Questions on proving and manipulating inequalities.

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2
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0answers
24 views

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

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 $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
2
votes
2answers
31 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= ...
3
votes
2answers
47 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
1answer
58 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 ...
0
votes
3answers
31 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!
2
votes
1answer
68 views

An inequality involving vectors

Let $n$ be a positive integer number. If $S$ is a finite set of vectors in the plane, let $N(S)$ denote the number of two-element subsets $\{\mathbf{v}, \mathbf{v'}\}$ of $S$ such that ...
-2
votes
1answer
52 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 ...
1
vote
0answers
39 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 ...
0
votes
2answers
27 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 ...
1
vote
1answer
24 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 ...
12
votes
3answers
175 views
+500

An inequality on sequences with each term dividing sum of two neighbouring terms

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is $$\frac{x_{i-1}+x_{i+1}}{x_i} = k_i $$ is an integer for ...
-1
votes
0answers
51 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
27 views

Trig question, inequality

How can I find the following product using elementary trigonometry? Suppose $0 \lt x \lt \frac{\pi}{2}$ is an angle measured in radians. Use the trigonometric circle and show that $\cos(x) \le ...
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 ...
0
votes
0answers
33 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
26 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 \\ ...
3
votes
2answers
67 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
4answers
58 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 ...
1
vote
1answer
40 views

Quadratic inequality with parameter

Hi I've got this inequality with parameter $a\in R$ $\frac{x+a}{x}\le x+2$ I've solved it but I've got different results than book. I've done it by dividing it into 2 cases. 1. x<0 2. x>0 and then ...
2
votes
4answers
78 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 ...
2
votes
1answer
31 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
1answer
68 views

Do there exist two vectors in a Hilbert space such that $(x,y)\geqslant k\|x-y\|^{-2}$?

Let $H$ be a Hilbert space, $(x,y)$ denote the inner product of the elements $x,y\in H$, $\|x\|$ denote the norm of $x\in H$, and $k>0$. Do there exist such $x,y\in H$ that $$ ...
4
votes
0answers
24 views
+50

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
1
vote
0answers
8 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
1answer
67 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 ...
1
vote
2answers
68 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 ...
0
votes
1answer
56 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: $$(x+y+z)^2+\frac{15}{2}\geq \frac{11}{4}(x+y+z+xy+yz+zx)$$
1
vote
1answer
36 views

Trigonometric inequality in a triangle

If $\alpha,\beta,\gamma$ are the interior angles in a triangle, the following inequality seems to hold: ...
-2
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0answers
38 views

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

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
48 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 ...
4
votes
1answer
240 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
40 views

Find value of $x$ for: $(1/3)(1-x) \geq 2(x-3)$

Find what value of $x$ satisfy: $(1/3)(1-x) \geq 2(x-3)$ First I multiplied both sides by $3$ so that $1/3$ became $3/3=1$. So I tried to find $x$ this way: $(1-x) \geq 6(x-3)$. I tried solving it ...
0
votes
0answers
56 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
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 $x \in (0,1)$. ...
1
vote
1answer
23 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.
0
votes
4answers
68 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
52 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 ...
1
vote
2answers
51 views

Inequality - Find what value of $t$ satisfies: $ (t/24) - (t+1) + (3t/8) < (5/12) (t+1)$

Inequality - Find what value of $t$ satisfies: $(t/24) - (t+1) + (3t/8) < (5/12) (t+1)$. Step 1: I multiplied both sides by $24$ and divided to get: $t-24(t+1)+9t < 10+24(t+1)$. Step 2: I ...
2
votes
1answer
44 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
0
votes
1answer
41 views

How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
1
vote
1answer
72 views

Prove the following inequality??

Someone can to help me with a hint in the following problem: Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that: ...
3
votes
1answer
55 views

Prove that $\sum_{cyc}\frac{a}{b(3+a-b)}\ge 1$

Let $a, b, c$ be positive real numbers such that $a + b + c = 3$. Prove that $$\sum_{cyc}\frac{a}{b(3+a-b)}\ge1$$ I tried applying the Cauchy-Schwarz inequality by doing: ...
0
votes
1answer
250 views

Combining inequalities into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
3
votes
0answers
49 views

An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that ...
4
votes
4answers
165 views

Inequalities, when does the sign change here?

I have encountered a problem with inequalities. I have been looking at examples provided by two websites which 'solve' inequalities, however when I try using my own method, the extremely simple ...
-4
votes
3answers
68 views

How to solve this: $|3-x|\ge2$

How to solve $|3-x|\ge2$ ? I know that if $|x| < y$, then $-y < x < y$. But in this case what to do? Thanks. Here, $|x|$ is the absolute value of $x$.
1
vote
2answers
72 views

$-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
2
votes
2answers
381 views

How prove this inequality $f(x)\ge f(0)$

Question: let $a>b>c>0,n\in N^{+},n\ge 2$ be given numbers,show that: ...
3
votes
1answer
62 views

What is the solution to $\frac1{a^2 +2} + \frac1{b^2 +2} + \frac1{c^2 +2} \le \frac{\sqrt2}{2}\frac{\sqrt a+\sqrt b+\sqrt c}{\sqrt{abc}}$

one of my friends asked me if I could solve him a mathematics problem. It looks like this: $$\frac1{a^2 +2} + \frac1{b^2 +2} + \frac1{c^2 +2} \le \frac{\sqrt2}{2}\frac{\sqrt a+\sqrt b+\sqrt ...
0
votes
1answer
521 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...