Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

-1
votes
0answers
18 views

Prove that x.y<1 when 0<x<1 0<y<1 [on hold]

i need a proof of this inequality: 0<x<1 0<y<1 and x,y ∈ (0.1) also x,y ∈ R x.y<1 only using of axioms. Thanks!
0
votes
1answer
16 views

Resolving a second degree inequation using its factored form

I'm trying to solve an exercise and I'm having troubles to get it right. We have two functions $f(x)$ and $g(x)$ defined as follow : $f(x) = 2x-3$ $g(x) = -x²+x-3$ Question : We have to demonstrate ...
-1
votes
2answers
50 views

Important: Proof of this inequality (?)

I'm really screwed since I don't know how to prove this inequality. I've tried creating an $\epsilon$ surrounding and everything but nothing seems to work. The task is: Prove $$\left(\frac{K+1}{N}...
0
votes
0answers
15 views

Proving an inequality involving absolute value; how do I justify using a conjunction (and) instead of a disjunction (or)?

I'm putting together the following the proof, and I have a question about one of the final steps. Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \...
-1
votes
0answers
38 views

Very curious identity and an inequality $a^{ab}f(a^{ab})+b^{ab}f(b^{ab})\leq a^{a^2}f(a^{a^2})+b^{b^2}f(b^{b^2})$

Related to my post Proving $a^{ab}+b^{ab}\leq a^{a^2}+b^{b^2}$ and an identity . I have found the following identity generalised : Let $x$ a real numbers with $x\neq 0$ then we have and $f(x)$ a ...
0
votes
0answers
42 views

Prove a challenging pseudo-cyclic inequality using basic inequalities [duplicate]

if a, b, c>0, then prove: $\frac{a+b}{\sqrt{a^{2}+a b+b^{2}+b c}}+\frac{b+c}{\sqrt{b^{2}+b c+c^{2}+c a}}+\frac{c+a}{\sqrt{c^{2}+c a+a^{2}+a b}} \geq 2+\sqrt{\frac{a b+b c+c a}{a^{2}+b^{2}+c^{2}}}$ I ...
1
vote
2answers
58 views

Using induction to prove a sequence $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$ is increasing

I have a sequence defined by $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$. I need to use induction to show that the sequence is increasing and $a_n<3$ for all $n$. Also to deduce that $a_n$ is ...
0
votes
3answers
34 views

Is the statement true?$|z+1|\ge|z|-1\ \forall z\in\Bbb{C}$

I have tried to prove it in the following manner- $||z|-1|=||z|-|1||\le|z-1|\ \forall z\in\Bbb{C}$ (by Triangle inequality) Now, can I write $|z-1|\le|z+1|$ in $\Bbb{C}$? If yes then the proof is ...
1
vote
1answer
43 views

How to prove Ky Fan Inequality using Forward-Backward Induction?

The classical version of the inequality is: $${\frac {{\bigl (}\prod _{{i=1}}^{n}x_{i}{\bigr )}^{{1/n}}}{{\bigl (}\prod _{{i=1}}^{n}(1-x_{i}){\bigr )}^{{1/n}}}}\leq {\frac {{\frac 1n}\sum _{{i=1}}^{...
1
vote
1answer
44 views

inequality about $I=\int_{2012}^{3012}\sqrt[3]{x}\,dx$

Consider $$ \begin{align} & L=\sqrt[3]{2012}+\sqrt[3]{2013}+\ldots +\sqrt[3]{3011} \\ & R=\sqrt[3]{2013}+\sqrt[3]{2014}+\ldots +\sqrt[3]{3012} \\ \end{align}\ $$ and $$ I=\int_{2012}...
0
votes
1answer
34 views

Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

Let $p>q>0$ and $C=\{f:[0,1] \to \mathbb{R} \mid f \text{ is continuous} \}$. Determine $$\max_{f \in C}\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$$ and the functions for which this maximum occurs. ...
0
votes
1answer
16 views

I want to understand the triangle inequality about contraction operator

I read the Chow's paper, Multigrid algorithms and complexity results, https://dspace.mit.edu/handle/1721.1/14254 I have a question on page 42. Let me write the Lemma 2.4.3 on the page. Lemma 2.4.3 ...
2
votes
1answer
72 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
0
votes
0answers
30 views

tight upper bound for $\left\lVert B^{T}B \right\rVert$

Given a matrix $A := \left( a_{i,j} \right) \in \mathbb{R}_{+}^{n \times n}$. Let $B := \left( b_{i,j} \right) \in \mathbb{R}^{n \times m}$ be a matrix such that \begin{equation} \sum\limits_{k=1}^{m} ...
1
vote
1answer
41 views

Proving the AM-GM Inequality with a given fact

Given $x + y + z \geq 3$ for all $(x, y, z) \in \mathbb{R}^{3}$ such that $x,y, z > 0$ provided $xyz = 1$, show that $$\frac{a_1+a_2+a_3}{3} \geq \sqrt[3]{a_1a_2a_3}$$ holds. I'm not really sure ...
1
vote
0answers
18 views

General Method to solve Power Sum Inequality

This is a general method to have power sum inequality . We work with $x_i> 1$ $n$ real numbers . We want to show this kind of inequality : Let $x_i> 1$ be $n$ real positive numbers and $...
0
votes
0answers
42 views

If $f:\mathbb{C}\to\mathbb{R}$ is continuous and $f\ll1$ then $\int_{|z|=1}f\ll4$

Show that if $f:\mathbb{C}\to\mathbb{R}$ is continuous and $f\ll1$ ($a\ll b$ means $|a|\leq |b|$), then $\int_{|z|=1}f\ll4$. Hint: Show first $\int f\ll\int_0^{2\pi} |\sin t|dt$. I'm not sure how to ...
1
vote
1answer
43 views

Minimizing $2$-norm subject to non-convex constraints

Let $|Ax|$ be the element-wise absolute value of $A x$, i.e., $|Ax|_i = |A(i,:)x|$. The inequalities are element-wise inequalities, i.e., $|A(:,i)x| \geq b(i)$. Also, let $\|x\|$ denote the $2$-norm ...
-3
votes
1answer
25 views

prove by induction that 2^n/n! < 4/n [on hold]

How do I do this? I know how to do the base case but I can't figure out how to do the next steps.
1
vote
1answer
47 views

Finding the Minimum Value of Sum of Positive Real Numbers

I'm not sure how to solve the following problem: $$d_1^2 + \ldots + d_n^2 = \sigma^2$$ $$d_i \geq 0$$ Find the minimum possible value of $$d_1 + \ldots + d_n$$ I have a hunch that its when every ...
0
votes
0answers
27 views

A question regarding inequality between weighted averages

Suppose you have three probability density functions $p_{1}(x)$, $p_{2}(x)$, and $p_{3}(x)$. Suppose further that the expectation values of $p_{1}(x)$ and $p_{2}(x)$ are unequal $\int_{0}^{1}p_{1}(x) ...
1
vote
1answer
23 views

How to use trace operator for inequalities dealing with an hermitian matrix and its inverse?

When I read a paper, I met these implications involving inequalities : $$R-a^Ha\ge0 \ \implies \ I-R^{-1/2}aa^HR^{-1/2}\ge0 \ \implies \ 1-a^HR^{-1}a\ge0$$ $R$ is an invertible Hermitian matrix ...
-1
votes
1answer
30 views

How to prove this inequality? It is true or false? [on hold]

I'm trying to prove this statement: Let $ \alpha, t >0 $, then there exists a positive constant $ c $ such that $$ \left( 1+\frac{t}{2}\right)^{-\alpha} \leq c \left( 1+t\right)^{-\alpha}. $$ Is ...
1
vote
1answer
17 views

Showing existance of tournament without transitive subtournament

Show that there exists a tournament on $n$ vertices that does not contain a transitive tournament on $2\log_2n+2$ vertices. My attempt: The number of tournaments of $n$ vertices is $2^{\binom{n}{2}}$...
1
vote
1answer
35 views

Proving $|E| \leq \frac{1}{2}(\sqrt{t-1}\;n^{3/2} + n$) in a graph with no copies of $K_{2,t}$

Show that if an $n$-vertex graph $G=(V,E)$ has no copy of $K_{2,t}$ then: $$|E| \leq \frac{1}{2}(\sqrt{t-1}\;n^{3/2} + n)$$ I know how to prove it for $n=2: \;$ Dentoe $|E|=m,$ $d(v)$ the degree of $...
3
votes
0answers
101 views
+50

Proving $a^{ab}+b^{ab}\leq a^{a^2}+b^{b^2}$ and an identity .

I was working on this Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$ when I have discovered the following identity : $$\Bigg|\Big(\frac{1}{2}\Big)^{\frac{x}{8}}\pm\Big(\frac{1}{4}\Big)^{\frac{x}{8}}...
0
votes
3answers
47 views

Prove the inequality $x^2+4y^2<1$ when $x-y=x^3+y^3$

I have to prove the following inequality: $$ x^2+4y^2<1$$ with this constrain $$x-y=x^3+y^3$$ and where $x$ and $y$ are positive real numbers. From the constrain we have also $0<x<1$. I ...
-1
votes
0answers
14 views

$\|f\|_{L^{\frac{10}{3}}(\mathbb T^3)} \leq C \|f\|_{H^1(\mathbb T^3)}^{3/5} \|f\|_{L^2(\mathbb T^3)}^{1-(3/5)}$? [on hold]

Let $\mathbb T^3$ be 3-dimensional torus. Can we expect that $\|f\|_{L^{\frac{10}{3}}(\mathbb T^3)} \leq C \|f\|_{H^1(\mathbb T^3)}^{3/5} \|f\|_{L^2(\mathbb T^3)}^{1-(3/5)}$?
0
votes
2answers
25 views

The number of integers 'n' which satisfy the inequality $(n^2 - 2)(n^2 - 20) < 0$ is? [on hold]

Tried hit and trail method . And not got the answer . Don't know anything about going forward onto the question .
0
votes
1answer
46 views

For which $a \in \mathbb{R}$ is $x^2 + 4|x-a| - a^2 \geq 0$ true for all $x \in \mathbb{R}$? [on hold]

As the title says, I have a problem answering the question: for which $a \in \mathbb{R}$ is $$ x^2 + 4|x-a| - a^2 \geq 0 $$ true for all $x \in \mathbb{R}$?
-1
votes
1answer
41 views

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$ $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to?

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$, $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to ? My thinking:- I have break the left hand side on $sinC + sinD$ and right hand ...
0
votes
0answers
33 views

How to get inequality ‎$‎0‎\leq ‎\gamma‎_g + ‎log ‎g(1)\leq ‎\frac{g^‎\prime_{-}(1) + g^‎\prime_{+}(1) ‎}{2g(1)}‎$‎?

‎Let ‎‎$‎g:‎\mathbb{R^+}‎‎‎\rightarrow‎‎\mathbb{R^+}‎$ ‎be a‎ ‎real‎ ‎function ‎such ‎that ‎‎$\log g(x)$ ‎is ‎concave and ‎$‎\displaystyle{\lim_{x\to\infty}}‎\frac{g(x+w)}{g(x)} = 1‎$ ‎for each ‎$‎w&...
0
votes
1answer
26 views

A complex integral with boundary on the norm

I was asked this question. I couldn't even manage to find a starting point on the question. Any help is welcomed. P.S: I know it is from a book but I do not know from which one. If you have any idea ...
1
vote
2answers
74 views

maximum value of $\sum (a-b)^2$

If $a^2+b^2+c^2=5$ and $a,b,c \in \mathbb{R},$ find the maximum value of $(a-b)^2+(b-c)^2+(c-a)^2$. My Try: $(a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2)-2(ab+bc+ac)$ $$=10-2(ab+bc+ac)$$ Now this ...
0
votes
1answer
19 views

Algebraic manipulation and inequality

Given real-valued terms $a,b,c \in {R}$ with the following conditions on them: $ 0 \leq a \leq 1 $, $|b|<1$, and $|c|< 1$. And given the terms $ X= \frac{1}{1-( (1-a)b + ac )}$ and $ Y =\frac{1}...
1
vote
1answer
32 views

$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v \in \{-1,1\}^n }{sup} |\underset{i,j}{\sum} a_{ij}u_iv_j|$?

let $(a)_{ij}$ be a $M\times N$ Matrix with real entries ,is that possible to prove that: for any $x \in [-1,1]^n, y \in [-1,1]^m$ we have: $$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v ...
-1
votes
0answers
43 views

If (x − 1)(x − 3)(x + 3)(x + 5) < 0, the range of its roots is included by [on hold]

It's an sat 2 question. I don't know where to start with. As the question talks about inequality how can it have a roots and it mention about the range. im my opinion the range of the roots always ...
2
votes
1answer
93 views

Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$

I have to prove the inequality $$ \frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}} $$ when $x^2+y^2=1$, using Cauchy-Schwarz Inequality. The RHS is equal to $\frac{12}...
-1
votes
1answer
19 views

How do i Solve this Linear Efficiency Problem in math?

I dont understand how to do these, anyway someone could explain to me and show me how to do it? Thankyou The environmental and health department of a country runs two refuse centers. Center 1 costs ...
-4
votes
1answer
17 views

Space explosion [on hold]

A shell of mass M is at rest at a point in space. It bursts into two fragments with a combined energy of E (ignore radiation losses). 1. Show that the relative speed of the two fragments after ...
0
votes
3answers
29 views

Infern sign of quadratic equation from coefficients

Consider the quadratic function $f(x)= ax^2 + bx + c$ where $x \in \{1,2,\ldots\}$. Presume that $b^2 - 4ac < 0$ holds and $a \neq 0$. We know that there does not exist a $x \in \mathbb R$ such ...
0
votes
0answers
15 views

Data processing inequality - KL divergence before and after a stochastic matrix

I'm studying the Blahut Arimoto algorithm using these notes and towards the end of section 6, an interesting quantity arises. The author does not talk about how to compute it so I was hoping I could ...
0
votes
1answer
28 views

Bound $|x^TAy|$ in terms of $\|A\|$ and $|x^Ty|$

Under what conditions on a square matrix $A$ of size $n$ do we have $|x^TAy| \le |x^Ty|$ for all $x,y \in \mathbb R^n$ ? Notes The above inequalities hold for $A \in \{0, I\}$, and so by simple ...
1
vote
2answers
69 views

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Let $a,b,c$ be non-negative real numbers such that $c \geq 1$ and that $a+b+c=2$. Find the minimum and maximum values of $$P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$$ To find the minimum of $P$ ...
-1
votes
1answer
47 views

How is $\log(x) \leq x-1$ for all $x>0$?

sorry if this is a very obvious question, but I'm writing a proof for the Kulback-Leibler inequality and the first step is to state $\log(x) \leq x-1$ for all $x>0$. I get it for $x>1$, but in ...
0
votes
1answer
68 views

Underapproximating the exponential function from below [duplicate]

I think that for positive natural numbers t and n we have $$ \left(1+\frac{n}{t}\right)^t\ \le\ e^n\,. $$ Is this true? I have constructed a proof of it (which would probably take some lengthy ...
0
votes
0answers
10 views

Inequality bounding $\|Qp\|_2^2$ where $Q$ is a Gaussian matrix and $p$ an independent Gaussian vector

I got the following expression, representing the r.h.s. of a tail bound on $Pr[\|Q^Tp\|_2^2 > tk]$, where the $k\times r$ matrix $Q$ has i.i.d. standard Gaussian entries, the vector $p$ is drawn ...
1
vote
4answers
75 views

Proving $\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca}$ for positive $a$, $b$, $c$

I'm at the end of an inequality proof that started out complex and I was able to simplify it to: $$\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca} \quad\text{where}\quad a, b, c ...
1
vote
1answer
44 views

If $x_1\geq x_2\geq…\geq x_n\geq0,$ $n\geq2$, $m\geq k\geq1$ then $x_1^mx_2^k+x_2^mx_3^k+…+x_n^mx_1^k\geq x_1^kx_2^m+x_2^kx_3^m+…+x_n^kx_1^m.$

Is the following inequality is true?? If $x_1\geq x_2\geq...\geq x_n\geq0,$ $n\geq2$, $m\geq k\geq1$ then $$x_1^mx_2^k+x_2^mx_3^k+...+x_n^mx_1^k\geq x_1^kx_2^m+x_2^kx_3^m+...+x_n^kx_1^m.$$ My ...
1
vote
1answer
34 views

Inequality in Sobolev Space ($L^p$ norm)

I want to find a constant $C$ that depends on the parameters $a$ and $p$ that satisfies the inequality $$\|f\|_p \leq a\|f'\|_1+C\|f\|_1$$ for all $f \in W^{1,1}(0,1)$. This is for arbitrary $p \in [...