Questions on proving and manipulating inequalities.

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0
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4answers
38 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
16 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
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1answer
38 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
101 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!
0
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0answers
15 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
64 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
42 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
13 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
2answers
100 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
106 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? ...
10
votes
1answer
205 views

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
53 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' ...
3
votes
2answers
96 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
28 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
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0answers
38 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
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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= ...
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0answers
78 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
55 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
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3answers
35 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
62 views

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

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
31 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
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0answers
50 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
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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
63 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
68 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
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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
39 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: ...
0
votes
3answers
52 views

If $a^2=b^2+c^2$ and $0<n<2$ prove $a^n<b^n+c^n$

If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove (a) if $n>2$ then $a^n>b^n+c^n$, (b) if $0<n<2$ then $a^n<b^n+c^n$. Part (a) was easy to prove: $a^2=b^2+c^2$ and ...
0
votes
1answer
58 views

Prove the following inequality: $(x+y+z)^2+\frac{15}{2}\geq \frac{11}{4}(x+y+z+xy+yz+zx)$

Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove the following inequality: $$(x+y+z)^2+\frac{15}{2}\geq \frac{11}{4}(x+y+z+xy+yz+zx)$$
4
votes
1answer
244 views

Proof of an inequality

If $a$, $b$, $c$ are positive real numbers, prove that $$\frac{\sqrt{a+b+c}+\sqrt{a}}{b+c} + \frac{\sqrt{a+b+c}+\sqrt{b}}{c+a} + \frac{\sqrt{a+b+c}+\sqrt{c}}{a+b} \geq ...
1
vote
1answer
31 views

How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
0
votes
4answers
76 views

Questions about solving inequality: $2 < \frac{3x+1}{2x+4}$

Solve the inequality: $2 < \frac{3x+1}{2x+4}$ Step 1: I simplified $\frac{3x+1}{2x+4}$ into: $3x+1-2x-4= x-3$. Step 2: $2>x-3$ Here I subtracted $2$ from both sides into: $x>-5$ or ...
0
votes
4answers
53 views

How to solve Absolute Value Inequality: |x-1| ≥ 3-x

I am learning the topic of solving absolute value inequality question. I had mostly understood the steps in order to solve for an inequality. However, I'm still clueless of a step to solve the ...
0
votes
0answers
64 views

Bound for this integral

Using the fact that $$\sqrt{(1+y^2)} - \sqrt{(1+x^2)} \geq \frac{x}{\sqrt{1+x^2}}(y-x)$$ for each $x,y\in \mathbb{R}$. We need to show that $$L(k)- L(h) \geq \int_a^b \frac{h'}{\sqrt{1+{h'}^2}} ...
1
vote
1answer
26 views

Galerkin Orthogonality in this FEM?

Problem Galerkin orthogonality is but I am not sure if it is in the right form. How can you use this orthogonality here? I think I should expand the last inequality first somehow.
2
votes
4answers
79 views

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$. Here are some of my ideas: Also by applying Mean Value theorem, we know that ...