Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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2answers
27 views

Prove $\sum_{k=1}^{n-1}\frac{(n-k)^2}{2k} \geq \frac{n^2\log(n)}{8}$

As the title says, prove $$\sum_{k=1}^{n-1}\frac{(n-k)^2}{2k} \geq \frac{n^2\log(n)}{8},$$ for $n>1$. This inequality is from Erdős, "Problems and results on the theory of interpolation". ...
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1answer
68 views

Generalize the inequality $(a_3-a_1)^2+(a_3-a_2)^2+(a_2-a_1)^2\ge b(a_1^2+a_2^2+a_3^2)$, $b>0$ with $a_1+a_2+a_3=0$.

To find a $b>0$ that satisfies the inequality of the title, one can use the condition $a_3=-a_1-a_2$, to obtain that $b=3$. I've been trying to find a way to show that $$(a_n-a_1)^2+(a_n-a_{n-1})^2+...
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0answers
10 views

From square norm to norm in an inequality

If I have an inequality involving the norm squared of kind: $$||\text{something}||^2<||\phi||^2$$ then can I say that in general $$||\text{something}||<a_1||\phi||$$ with $0<a_1<1$?
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1answer
96 views

Integration $\int_{\frac{1}{10}}^{\frac32}\frac1k da$

This is the question I tried to solve and got answer as wrong 3 times. $\color{blue}{\text{I feel the correct answer should be 0.23 but it says correct is 0.557}}$. My try: I split the integral at $a=\...
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0answers
20 views

Minimum number of inequalities characterizing a point

Suppose I have a point $x \in \mathbb{R}^d$. My question is, what the minimum number of inequalities $\phi_i^\top x \geq y_i$ is that is needed to uniquely characterize $x$, where $\phi_i \in \mathbb{...
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0answers
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On an inequality related to Selberg's sieve

I am reading section 9.3 of GTM206, which is about the derivation and application of Selberg's sieve, and I was stuck at proving $$ \sum_{n\le x}\left(\sum_{d|(n,P_z)}\lambda_d\right)^2\le\sum_{d_1,...
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2answers
86 views

Prove or disprove: $7 \lt \sqrt{3} + \sqrt{27}$ [duplicate]

In an admission test to enroll in a Earth's Science Bachelor Degree course there is this question: Sort in increasing order $7$, $\sqrt{47}$ and $\sqrt{3} + \sqrt{27}$. Now, I know that $7=\sqrt{49}$...
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0answers
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Upper bound by diagnoal matrix

I got stuck in proving the following inequality. Let $Y∈R^{n×p}$ with nonzero singular values $(n>p)$, $Λ=diag(λ_1,λ_2,⋯,λ_n)$ with $λ_1>λ_2>⋯>λ_n>0$. $e_n=[0,0,⋯,0,1]^⊤∈R^n$, \begin{...
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0answers
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inequality about finite sums

This is a problem from Selected Problems in Analysis, Makarov, B. M. et al. Suppose $(a_n:n=1,\ldots,n)$ is a sequence of real numbers. For any $0\leq m\leq n$, define $$\widetilde{S}_m=\sum^m_{k=1}...
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0answers
25 views

Inequality for probabilities of events, not all of which are independent [closed]

Here is the question. If A_1, A_2, ... , A_n are not all independent, prove that $$- (1-1/n)^n \leq P(A_1 A_2 ... A_n) - P(A_1) P(A_2) ... P(A_n) \leq (n-1)n^{-n/(n-1)}$$ I'd appreciate any help.
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1answer
19 views

union of two spaces [closed]

I would like to prove that the union of two subspaces is equal to a subspace. note $X$ a measurable space and $f,g,h : X \rightarrow \mathbb{R}$ are mesurable functions with also $r ,t >0$ I would ...
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1answer
24 views

union of two inequalities

I would like to see if for two functions if it was possible to prove that for three functions $f, g, h$ that the union of $| f -g| \geq t$ and $|g -h| \geq r$ where $ t,r \geq 0 $ would be equal to $|...
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0answers
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An Inequality concerning double integral

Prove $$ \frac{\pi}{8}\left(1-\cos\frac{2}{\pi}\right)\le \iint_D \sin (x^2)\cos (y^2) dxdy \le \frac{\pi}{8}(1-\cos 1),$$where the integrating region $D$ is enclosed by $x=0, y=0$ and $x+y=1$. We ...
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2answers
73 views

Almost there?? Help with inequality.

I was recently doing a combinatorics problem from MMO given in book Arthur Engel. I might have solved it (though unlikely) as I arrived at following inequality:- Both $x$ and $y$ are positive. the ...
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2answers
29 views

Please help me prove following inequality

$(1+x)^{\theta}(1+y)^{1-\theta}-x^{\theta}y^{1-\theta}\geq1$ where $x,y\geq 0$ and $\theta\in(0,1).$
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3answers
80 views

Minimum Value of $x_1^2+y_1^2+x_2^2+y_2^2-2x_1x_2-2y_1y_2$

Minimum Value of$\quad$ $x_1^2+y_1^2+x_2^2+y_2^2-2x_1x_2-2y_1y_2$ subject to condition $x_1,y_1$ and $x_2,y_2$ lies on curve $xy=1$. It is also given that $x_1\gt0$ and $x_2\lt0$ My Approach: $AM\...
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0answers
28 views

Reverse of Young's convolution inequality

I tried to solve the following exercise with no success. Any suggestions? Let $p\in (1,+\infty)$. Show that for every $e>0$ exist $f \in L^p(\mathbb{R}^n)$ and $g\in L^1(\mathbb{R}^n)$ such that $$|...
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0answers
28 views

Condition for 1 to be between the roots of a function.

Q. Find the range of value of x for which 1 lies between the roots of the equation. $3y^2-(3sinx)y -2cos^2x=0$ By IVT, we know that $f(1)<0$, $3y^2-(3sin1)y-2cos^2(1)=0$ should have roots which are ...
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1answer
60 views

I have an equation I need to prove but I don't really have an idea how to: [closed]

$$ \left||y|\,x^2-|b|\,a^2\right| \leq 2(|x-a|+|y-b|), \quad\forall x,y,a,b \in(-1,1) $$ It is part of a much longer proof and I'm not entirely sure that it is necessary (or reasonable).
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1answer
43 views

how to prove this inequality equation?

I am trying to prove the following inequality for $R>1$ R is a constant: $$\Bigl(1-\frac{1}{n+1}\Bigr)^R\ge1-\frac{R}{n}$$ I am not sure if it's correct or not, but I tried to prove it with ...
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1answer
29 views

Does the following inequality hold - the inner product divided by the product of norms?

Let $\cdot$ denotes the dot product and $||\boldsymbol{x}||$ denotes the $L^2-$norm of the vector $\boldsymbol{x}$. Suppose $\boldsymbol{a,b,c}$ are vectors in $\mathbb{R}^3$. Does the following ...
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1answer
59 views

Two conjectures about the prime counting function : $\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$

Let $x\geq 100$ then we have as conjecture : $$\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$$ I have tested at $x=100$ to $x=5000000000$ without any counter-example. The first fact : It seems that the ...
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1answer
40 views

A nonnegative function $f(a,b)\geq 0$

Let $c$ be a fixed positive real umber such that $c\geq 1.$ If $a\geq c^m, b\geq c, $ where $m$ is any positive integer, then is $$f(a,b)=c(ab+1)(a-c^m)(b-c)+c(ab+1)(a-c^m)(b+1)+c^m(b-c)(ab+1)(a+1)-(...
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1answer
55 views

Existence of real numbers satisfied that …

Are there some $a,b,c$ such that $\dfrac{a}{b^2-ac}=\dfrac{b}{c^2-ba}=\dfrac{c}{a^2-bc}=\dfrac{1}{2019}$¿ My student asked me this question? And I didn't have any direction? I wonder what is role of 1/...
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1answer
56 views

Prove that $\frac{1}{kn} + \frac{1}{kn + 1} + \dotsb + \frac{1}{kn + n - 1} > n \left(\sqrt[n]{\frac{k+1}{k}} - 1 \right)$

Let $k,n \in \mathbb{Z}^+$ with $n > 1$. Prove that $$\frac{1}{kn} + \frac{1}{kn + 1} + \dotsb + \frac{1}{kn + n - 1} > n \left(\sqrt[n]{\frac{k+1}{k}} - 1 \right)$$ I roughly observe that AM-...
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1answer
28 views

Are the following systems of inequalities the same?

Suppose $x, y \in \mathbb{R}$, and $\mathcal{S_1}$ is a system of inequalities: \begin{align*} \mathcal{S_1} &= \begin{Bmatrix} x - y \geq 1\\ -x + 2y \geq 1\\ 3x - 5y \geq 2 \end{Bmatrix}\\ &...
3
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1answer
49 views

Proving Young's Inequality for Inner Product Spaces

I am trying to prove the following inequality, I have proven it for when my field is $\Bbb R$ but I am having trouble when my field $\Bbb F = \Bbb C$ $$|\langle u,v\rangle| \leq \frac{\lambda^2}{2} \...
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0answers
26 views

why the largest value of $k$ for any number of the form $𝑚^𝑘<B$ is $\lfloor\log B \rfloor$

Given an integer $n>2$, let $B=\lceil\log^5n\rceil$. I am trying to understand why the largest value of $k$ for any number of the form $m^k \le B$, $m\ge2$ is $\lfloor\log B \rfloor$. Logarithms ...
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0answers
19 views

what are 3 solutions to the problem: x ≤ 50 [closed]

the boy cannot answer more than 50 questions
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0answers
26 views

Is there a mistake in the convergence proof in Watkins technical note of Q-learning?

I am reading the technical note about Q-learning written by Christopher J.C.H Watkins(Watkins, 1992). The proof of Theorem B.4 quite confuses me. The author assumes that $|\mathcal R_x^i(a) - \mathcal ...
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1answer
25 views

How to prove this inequation?

How to prove this inequation. $$(\frac{n}{m})^m\le C_n^m\le(\frac{en}{m})^m$$ where $0<m\le n$ and $C_n^m$is the Combinatorial number. I tried to prove the left part by using the convexity of $\ln$,...
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0answers
29 views

Show $\operatorname{tr}(X^T X Y^{T}\Lambda Y\Phi) \geq \lambda_n\operatorname{tr}(X^T X Y^{T} Y\Phi)$

Let $X,Y\in R^{n\times p}$, $\Lambda = \operatorname{diag}(\lambda_1,\lambda_2,\cdots,\lambda_n)$, $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\gt 0$, $\Phi=\operatorname{diag}(\mu_1,\mu_2,\cdots,\...
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1answer
17 views

Inequalities: How to show $W_t \ge M_t$ for $t \in [0, T]$?

$\mu$ and $\lambda$ are known and $\mu > \lambda > 0$. For $t \in [0, T]$, we have \begin{align} \frac{dW_t}{dt} = \mu W_t - \mu H_t, H_t > W_t , W_T = 0 \\ \frac{dM_t}{dt} = \lambda M_t - \...
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1answer
20 views

Does the following norm inequality hold?

Let $\times$ denotes the cross product and $\cdot$ denotes the dot product. $\boldsymbol{e',e'',e''',r',r'',r'''}$ are vectors in $\mathbb{R}^3$. Does the following inequality hold? $$||\boldsymbol{e'}...
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1answer
27 views

Problem with proving an inequality by mathematical induction [duplicate]

I wonder how to solve this problem using mathematical induction: $1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}} > \sqrt{n}, n\geq2$ I showed true for $n = 2$ Assumed true ...
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1answer
20 views

Equalities involving power of natural numbers

If I have $N\in\mathbb{N}$ and $\alpha>0$ is the following true? $$(2N)^\alpha-(2N+2)^{\alpha}<0$$ and $$-(2N+1)^\alpha+(2N+3)^{\alpha}>0$$ I think yes since $2N<2N+2$ and $2N+3>2N+1$.
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35 views

Find $c_{*}$ such that ${\sum_{i=1}^{n}c_{i}a_{i}\ge c_{*}\sum_{i=1}^{n}a_{i}}$ where $c_{i}$ are positive and increasing, $a_{i}\in\mathbb{R}$

Given ${\displaystyle \sum_{i=1}^{n}c_{i}a_{i}}$ where $0<c_{1}<c_{2}<...<c_{n}$ and $a_{i}\in \mathbb{R}$. Is it possible to find $c_{*}$ such that ${\displaystyle \sum_{i=1}^{n}c_{i}a_{i}...
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0answers
17 views

Inequality about absolute value and tolerance, as applied to real analysis

It seems that the following statement about absolute value inequalities are very useful in real analysis. I can prove it using case analysis, but would expect a more direct proof, such as via the ...
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3answers
108 views

$\mathbb{P(|\mathbf{X}|\leq 2) = 1}$ if $\mathbf{X}$ has bounded moments?

Suppose that a random variable $\mathbf{X}$ has bounded moments: $\mathbb{E}(\mathbf{X}^k)\le k^{2}2^k$. I would like to show that $\mathbb{P(|\mathbf{X}|\leq 2) = 1}$. I am considering using Markov's ...
3
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4answers
81 views

Approximate $\log_{10}$ values without a calculator

I've got this problem: $1,000,000^{{1,000,000}^{1,000,000}} < n^{n^{n^n}}$ What is the first positive integer value of n for which this inequality holds? I managed to reduce it to this: $6+\log_{...
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1answer
28 views

Simple cyclic inequality, similar to Shapiro's

The numbers $x_1$, $x_2$ and $x_3$ are poritive. Prove, that $$\sum_{cyc}\frac{x_1}{x_1 + x_2} \leq \frac{3}{2}$$ Or show a counterexample. It looks simple, but I'm having a hard time proving / ...
3
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3answers
75 views

Taylor's Theorem and natural logarithmic inequality

Use Taylor's theorem to prove that, for $x>0$. $$ \ln x+\frac{1}{x}-\frac{1}{2 x^{2}}<\ln (x+1)<\ln x+\frac{1}{x} $$ The RHS is indeed obvious: algebraic manipulations yield to show $e^{\frac{...
0
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1answer
41 views

$\sum_{i=1}^{n}{\frac{x_i}{\sqrt{1-x_i}}} \geq \sqrt{\frac{n}{n-1}}$ for $x_i \in \mathbb{R}_{++}$ such that $\sum_{i=1}^{n}{x_i}=1$

Let $x_1,\dots, x_n\in \mathbb{R}_{++}$ such that $\sum_{i=1}^{n}{x_i}=1$. Prove that $$ \sum_{i=1}^{n}{\frac{x_i}{\sqrt{1-x_i}}} \geq \sqrt{\frac{n}{n-1}} $$ I tried using AM-GM and Cauchy-Schwarz ...
2
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1answer
66 views

Proving that $-\frac{\pi}{2}\le\int_{-1}^{1}\arctan (x)dx\le \frac{\pi}{2}$

I have the following question: Prove that: $$-\frac{\pi}{2}\le\int_{-1}^{1}\arctan (x)dx\le \frac{\pi}{2}.$$ I know that the function is odd and therefore, the given integral is 0, and the ...
0
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1answer
38 views

$\left|\frac{x^n}{n+1}\right|\leq |x^n|=|x|^n$

Is it right to say that $\left|\frac{x^n}{n+1}\right|\leq |x^n|=|x|^n$ if $n$ is a natural number $\geq 0$? This is not an isolated question but it is related to this my Solution verification of $\...
5
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2answers
81 views

Checking $( \binom{n}{k} - \binom{n-k}{k}) / \binom{n}{k} < \frac{k^2}{n}$

Let $n$ and $k$ be integers with $2\leq k \leq n$. I want to check the inequality $$\left( \binom{n}{k} - \binom{n-k}{k} \right) / \binom{n}{k} < \frac{k^2}{n}$$ which occurs in a paper I am ...
0
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1answer
22 views

$X\sim\text{Bin}(n,p)$. Lower bound on $\Pr(X=np)$ that is a function of $m=np$, and not dependent on $n$

Consider the binomial distribution with parameters $n$ and $p$, $\text{Bin}(n,p)$. It has mean $m=np$, and the probability that a random variable $X\sim\text{Bin}(n,p)$, equals its expected value, is $...
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0answers
76 views

Prove $\frac{x-\ln x-1}{x+1}\ge \frac{e^{x-1}-x}{e^x}$ where $x>0$.

Prove $$\frac{x-\ln x-1}{x+1}\ge \frac{e^{x-1}-x}{e^x}$$ where $x>0$. This is ture, as WolframAlpha shows. But it's hard to prove.
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1answer
44 views

In a C*-algebra , if $a\leq b$, then $a^{\alpha}\leq b^{\alpha}$, for $0<\alpha\leq1$ [closed]

I need help to find a reference to prove the the following theorem: Theorem 6.9. Let $A$ be a $C^*$ algebra. If $a,b\in A_+$ and $a\leq b$ then $a^\alpha\leq b^\alpha$ for $0\leq \alpha\leq 1$. On ...

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