Questions on proving, manipulating and applying inequalities.

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3
votes
1answer
54 views

Rewriting $|x-10|+|y-5|\leq 7$ so that absolute values disappear - Algebra

Equation 1: $|x-10|+|y-5|\leq 7$ I want to rewrite this equation into equations that do not have the absolute value. $|A|\leq b$ can be written as $A \leq b$ $A \geq -b$ I want to apply the ...
1
vote
2answers
27 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
5
votes
4answers
95 views

Show that $2^{\sqrt{2}}>1+\sqrt{2}$

Given that $\sqrt{2}>1.4$ and $(1+\sqrt{2})^5<99$, I need to show that $2^{\sqrt{2}}>1+\sqrt{2}$ From the given inequalities, I deduce that $(1+\sqrt{2})<\sqrt[5]{99}$ and ...
1
vote
0answers
19 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
3
votes
0answers
59 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
2
votes
2answers
49 views

O.I.M. polygon inequality

I am trying to prove an inequality which was used to prepare the Romanian O.I.M. team. I seem to lack ideas on how to tackle this problem. We take a convex polygon $P_1\ldots P_{n+2}$ and consider ...
0
votes
1answer
21 views

Is there an upper bound for expectation of product of two measurable function on a random variable?

I wonder if there is an useful upper bound for $\mathbb{E}_{x\sim p(x)}[f(x)g(x)]$ in the following form: $$ \mathbb{E}_{x\sim p(x)}[f(x)g(x)] \leq \mathbb{E}_{x\sim p(x)}[f(x)]\times xxxxxx $$ The ...
3
votes
0answers
19 views

The Hardy-Littlewood-Sobolev Inequality

Let $f:\mathbb R^n \to \mathbb C$, $n\ge 2$. I saw the line that the inequality $$ \left\| |x|^{-1} * |f|^2 \right\|_{L^\infty} \le C\|f\|_{L^{\frac{2n}{n-1},2}}^2 $$ with some constant $C>0$. Here ...
1
vote
2answers
75 views

how to solve the a,b,c inequality?

$a,b,c>0,a+b+c=3,$ prove that: $$\frac{a^3}{a^2+2b+c}+\frac{b^3}{b^2+2c+a}+\frac{c^3}{c^2+2a+b}\geq\frac{3}{4}$$
1
vote
2answers
51 views

$2^{m+1}-2^n\geq(m-n)^2$

So basically, I'm trying to see if this equality holds for large enough $n$ and such that $m\geq n$. However, Wolfram Alpha won't show me much, so I'm here to see if anyone has a solution for this. ...
4
votes
3answers
43 views

Multiplicating inequalities

I have two inequalities: $|x|\leq\sqrt{x^2+y^2}$ and $|y|\leq\sqrt{x^2+y^2}, \forall x,y \in \Bbb R$, can I multiply these inequalities to get $|xy|\leq x^2+y^2$? If yes, what is the justification? ...
2
votes
1answer
34 views

Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
-1
votes
1answer
29 views

Supremum vs Integral [on hold]

Let $h$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}h(t)\leq\int_{r}^{\infty}h(t)\frac{dt}{t} $$
1
vote
0answers
9 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. Let its CDF be given by $F(\theta) := ...
2
votes
1answer
47 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
1
vote
2answers
88 views

Prove that Euclidean distance in $\mathbb{R}^n$ is a distance

I'm trying to show that: $$\forall x,y\in\mathbb{R}^n, d(x,y)=\left(\sum_{i=1}^n(x_i-y_i)^2\right)^{1/2}$$ is a distance. However I have not proved Cauchy-Schwarz yet and I'm pretty sure I wouldn't ...
3
votes
1answer
78 views

Solving the trigonometric equation $\tan^2x+\cot^2x=2-\cos^{2014}(2x)$

I was solving the trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. $$ L.H.S=\tan^2x+\cot^2x =\tan^2x+\frac{1}{\tan^2x} ...
6
votes
1answer
338 views

Application of Jensen's inequality to $x^x+y^y+z^z$

Claim: If $x, y, z >0$ and $x+y+z = 3\pi, $ then $x^x + y^y + z^z > 81.$ My attempt: Let $f(w) = w^w$, so $f$ is convex on $(0, \infty).$ By Jensen's inequality, $f(x\frac{x}{3\pi}+ ...
5
votes
2answers
246 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.
1
vote
0answers
18 views

Martingale and quadratic variation inequality

I have the following inequality $$\mathbb{E}(\mid[M^{\Pi^m},M^{\Pi^m}]_T^{1/2}-[M^{\Pi^n},M^{\Pi^n}]_T^{1/2}\mid^p)\leq \mathbb{E}([M^{\Pi^m}-M^{\Pi^n},M^{\Pi^m}-M^{\Pi^n}]_T^{p/2}),$$ where $M$ is a ...
0
votes
2answers
57 views

For which $x, y\in\mathbb{R ^+}$ do we have $|xy-\frac{1}{xy}|\le|x-\frac{1}{x}|+|y-\frac{1}{y}|$?

I need to find all $x, y\in\mathbb{R^+}$ such that the following inequality holds. $$\Big| xy-\dfrac{1}{xy}\Big|\le\Big|x-\dfrac{1}{x}\Big|+\Big|y-\dfrac{1}{y}\Big|$$ If I substitute $x=2$ and $y=3$ ...
2
votes
0answers
15 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
10
votes
3answers
122 views

Struggling with an inequality: $ \frac{1}{\sqrt[n]{1+m}} + \frac{1}{\sqrt[m]{1+n}} \ge 1 $

Prove that for every natural numbers, $m$ and $n$, this inequality holds: $$ \frac{1}{\sqrt[n]{1+m}} + \frac{1}{\sqrt[m]{1+n}} \ge 1 $$ I tried to use Bernoulli's inequality, but I can't figure it ...
1
vote
1answer
20 views

Property of an almost additive sequence of functions

We say that a sequene of functions $\Phi=(\phi_n)_n$ is almost additive if there exists a constant $C > 0$ such that for every $n,m \in \mathbb{N}$ and $x\in \Lambda$ we have \begin{equation*} -C + ...
-2
votes
3answers
73 views

inequality question? [on hold]

$x= \sqrt{25}$ $y^2= 49$ What is the relationship between $x$ and $y$ ? $x>y$ $x<y $ $x\ge y $ $x\le y$ No relation can be established.
2
votes
1answer
35 views

A form of Nash's inequality, $\|f\|_2\le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta$

For $f\in \mathcal{S}(\mathbb{R})$ can anyone help me prove the following Nash inequality, $$\|f\|_2 \le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta.$$ I believe $\alpha$ and $\beta$ should be $2/3$ and ...
0
votes
0answers
18 views

Partial sum of exponential series strictly increases after certain step

While trying to show that partial exponential series evaluated at two different values are strictly increasing provided that sufficient number of terms are applied I stuck at a problem. Given two ...
0
votes
1answer
31 views

Help to manipulate and rearrange this inequality

I am working through a proof and I am trying to understand all of the steps. It uses one inequality to show another: Let $a_1, \ldots, a_k$ be given real numbers and $p_1, \ldots, p_k$ where $p_i \geq ...
1
vote
0answers
14 views

Intersecting Simplices with Normballs

Let $e$ be the vector of all ones 's consider the standard simplex $$\Delta_m:=\{x\in\mathbb{R}^m_+: \langle x, e\rangle=1\}.$$ Then the truncated simplex $\Delta_m^d$ is given as ...
3
votes
3answers
102 views

Is the inequality $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ true?

I'm having some trouble deciding whether this inequality is true or not... $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ for $x, y \in \mathbb{R}.$
0
votes
1answer
44 views

How prove that $6x - 6 < (1 + 4 \sqrt{x} + x) \log x$

How prove that $6x - 6 < (1 + 4 \sqrt{x} + x) \log x$ for $x>1$ $((1 + 4 \sqrt{x} + x) \log x)'=\textstyle 1 + 4 x^{-1/2} + x^{-1} + (2 x^{-1/2} + 1) \log (x)$
3
votes
1answer
52 views

Proving that if $0<a<b$, then $a<\sqrt{ab}<\frac{a+b}{2}<b$ [duplicate]

In proving these inequalities, what I did was to take each of the "<" relations and prove them. Is this a valid way of proving if we have got several inequalities as in this problem? So here is my ...
-1
votes
2answers
48 views

I need help to show that some function is nonnegative

This is a function of $x\in(0,1]$ $$(a_0+v_0 )\left(a_1+\frac{1}{K}\right)\left(a_0+(1-x) \frac{1}{K}\right)-(a_1+v_1 ) \left(a_0+\frac{1}{K}\right)(a_0+(1-x) v_0 )$$ The conditions are: ...
1
vote
0answers
14 views

Inner product inequalities with a diagonal matrix defining the inner product

This question came about from analyzing symmetric positive definite bilinear form decompositions and trying to understand what conditions would ensure certain inequalities hold. Suppose we have 3 ...
2
votes
1answer
38 views

Showing inequality in integrating polynomials

Let the polynomial $|P(x)| = a_0 + a_1x + \dots + a_nx^n$ have coefficients satisfying the relation $$ \sum_{i=0}^{n} a_i^2 = 1.$$ Prove that $$\int_{0}^{1} |P(x)| \ dx \leq \frac{\pi}{2}.$$ Show ...
-1
votes
1answer
34 views

Does these inequalities hold in General for probability distribution? [closed]

Let $Q(y)$ be a probability density of $y \in [-1,1]$. Then for $t> 0$, the inequalities are $\displaystyle \int_{0 \leq y <t} y^2 Q(y) \, dy \leq t^2 \int_{0 \leq y <t} Q(y) \, dy $. ...
0
votes
1answer
48 views

If $f \le g$ and f, g are integrable, decreasing functions, then$\int_{x}^{\infty} f \le \int_{x}^{\infty} g$?

If $f \le g$ and $f, g$ are integrable, decreasing functions, then $\int_{x}^{\infty} f \le \int_{x}^{\infty} g$? Intuitively, I suppose it holds, but I have not found any such theorem in the ...
3
votes
1answer
45 views

Comparing $\text{tr}(A^{-1})$ and $\text{tr}(A(B+A)^{-2})$ for pd $A$ and psd $B$

Suppose that $A$ is positive definite and $B$ positive semidefinite, both with dimension $n\times n$. Is there some inequality between $$ \text{tr}(A^{-1})\quad\text{and}\quad\text{tr}(A(B+A)^{-2})? ...
0
votes
1answer
48 views

Show that for any random variable $X$, and any $a > 0$, $P(|X| > a) \leq {EX^4 \over a^4}$.

Show that for any random variable $X$, and any $a > 0$, $$P(|X| > a) \leq {EX^4 \over a^4}.$$ Maybe I need to use Markov's Inequality, but I don't know how.
2
votes
1answer
42 views

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
2
votes
1answer
62 views

Inequality - Cauchy Schwarz

Let $a, b, c, d > 0 \in \mathbb{R}$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Show that: $S = \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq 4$ My approach: I used the ...
2
votes
1answer
54 views
+50

Checking logarithm inequality.

Which one of the following is true. $(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\ (c.)\ \log_{17} 298>\log_{19} 375 \quad \quad ...
1
vote
2answers
37 views

Prove the following inequality from jensen's inequality

By using the concave function $f(x)=\ln(x)$ inside the jensen inequality, I get the result: $$\sqrt[n]{t_1t_2\cdots t_n}\leq \frac{t_1+\cdots+t_n}{n}$$ Where $t_1,\ldots,t_n\in \mathbb{R}_{>0}$ ...
0
votes
3answers
49 views

Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.

I need help justifying that $|x-a|$ is continuous and non-differentiable at $x=a$. I would also like to prove that it achieves a minimum at $x=a$, but I do not know if that is already clear enough.
0
votes
1answer
20 views

Inequality for the difference of two products

Suppose $a_1,\ldots,a_k$ and $b_1,\ldots,b_k$ are complex numbers bounded in absolute value by $1$. Is it true that $$ \left| \prod_{i=1}^k a_i - \prod_{i=1}^k b_i\right|\leq \sum_{i=1}^k |a_i-b_i|? ...
1
vote
4answers
73 views

Show that $\frac{a+b}{1 + ab} < 1$ for $a,b < 1$

I'm currently solving a physics problem which comes down to show that $$ \frac{a + b}{1 + ab} < 1 $$ for $0 < a,b < 1$. I tried some numbers and it seems to hold. I tried replacing $1$ by ...
2
votes
1answer
61 views

Point on the Plane, a Triangle, and a Lower Bound of a Ratio Sum

Let $ABC$ be a triangle on the Euclidean plane. At which point $P$ on the plane does the ratio sum $\frac{PA}{BC}+\frac{PB}{CA}+\frac{PC}{AB}$ attain its minimum value? Prove also that, for any ...
2
votes
0answers
54 views

Upperbound for $\sum_{i=1}^n\frac{1}{x_i^2}$?

Suppose that $x_i>0$, $i=1,\ldots,n$. I'm looking for an upperbound (doesn't have to be particularly tight) of $\sum_{i=1}^n\frac{1}{x_i^2}$ in terms of some symmetric function of ...
0
votes
1answer
23 views

Need to find least value of an algebraic expression without helper constraints.

I am trying to solve this problem: Given $a>b>0$, find the least value of $a + \frac {1}{b(a-b)}$ Initially I was confused and things got better when I re-wrote $a + \frac {1}{b(a-b)}$ as ...
1
vote
0answers
38 views

Conditional expectation with Cauchy-Schwarz Inequality

Consider real-valued random variables $X$, $Y$, and $Z$; and a scalar, positive constant $k$. I want to prove the following \begin{equation} E[1|X+Y<Z<X+Y+k]E[X^2|X+Y<Z<X+Y+k]\ge ...