Questions on proving and manipulating inequalities.

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Inequality involving $f(x)=x^p$ with $0<p<1$

Let $f(x)=x^p$ with $0<p<1,p\in \Bbb{R}$ defined in $[0,+\infty[$. Knowing that $f$ is crescent, show that, $\forall a,b\in \Bbb{R}$, $$(|a|+|b|)^p\leq|a|^p+|b|^p$$
1
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1answer
37 views

Prove simple inequality

Prove that for any $a, b \in \mathbb{R}$ the following inequality holds true: $$a^2+b^2+16\geq ab+4a+4b$$ I tried some reductions, but with no success. Any hint appreciable! Thanks in advance.
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3answers
23 views

How to determine different absolute value equation cases?

This is a question from this post. From: $$ |3x|=\left\{ \begin{align} 3x & \text{ , if }x\geq 0 \\ -3x & \text{ , if }x <0 \end{align} \right\} $$ $$ |4x+1|=\left\{ \begin{align} ...
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0answers
56 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
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2answers
16 views

Inequalities and floors.

I've been presented with a question that I actually don't understand. The question is: Given $$\lfloor a\rfloor\leq a<\lfloor a\rfloor+1$$ Write an inequality for $\lfloor a\rfloor$ I'm fine ...
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0answers
16 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
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1answer
34 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
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3answers
40 views

Fail in the reasoning?

I'm trying to prove that this is a true statement: $$\forall x\in\mathbb{R}\hspace{22mm}2<x<3\implies3<\frac{2x}{x-1}<4$$ My reasoning is if $3<\frac{2x}{x-1}$ adding to that the ...
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0answers
17 views

subtraction, multiplication and division of inequalities

if a>b and c>d I know that a+c>b+d for all real a,b,c,d what are the corresponding rules for the subtraction, multiplication and division of two inequalities and for what values of a,b,c,d are they ...
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0answers
45 views

Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
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0answers
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proving an inequality [on hold]

If $0 < c < 1$, prove that there is a positive number $h$ such that $\displaystyle 0 < c^n < \frac{1} { 1+nh}$ . With the same number $h$, formulate a non-trivial inequality for $(nc)^n$.
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0answers
11 views

Is my proof correct : $\sum\limits_{i=0}^b a_in^i = \Theta(n^b)$

I want to prove : $$\sum\limits_{i=0}^b a_in^i= \Theta(n^b), a_o,a_1,...,a_b\in R, a_b \ne 0$$ So first, I prove that $\sum\limits_{i=0}^b a_in^i= O(n^b)$. First, Let's find $c$ $$ 0\le a_o + a_1n + ...
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1answer
30 views

Maximal inequality for a sequence of partial sums of independent random variables

Let $X_n$, $n=1,2,3,...$ be a sequence of independent (not necessarily identically distributed) random variables, let $S_n=\sum_{i=1}^nX_i$. Prove the following maximal inequality: for all ...
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0answers
31 views

How prove there exsit $C_{0}$ such $e^{-\pi^2 k^2\beta}-(\cos{(k\alpha\pi)}+\pi k\sin{(k\alpha\pi)})>C_{0}$

Question: let $$\delta_{\alpha\beta}(k)=e^{-\pi^2 k^2\beta}-(\cos{(k\alpha\pi)}+\pi k\sin{(k\alpha\pi)})$$ if $\alpha\in Q$ and $\alpha>0,\beta>0,k\in N^{+}$, show that there ...
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2answers
53 views

Alternate proof for $a^2+b^2+c^2\le 9R^2$

As I studying geometric inequalities, one of those famous inequalities is $$a^2+b^2+c^2\le 9R^2$$ I did some research and I found that there is a proof (not exactly the this inequality but an useful ...
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1answer
16 views

generalization of midpoint-convex

Let f : (a,b) → R is a midpoint-convex function (I didn't say continuity). Here I'd like to verify following inequality ""directly"". f( (x1+x2+x3)/3 ) ≤ (f(x1)+f(x2)+f(x3))/3 .. I can easily ...
2
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1answer
45 views

euler triangle inequality proof without words

today i was studying geometric inequalities and I saw this inequality $$R \ge 2r$$ unfortunately the book did not provided any prove or further explanations. So I just did a little research about it. ...
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1answer
24 views

determinant in terms of quadratic form evaluated at a point

Say $A$ is a $n$ by $n$ positive definite matrix. Let $b$ be a column vector in $\mathbb{R}^n$. Consider the following quantity: $$b^TA^*b$$ where $A^*$ is the cofactor matrix of $A$. A simple ...
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2answers
22 views

Proving of Inequalities

How to prove: If $a>0$ and $b>0$, and $a^2>b^2$, then $a>b.$ I've tried different methods but I really can't prove this one. Thank you for your help!
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1answer
10 views

Inequality in Evans PDE section 5.7

I'm stuck in the proof of the Compactness Theorem in Evans PDE 2nd edition book. On page 287, last line, how do you get the inequality $$ \epsilon ...
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2answers
38 views

The sum of binomial coefficients up to $k\le n/4$ does not exceed the $k$th coefficient

How would you prove the following (for when $k\leq\frac{n}{4}$)? $$\sum_{i=0}^{k-1} \binom ni \le \binom nk$$
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1answer
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Inequality proof (AM-GM, Cauchy-Schwarz?)

I'm not able to proof this inequality for all positive real numbers. Also I need to know when does the equality hold. $ \frac{1}{a^{2}-ab+b^2}+\frac{1}{b^{2}-bc+c^2} +\frac{1}{c^{2}-ca + a^2} \leq ...
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0answers
13 views

Application of Frobenius inequality [on hold]

What are the most interesting applications of the Frobenius inequality?
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4answers
49 views

Intuitive way of solving this inequality: $x^2>y^2 \wedge x>0 \Rightarrow x>y$

I have to prove that $x^2>y^2 \wedge x>0 \Rightarrow x>y$. I decided to do this: $x^2>y^2 \Leftrightarrow \sqrt{x^2}>\sqrt{y^2}\Leftrightarrow |x|>|y| \Leftrightarrow x>|y| \vee ...
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1answer
43 views

Analog of the Chebyshev's inequality

Lets consider two random variables $\xi$ and $\eta$, which satisfy such conditions: $$E\xi = E\eta = 0,~~~ D\xi = D\eta = 1;$$ $$\operatorname{cov}(\xi, \eta) = \rho.$$ How can we prove? $$E\bigl( ...
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0answers
43 views

How do I find which value is greater? [on hold]

Q) Indicate which of the following quantities contains the greatest number of atoms and why? ( a ) $6.022\times 10^{23}$ Nickel atoms ( b ) $25.0\space\text{g}$ of Nitrogen ( c ) ...
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2answers
83 views

Show that $\lvert \cos (1)\lvert $ + $\lvert \cos (2)\lvert $ + $\lvert \cos (3)\lvert \geq \frac{3}{2}$

Show that $\lvert \cos (1)\lvert $ + $\lvert \cos (2)\lvert $ + $\lvert \cos (3)\lvert \geq \frac{3}{2}$. I've been trying to figure out an analytic way of showing this is true for a while now, but ...
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2answers
20 views

Show inequality for the number of different prime factors

We consider the function $k(n) $ that represents the number of the different prime factors of $n$.We want to show that for $n>2$ $$k(n) \leq \frac{\log n}{\log \log n}(1+O((\log \log n)^{-1})) ...
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0answers
31 views

Inequality in $\mathbb{Z}^2$

Let $k=(k_1,k_2)\in\mathbb{Z}^2$. Denote $|k|\leqslant n$ when $|k_1|,|k_2|\leqslant n$. I need help to show $$|\sum_{k+l+m=0}_{|k|,|l|,|m|\leqslant ...
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1answer
41 views

When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn. For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy ...
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0answers
18 views

Expand function using Maclaurin's series(infinite form)

Expand the function f(x)=log(1+x) in powers of x in an infinite series stating the validity of such expansion for x belonging to (-1,1]. The question actually asks to show that cauchy's remainder or ...
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2answers
40 views

Square roots in inequalities

If I have the inequality $$x^2 > b^2,$$ is this always equal to $\;|x| > |b|\quad ?$
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2answers
43 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
3
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1answer
66 views

three variables inequality

$x,y,z$ are positive real numbers such that $$xyz=1$$ Prove that $\dfrac{1}{y(x+y)}+\dfrac{1}{z(y+z)}+\dfrac{1}{x(z+x)} \geqslant \dfrac{3}{2}$.I have no idea how to solve this problem. I've tried ...
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0answers
13 views

How to compare the dimensions of two blocks?

Consider I have the dimensions of two boxes (length x width x height), what would be the easiest way to compare them, allowing N% error? For example, 20x30x40 would be the same box as 40x30x20, so ...
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1answer
26 views

LQ decomposition and inequalities

Suppose I have an element-wise inequality: $Ax \ge b$, where $A$ is a rectangular matrix with full row rank, and $x$ and $b$ are appropriately sized column vectors. I need to check if the inequality ...
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4answers
55 views

Prove $\frac{n}{n+1}<\frac{n+1}{n+2}$

How can we prove the following inequality: $$\frac{n}{n+1} < \frac{n+1}{n+2}$$ I understand how to do proof by inductions and contradictions, but I am getting stuck on this question. My proof ...
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2answers
49 views

Easy question on integrals

I have some problems understanding this inequality: $$\int_{x-\varepsilon x}^x \vartheta\left(t\right)dt \leq \vartheta\left(x\right)x\varepsilon$$ where $\vartheta\left(x\right)$ is the Čebyšëv (or ...
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1answer
56 views

I'm new to proving inequalities. How does one prove this?

If a, b, and c are non-negative real numbers and $a + b + c = 2 $, prove that $ 2 \ge a^2 b^2 + b^2 c^2 + c^2 a^2 $
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1answer
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Prove $\sin(x)< x$ when $x>0$ using LMVT

According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that ...
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2answers
55 views

Regarding the $\sigma (n)$ function.

This question relates to Robin's Inequality. Is $\sigma{(n^2)}$ < (2 n) $\sigma{(n)}$ ? For what integer values of n is this satisfied?
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1answer
55 views

Why does $2x_1x_2y_1y_2 \leq x_1^2y_2^2+x_2^2y_1^2$?

When I tried to prove the triangle inequality $|z_1+z_2| \leq |z_1| + |z_2|$ algebraically for complex variables $z_1$ and $z_2$, I came across this inequality and found that this is always true no ...
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2answers
107 views

Proving the bound $\left ( 1+\frac{x}{n} \right)^n \leqslant 3^x$, $\forall x \in \mathbb{R^+}$

I'm trying to directly prove the above bound. I have tried expanding it $$\left ( 1+\frac{x}{n} \right)^n = \sum_{k\geqslant 0} \binom{n}{k}\left ( \frac{x}{n}\right)^k$$ $$= \sum_{k=0\dots n ...
1
vote
1answer
29 views

if $x,y,z\ge0$ Prove $x^3+y^3+z^3+3xyz\ge xy(x+y)+yz(y+z)+zx(z+x)$

if $x,y,z\ge0$ Prove $$x^3+y^3+z^3+3xyz\ge xy(x+y)+yz(y+z)+zx(z+x)$$ Things I have done: At first i thought this can be solved by AM-GM but I was wrong. I tried to move $RHS$ to $LHS$ and result ...
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2answers
30 views

Relation of length of a projection of a point to a line

In the given figure, can it be said that $x \leq a + b - d$?
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1answer
34 views

if $a$ is a real number that $a\neq1$ and $a^5-a^3+a=2$,Prove $3<a^6<4$

if $a$ is a real number that $a\neq1$ and $a^5-a^3+a=2$,Prove $3<a^6<4$ Things I have done: using AM-GM for $a\ge0$ $$a^5+a\ge2a^3$$ The equality occurs only in $a=1$ which is not true as ...
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1answer
21 views

When can I expect the derivative of an inequality to always hold true?

Let $f,g$ be real-valued functions. Suppose I have an inequality $$f(x)>g(x)$$ for $x\in D$, where $D\subseteq\mathbb{R}$ is some domain. After some "tinkering" I see that we can not always expect ...
2
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2answers
52 views

show $\frac{1}{15}< \frac{1}{2}\times\frac{3}{4}\times\cdots\times\frac{99}{100}<\frac{1}{10}$ is true

Prove $\frac{1}{15}< \frac{1}{2}\times\frac{3}{4}\times\cdots\times\frac{99}{100}<\frac{1}{10}$ Things I have done: after trying many ways and failing, I reached the fact ...
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0answers
22 views

Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
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4answers
66 views

Proving $4^n > n^4$ holds for $n\geq 5$ via induction.

I know that it holds for $n=5$, so the first step is done. For the second step, my IH is: $4^n > n^4$, and I must show that $4^{n+1} > (n+1)^4$. I did as follows: $4^{n+1} = 4*4^n > 4n^4$, ...