Questions on proving and manipulating inequalities.

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-1
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0answers
24 views

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

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
30 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 ...
-2
votes
0answers
24 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: $$(x+y+z)^2+\frac{15}{2}\geq \frac{11}{4}(x+y+z+xy+yz+zx)$$
2
votes
1answer
68 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
28 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
64 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
51 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
51 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
13 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.
1
vote
4answers
66 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 ...
1
vote
2answers
46 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 ...
0
votes
1answer
35 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
31 views

Trigonometric inequality in a triangle

If $\alpha,\beta,\gamma$ are the interior angles in a triangle, the following inequality seems to hold: ...
2
votes
1answer
42 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 ...
3
votes
1answer
51 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: ...
3
votes
0answers
42 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 ...
1
vote
1answer
69 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: ...
4
votes
4answers
163 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 ...
1
vote
2answers
68 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 ...
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 ...
2
votes
0answers
15 views

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 ...
0
votes
1answer
39 views

Use mean value theorem on $f(x) = x^{1/5}$, to show that $2< \sqrt[5]{33}<2.0125$

The problem specifically aks us to use mean value theorem on the interval $[32, 33]$ It has always puzzled me that mean value theorem can be used to prove Inequalities. Can anyone show how mean ...
0
votes
2answers
25 views

Spivak Absolute Value Problem (Prologue 9-v)

I'm working on the following problem Express the following with at least one less pair of absolute value signs $$|(| \sqrt2 + \sqrt3| - |\sqrt5 - \sqrt7|)|$$ Now I can see that the ...
0
votes
0answers
29 views

What is the meaning behind this mathematical rebus? [on hold]

I believe I saw this 'inequality' in someone's profile description here on Math Stack Exchange. I think it expresses a message or has a meaning. What is it?
3
votes
1answer
33 views

Range of $f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}$ for a specified domain

We are asked to find the range of the function $$f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}, \;\;\text{for}\;0\le x\le2\pi$$ I tried to find the range of each basic function of cos and sin then ...
1
vote
1answer
27 views

Equality case in the Frobenius rank inequality

In many linear algebra books, the following rank inequalities are found: Frobenius inequality Let $A$, $B$ and $C$ be three matrices such that the product $ABC$ is defined. Then ...
3
votes
8answers
148 views

How to show that $(a+b)^p\le 2^p (a^p+b^p)$ [duplicate]

If I may ask, how can we derive that $$(a+b)^p\le 2^p (a^p+b^p)$$ where $a,b,p\ge 0$ is an integer?
1
vote
1answer
52 views

A simple complex inequality

I feel this is not hard, but no way to prove it $|\sqrt{z^2 -4}-z|\le 2$ Any body can help? Thanks! The total statement should be one of the branchs of square root should satisfy this ...
2
votes
1answer
76 views

Inconventional Integral inequality

$$\int_a^bw(x)|f(x)||g(x)|\;dx \le \left(\int_a^bw(x)\;dx\right) \max_{a\le x\le b}|f(x)|\cdot \max_{a\le x\le b}|g(x)|$$ I don't really understand this integral inequality. How do I go about ...
-8
votes
2answers
60 views

How to solve an irrational inequality?

How to solve the following inequality: $$\sqrt{1-2x} < \sqrt{4 - x}$$ I don't understand why "$(1-2x)$ have to be $\ge 0$". If it was the rule for numbers inside a square root, I was checking ...
8
votes
4answers
171 views

Inequality question­

$$a,b,c,d\ge 0$$ $$a\le 1$$ $$a+b\le 5$$ $$a+b+c\le 14$$ $$a+b+c+d\le 30$$ Prove that $\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}\le 10$. We can subtract inequalities to get the answer, but that is ...
8
votes
1answer
97 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
vote
2answers
73 views

prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$

Prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$ I already showed that $a_n$ diverges to infinity like this: I used to the lemma which says that if ...
0
votes
0answers
50 views

Prove the inequality $\sum_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $

A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} ...
3
votes
0answers
38 views

A Cauchy-Schwartz type inequality

Given positive integers $k<n$ and positive real numbers $x_1$, $x_2, \dots, x_n$. Denote $$ A={x_1\over x_2+x_3+\dots+x_{k+1}}+{x_2\over x_3+x_4+\dots+x_{k+2}}+\ldots+{x_n\over x_1+x_2+\dots+x_k}$$ ...
2
votes
3answers
110 views

Proving that one of $a(1-b), b(1-c), c(1-a) \le \frac{1}{4}$

how can a prove that at least one of those is less than or equal to 1/4. $$\forall a,b,c\in \mathbb R^+, \ a(1-b)\leq 1/4 \lor b(1-c) \leq 1/4 \lor c(1-a) \leq 1/4.$$ help please!
0
votes
1answer
70 views

Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
0
votes
3answers
25 views

Can the proof for the following 4 cases be simplified to 2 cases?

Let $X$ and $Y$ be finite and disjoint sets. Suppose we are required to prove the following: $|X|\ge 0 \text{ and } |Y|\ge 0 \Rightarrow Q $ where $Q$ is some statement. Therefore, I know I need to ...
0
votes
1answer
34 views

Geometric Application of Cauchy-Schwarz Inequality Problem

I have been struggling with this problem, and would like to prove the inequality using the Cauchy-Schwarz Inequality: The vertices of a fixed triangle are $A$,$B$ and $C$, and $P$,$Q$ and $R$ lie on ...
2
votes
1answer
14 views

Bounds on a recursively defined sequence

I have a sequence defined by $h_0=h_1=1$, $h_2=2$ and $h_{n+1}=(n+1)h_n + \frac{n(n-1)}{2}$. The paper I'm reading claims $n! \le h_n \le 2(n!)$. It is easy to show the first inequality by induction. ...
1
vote
2answers
46 views

Questioning a Proof of Khinchine Inequality

[Khinchine Inequality] Let $a_1,\ldots,a_n\in R$, $\varepsilon_1,\ldots,\varepsilon_n$ be i.i.d. Rademacher random variables: $P(\varepsilon_i=1)=P(\varepsilon_i=-1)=0.5$, and $0<p<\infty$. Then ...
4
votes
3answers
398 views

An inequality in numbers

Which number is larger? $\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits}$ or $\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots67}_\text{68 ...
3
votes
0answers
61 views

Upper bound for the sums of powers of factors

Fix $\alpha \in \,]0,1]$. Is it true that for each sufficiently large positive integer $n$, if $n = x_1 \cdots x_j$, for some integers $x_1, \ldots, x_j \geq 2$, with $j \geq 2$, then $$x_1^\alpha + ...
2
votes
1answer
32 views

Number of integers satsifying inqualities with logarithm

I am trying to solve the problem of finding the integers x satisfying the inequalities: $2\lt log_x45\lt3$ I realize this is a very basic question on logarithms and I have the key with the answers 4, ...
0
votes
1answer
45 views

Inequality: $2\sqrt{xz}+2\sqrt{yz}+2\sqrt{xy}\geq 3x+3y+3z-3$

Let $x,y,z$ be nonnegative real numbers satisfying $x^2+y^2+z^2=1$. Prove that: $2\sqrt{xz}+2\sqrt{yz}+2\sqrt{xy}\geq 3x+3y+3z-3$ I have tried squaring both sides to get: ...
0
votes
1answer
65 views

How find this maximum $\sum_{i=1}^{n}(x^3_{i}-x_{i}x_{i+1}x_{i+2})$,$x_{n+1}=x_{1},x_{n+2}=x_{2}$

let $n$ is give postive integer numers,and $x_{i},i=1,2,\cdots,n$ be real numbers,and such $$0\le x_{i}\le i,i=1,2,\cdots,n$$ Find the maximum of the value ...
5
votes
4answers
143 views

How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$

Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$ My idea: I guess $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le ...
2
votes
3answers
163 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
2
votes
1answer
79 views

Homework on basic inequalities.

Let $a_j$ be a sequence of positive reals. Show that (a) $\left(\sum_{j=1}^\infty a_j\right)^\theta \le \sum_{j=1}^\infty a_j^\theta$ for any $0\le\theta\le1$. (b) $\sum_{j=1}^\infty a_j^\theta \le ...
4
votes
2answers
73 views

Binomial expansion inequality

In a paper I am reading, there is a step that seems to come from the following inequality: $$(1+x)^\alpha \le 1+2^\alpha x,$$ where $0<x<1$. (Also, $3\le \alpha \le 9/2$ in the context of the ...