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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

3
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1answer
25 views

How to prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$?

Suppose $A(t)>0(t\ge 0)$, $a, b>0$, let $$ A'(t)\le aA-bA^2. $$ Prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$. Using Taylor formula $$ A(0)=A(t)-tA'(t)+o(t)\ge (1-ta)A(t) +tbA^2(t)+o(t). ...
0
votes
1answer
25 views

Inequality w.r.t determinant of a non-negative definite matrix.

I am reading a paper where the author mentioned the following property without proof. Neither can I prove it nor can I find the proof in various textbooks. For any non-negative definite (i.e. ...
-1
votes
1answer
38 views

Prove that $\binom{n}{k} < \frac{1}{e}\binom{en}{k}^k$ where $e$ is the euler's constant

$\forall n,k\in \mathbb{Z^+}$, prove that $\binom{n}{k} < \frac{1}{e}\binom{en}{k}^k$ where $e$ is the euler's constant. I tried my best to solve it and thought of expanding both sides but it helps ...
2
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0answers
18 views

Given $p_1^{a_1}+ \dots + p_r^{a_r} \leq n$, show that : $r \leq 2\sqrt{\frac{n}{\log(n)}}(1+o(1))$

I post here because I really don't succeed to prove this : Given $p_1^{a_1}+ \dots + p_r^{a_r} \leq n$, show that : $r \leq 2\sqrt{\frac{n}{\log(n)}}(1+o(1))$ We know that, if we note $q_i$ the $i$...
0
votes
0answers
11 views

Upper bound for multidimensional integral

I am stuck with the following inequality: $$\int_{\mathbb{S}^k}(2\pi)^{k+1}\prod_{j=1}^{k+1}\bigg(2\frac{\sin\big(b(x_j-y_j)\big)}{x_j-y_j}\bigg)^2\,d\sigma(y) \leq c(k)b^{2(k+1)}\bigg(\frac{1}{b}\...
-1
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2answers
49 views

Prove that $\frac{a^3 b}{(a+b)^4} \le \frac{27}{256}$

For $a$ and $b$ positive , prove that $$\frac{a^3 b}{(a+b)^4} \le \frac{27}{256}$$ I tried using weighted means and arrived at the result: $$\frac{a^3 b}{(3a+b)^4} \le \frac{1}{256}$$ I would be ...
0
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0answers
10 views

Exponents laws for Inequalities

I'm solving a problem involving inequalities and I arrived at a stage where I got $a^{\alpha+\beta}\leq b$ and I want to show $a<b$ where $a,b\geq0$, $a\leq b$, and $\alpha,\beta \epsilon (0,1)$....
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0answers
12 views

An integral inequality on a disc about twice derivatives.

Denote $f:\mathbb{R^2}\rightarrow\mathbb{R}$ has a continuous twice derivative on the disc $D=\{(x,y):x^2+y^2\leq1\}$, which satisfy $f(\partial D)=\{0\} $, where $\partial D=\{(x,y):x^2+y^2=1\}$, ...
2
votes
2answers
57 views

Prove by mathematical induction $4^n > n+1$

Prove the following by mathematical induction: $4^n > n+1$, for all integers $n ≥ 1$ Step 1: $n=1$: LHS $= 4^{(1)} = 4 $ RHS $= (1) + 1 = 2$ LHS > RHS. ∴ $P(1)$ is true. Step 2: Assume $P(k)$...
0
votes
1answer
13 views

Is the number of unique values in a matrix bounded by the product of the corresponding statistic for rows and columns?

Suppose we have an $r\times c$ matrix of natural numbers, $M$. Suppose also that the number of unique values in any given row is at most $n$, and the number of unique values in any given column is at ...
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0answers
18 views

If ln(2cosh(x)) > x, then what is ln(cosh(x)) >? [on hold]

Not sure how to find this inequality, any help would be appreciated
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0answers
16 views

Properties of “power inequalities”? [on hold]

Wikipedia defines power inequalities, as inequalities that use exponentiation, and lists a bunch of examples, but gives no names that can be Googled or proofs or citations except for one. Are there ...
2
votes
1answer
43 views

Proof by mathematical induction with the problem $40(2n)! ≥ 30^n$

I want to start by saying that I have far less trouble handling a non-inequality induction problem. I really don't understand the steps to take to get to the desired end product with these inequality ...
-1
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0answers
28 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
22 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
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2answers
63 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
2answers
31 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) \...
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0answers
45 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
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0answers
45 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
62 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
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3answers
38 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
55 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
37 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
17 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
74 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
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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
42 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
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0answers
20 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
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0answers
44 views

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

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
62 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
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1answer
26 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
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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
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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 ...
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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
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1answer
19 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
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1answer
36 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
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0answers
116 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
48 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
26 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 .
-1
votes
1answer
42 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
27 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
75 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
33 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}...