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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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Misconception about proof by cases

Let $x,y \in \mathbb{R}$, s.t. $y>0$. Then $|x|<y \iff -y < x < y$. I'm confused about how to join the cases in this proof into a single interval. By definition, $|x|= x$ if $x \geq0$ ...
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1answer
22 views

How to prove the following norm inequality

If $x$, $y$ and $a$ are vectors in $\mathbb{R}$, is the following inequality true? $||y - a|| - ||x-a|| \ge (\frac{x-a}{||x-a||})^T(y-x)$ I cannot come up with a counterexample, but I also do not ...
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2answers
33 views

elementary inequality involving exp and ln

Suppose $x$ and $y$ are real numbers satisfying $x\geq 0$ and $y\geq 1$. Is the following inequality true? $xy \leq e^x + y \ln (y)$ If so, is there a reference or proof?
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3answers
24 views

Solving Logarithmic Inequality

Solve $\log$ $\left(5^{\frac{1}{x}}+5^{3}\right)<\log$ $6$ $+$ $\log$ $ $$5^{\left(1+\frac{1}{2x}\right)}$ It is possible to simplify the inequality using the quotient rule property of logarithms, ...
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1answer
42 views

Proving $F(k)=(16X^2- 24X+18)k^2-11kX+1\geqq0$

Given that $1\leqq X\leqq k$, prove that$$F(k)=(16X^2-24X+18)k^2-11X+1\geqq0.$$ Original problem: Given that $a$, $b$, $c$ are three non−negative numbers and $a+b+c=3$, prove that$$(2+a^2)(2+b^2)(2+c^...
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1answer
55 views

Showing that for $x \ge 7$, $x\# \ge x^2+x$

Let $x\#$ be the primorial of $x$. I am trying to show that if $x \ge 7$: $$x\# \ge x^2+x$$ Is there a straight forward argument? Here's what I came up with: (1) From Bertrand's Postulate, for ...
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1answer
36 views

There is no function in $L^1$ whose Fourier transform is 1/log(x)

I need to prove that there is no odd function on $L^1$ whose Fourier transform is a continuous function $g:\mathbb{R}\to\mathbb{R}$ such that $g(\xi)=1/\log(\xi)$ for $\xi\geq 2$. I am suggested to ...
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1answer
50 views

Can I split this inequality like this?

Recently I had solved this number theory problem but after I solved it I was a bit uncertain whether my approach was correct so I approached AOPS. The problem is : Prove that $[x] + [y] + [x + y] \...
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1answer
53 views

Prove that : $\exists m \in ({\pi/4},{\pi/3})$ such that

I need to prove that $\exists m \in ({\pi/4},{\pi/3})$ such that $$\int_\frac{\pi}{4}^{\pi/3}\frac{1}{x\tan x}dx≤\frac{\pi}{12}\left(\frac{\ln(m)}{m^2}-\frac{\ln(m)}{\sin^2 (m)}+\frac{1}{m\tan m}\...
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2answers
48 views

How to show that $ab+bc+ca\le \frac34$

Let $a,b$ and $c$ be positive real numbers such that $(a+b)(b+c)(c+a) = 1$ , hen show that $$ab+bc+ca\le \frac34$$ I believe I need to use AM-GM inequality and use the fact $(a+b)(b+c)(c+a) = 1$ ...
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18 views

Multivariate Kolmogorov distance bounded by Wasserstein distance

I'm trying to find a bound for the multivariate Kolmogorov distance in terms of the Wasserstein distance. Denoting by $F$ and $G$ two cumulative distribution functions (cdf) on $\mathbb{R}^n$ the ...
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3answers
82 views

Prove inequality $1+ a^2 + b^2+ c^2+ 4\,abc \geq a + b+ c+ ab+ bc+ ca$

Given that $a,\,b,\,c$ are $3$ non$-$negatve numbers$,$ prove$:$ $$1+ a^{\,2}+ b^{\,2}+ c^{\,2}+ 4\,abc\geq a+ b+ c+ ab+ bc+ ca$$ Let$:$ $X= a+ b+ c$$,$ we have to prove$:$ $$\left ( \frac{1}{X^{\,3}}-...
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2answers
53 views

to prove $x^2 + y^2+1\ge xy + y + x$

$$x^2 + y^2+1\ge xy + y + x$$ $x$ and $ y$ belong to all real numbers my attempt $(u-2)^2\ge0\Rightarrow \frac{u^2}{4}+1\ge u $ let $u=x+y\Rightarrow \frac{(x+y)^2}{4}+1\ge x+y$ $\Rightarrow (x+y)...
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1answer
40 views

Triangle Inequality and absolute value

I'm curious if the triangle inequality (and reverse triangle inequality) still hold if we only take the absolute value of one term. For example: $$||a| - b| \le |a - b|$$ If $b \ge 0$, then $|b|$ is ...
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1answer
69 views

Find : $\lim_{n\to\infty}\sum_{k=0}^n\frac{\binom{n}{k}}{n^k(k+1)}$ [on hold]

I'm try to find this lim $\lim_{n\to\infty}\sum_{k=0}^n\frac{\binom{n}{k}}{n^{k}(k+1)}$ Is this limits can be done by integral !? Or inequality Someone help me hints me Thanks!
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2answers
35 views

Infinity norm is actually a norm : triangle inequality

I have to prove the following assertion : Let $V$ be a finit dimentional vector space with dimension $n$ over the field $K$ which is the field of real numbers or complex numbers. Let the map defined ...
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1answer
45 views

lower bound for $\vert a+b\vert^\alpha$

Let $a,b$ be two positives real numbers and $\alpha >1$, by convexity we know that $$\vert a+b\vert^\alpha\le 2^{\alpha-1}(\vert a\vert^\alpha+\vert b\vert^\alpha).$$ But is it possible to have a ...
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20 views

How to prove/verify the following inequality consisting of polynomial and exponential holds?

How to verify the inequation $C^{H}e^{-\lambda^2 C} \leq \delta$ holds for $C = \frac{1}{\lambda^{2}} (2H log \frac{H}{\lambda^2} + log\frac{1}{\delta})$? Namely, we treat $H \geq 1$, $\lambda > 0$,...
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3answers
41 views

Discrete math - The ceiling of a real number x, denoted by$ ⌈𝑥⌉$, is the unique integer that satisfies the inequality

I have a discrete math question below with a solution written by my teacher. I'm really lost as to what answers I'm trying to find exactly. I don't understand how the teacher got 1 and -1 for the ...
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1answer
41 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). ...
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1answer
33 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. ...
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1answer
47 views

Inequality involving $e$

$\forall n,k\in \mathbb{Z^+}$, prove that $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$ where $e$ is the euler's constant. I tried my best to solve it and thought of expanding both sides but ...
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2answers
59 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 $p_1^{a_1}+ \dots + p_r^{a_r} \leq n$, $p_i$ distinct prime numbers and $a_i \in \mathbb{N} $, $a_i \geq 1$, $r \in \mathbb{N} $, ...
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2answers
53 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 ...
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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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15 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 $\partial D=\{(x,y):x^2+y^2=1\}$, ...
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2answers
68 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 $P(k)$...
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1answer
16 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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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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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 ...
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1answer
49 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 ...
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0answers
47 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!
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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 ...
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2answers
66 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}...
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2answers
34 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
49 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 ...
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46 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 ...
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2answers
65 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 ...
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3answers
39 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 ...
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1answer
57 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}}^{...
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1answer
45 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}...
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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. ...
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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
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1answer
76 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 ...
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1answer
45 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 ...
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0answers
21 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 ...
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1answer
81 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 ...
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1answer
26 views

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

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.
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1answer
48 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 ...