50

How prove $\sum_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum_{cyc}h_{a}}$

Question: let $\Delta ABC$,and the altitude is $h_{a},h_{b},h_{c}$,where $AB=c,BC=a,AC=b$ and for any $P$ show that $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
3
votes
2answers
58 views

prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
0
votes
3answers
36 views

How do I prove $x^n < x^m$ when $m > n$ and $x > 1$

Title I made an attempt at it here: $x^n < x^m$ when $m > n$ and $x > 1$, $m$ and $n$ are naturals so divide both sides by $x^n$ so $1 < x^{m-n}$ but here i am stuck. Please help!
-3
votes
1answer
63 views

Prove that: $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac{1}{3}$ [closed]

Let $a,b,c$ be positive real numbers satisfying $a+b+c+d=4$. Prove that: $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac{1}{3}$ p/s: I have no idea about the problem ...
0
votes
2answers
32 views

Solving inequalities with modulus in addition

How would you solve an inequality with modulus in addition? Question is: $$|2x-1| + |x-3| \geq 10$$ How to start here? What I tried: Well you can obviously solve the equation for each possibility ...
2
votes
0answers
51 views

How prove this inequality $\frac{3(x^2+y^2+z^2)}{(x+y+z)^2+2(yz+xz+xy)}\ge\sum_{cyc}\frac{x^2}{x^2+(y+z)^2}$

Question: let $x,y,z\ge 0$.prove or disprove $$\dfrac{3(x^2+y^2+z^2)}{(x+y+z)^2+2(yz+xz+xy)}\ge\sum_{cyc}\dfrac{x^2}{x^2+(y+z)^2}$$ My idea: let $x+y+z=1$, then we can only ...
1
vote
1answer
25 views

Proving that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance in $\mathbb{R}^2$

I was asked to prove that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance function in $\mathbb{R}^2$. I've got myself stuck with proving the triangle inequality. Can someone give me an hint ...
-1
votes
0answers
52 views

How prove that $\ln2>{\left(\frac{2}{5}\right) }^{\frac{2}{5}}$? [duplicate]

How prove that inequality $\ln2>{\left(\frac{2}{5}\right) }^{\frac{2}{5}}$?
0
votes
1answer
63 views

Prove that: $\sum \frac{a^2+2bc}{(b+c)^2}\geq \sum \frac{3}{2}\frac{a}{b+c}$

Let $a, b, c > 0$.Prove that: $\sum \frac{a^2+2bc}{(b+c)^2}\geq \sum \frac{3}{2}\frac{a}{b+c}$ p/s: I tried to solve the problem by $S.O.S$. But I cannot solve it !! I have: The inequatily ...
1
vote
4answers
41 views

Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

I'm trying to prove this by induction but I'm running into some trouble. The base case is $0$, so, $a^0 = 1$, the inequality holds true Being new to induction, I don't exactly know what to do for ...
1
vote
0answers
51 views

Almost Jensen's Inequality

Let $a,b$ and $c$ three positive reals numbers such that $abc=1$. Define the function $f$ by $f(x)=\frac{^1}{1+(n-1)x^n}$ where $n$ is a positive integer. Prove that ...
2
votes
1answer
31 views

Morrey's inequality

From PDE Evans, 2nd edition, page 281: Now \begin{align} \int_0^s \int_{\partial B(0,1)} |Du(x+tw)| \, dS(w) dt &=\int_0^s \int_{\partial B(x,t)} \frac{|Du(y)|}{t^{n-1}} \, dS(y) dt \\ ...
2
votes
4answers
64 views

Calculus - inequality problem.

I have this inequality : $$|g(x)-B|<\frac{|B|}{2}$$ $$-\frac{|B|}{2}<g(x)-B<\frac{|B|}{2}$$ $$B-\frac{|B|}{2}<g(x)<B+\frac{|B|}{2}$$ I don't understand how it possible to conclude ...
3
votes
2answers
69 views

An inequality I am stuck on

This is somehow related to this problem but I don't have any idea about it. $a,b,c,d$ are positive reals such that $a+b+c+d=4$ $$\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}\le ...
2
votes
1answer
35 views

Inequality involving Jensen (Rudin's exercise)

Exercise (Rudin, R&CA, no. 3.25). Suppose $\mu$ is a positive measure on the space $X$ and let $f \colon X \to (0,+\infty)$ be such that $\int_X f \, d\mu=1$. Then for every $E \subset X$ ...
1
vote
0answers
9 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
1
vote
2answers
70 views

Proving inequality $3^{n^2} > (n!)^4$

Prove that $3^{n^2} > (n!)^4$ for all positive integers $n$. I tried to use induction on this problem but failed to do so. I instead tried to prove $3^{2n+1}>(n+1)^4$, but couldn't come up ...
1
vote
1answer
68 views

Find the maximun value of the expression $P=\sum \sqrt[3]{\frac{a^{2}+a}{a^{2}+a+1}}$

Let $a,b,c$ be positive real numbers such that $abc\leq 1$ .Find the maximun value of the expression ...
-2
votes
0answers
40 views

Prove that: $a^6+b^6+c^6 \leq x^3+y^3+z^3$ [closed]

Let $a,b,c,x,y,z$ be positive real numbers such that: $\left \{\begin{matrix} x \geq y \geq z,a\leq x \\ a^2+b^2 \leq x^2+y^2 \\ a^3+b^3+c^3 \leq x^3+y^3+z^3 \end{matrix}\right.$ Prove that: ...