Questions on proving, manipulating and applying inequalities.

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0
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0answers
4 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) := ...
1
vote
0answers
28 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 ...
2
votes
3answers
76 views

If $x,y,z \in \Bbb{R}$ such that $x+y+z=4$ and $x^2+y^2+z^2=6$, then show that $x,y,z \in [2/3,2]$.

If $x,y,z\in\mathbb{R}$ such that $$x+y+z=4,\quad x^2+y^2+z^2=6;$$then show that the each of $x,y,z$ lie in the closed interval $[2/3,2]$. I have been able to solve using $2(y^2+z^2)\geq(y+z)^2$. Is ...
3
votes
4answers
76 views

prove $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

$a,b,c$ are positive real numbers.prove:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$ Additional info: We can't use induction. We should mostly use Cauchy inequality.other ...
1
vote
2answers
51 views

Prove $a^3+b^3+c^3\ge a^2+b^2+c^2$ if $ab+bc+ca\le 3abc$

if $a,b,c$ are positive real numbers and $ab+bc+ca\le 3abc$ Prove:$$a^3+b^3+c^3\ge a^2+b^2+c^2$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwarz and AM-GM because ...
1
vote
0answers
50 views

Show inequality for modified bilinear form

Let $\Omega_h$ denote to the domain that is bounded by a polygon, and $V_h$ denote to the space of all $v\in C^0(\Omega)$ such that $v_{|T}$ is linear on any (curved) triangle T and $v=0$ in the ...
5
votes
2answers
241 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.
7
votes
3answers
127 views
+50

$L^{2}$ Approximation Error of Fourier Series of Union of Disjoint Arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
3
votes
1answer
77 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} ...
1
vote
2answers
92 views

Prove $\sum_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$ if $x+y+z=1$

if $x,y,z$ are positive real numbers and $x+y+z=1$ Prove:$$\sum_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$$ where $\sum_{cyc}$ denotes sums over cyclic permutations of the symbols ...
2
votes
1answer
33 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 ...
1
vote
1answer
82 views

Prove $(x+y)(y+z)(z+x)\ge\frac{8}{3}(x+y+z)\sqrt[3]{x^2y^2z^2}$

if $x,y,z$ are positive real numbers,Prove:$$(x+y)(y+z)(z+x)\ge\frac{8}{3}(x+y+z)\sqrt[3]{x^2y^2z^2}$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwarz and AM-GM ...
1
vote
0answers
17 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 ...
1
vote
2answers
82 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 ...
1
vote
2answers
55 views

Prove or disprove $xyz+\frac{8}{27}\ge xy+yz+zx$ if $x+y+z=1$

if $x,y,z$ are positive and $x+y+z = 1$,Prove:$$xyz+\frac{8}{27}\ge xy+yz+zx$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwarz and AM-GM because I have background ...
6
votes
1answer
331 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}+ ...
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 ...
10
votes
3answers
119 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
112 views

If $x^2+y^2+z^2=3/2$, then $\sum\frac{x^2}{x(4x-3)+z^2+y^2} \le1+\frac{\sqrt2}2$

If $x$, $y$, and $z$ are real numbers satisfying $$ x^2 +y^2 +z^2 = 3/2 $$ then prove that $1+\frac{\sqrt{2}}{2} \geq$ the cyclic sum of $$ \frac{x^2}{x(4x-3)+z^2+y^2} . $$ I've tried ...
16
votes
1answer
332 views

On the inequality $\frac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
0
votes
1answer
31 views

Is the function (sum-of-squares) / sum convex on nonnegative input?

Let $$f \colon \mathbb{R}_{> 0}^n \to \mathbb R$$ be defined by $$f(x_1,\dotsc,x_n) = \begin{cases} 0 &\text{if }x_1 = \dotsb = x_n = 0\text{,}\\ \frac{\sum_i x_i^2}{\sum_i x_i} ...
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 ...
0
votes
1answer
50 views

Inner product inequality

Is there any inequality for $x^TAy$ where $x$ and $y$ are vectors and $A$ is positive definite matrix. For example: $x^TAy\ge k||x||||y||$ where $k$ is a coefficient of (min or max) eigenvalue of ...
0
votes
2answers
55 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 $y=3$ ...
5
votes
2answers
405 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
2
votes
1answer
44 views

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 ...
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}|| ...
2
votes
4answers
55 views

Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
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 ...
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
66 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.
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
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
1answer
42 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 $x>1$ $((1 + 4 \sqrt{x} + x) \log x)'=\textstyle 1 + 4 x^{-1/2} + x^{-1} + (2 x^{-1/2} + 1) \log (x)$
-1
votes
1answer
34 views

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

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 $. ...
37
votes
19answers
3k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
3
votes
4answers
155 views

Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+…+\frac1{2^n - 1} \le n$

$\dfrac{n}{2} \le \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n - 1} \le n $ I've Tried for hours but didn't got any striking idea. I don't have any efforts to show rather than induction. ...
5
votes
1answer
999 views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
13
votes
5answers
317 views

Proving $\frac{n^n}{e^{n-1}}<n!<\frac{(n+1)^{n+1}}{e^{n}}$ by induction for all $n> 2$.

I am trying to prove $$\frac{n^n}{e^{n-1}}<n!<\frac{(n+1)^{n+1}}{e^{n}} \text{ for all }n > 2.$$ Here is the original source (Problem 1B, on page 12 of PDF) Can this be proved by ...
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
265 views

Proving the expected value of the square root of X is less than the square root of the expected value of X

How do I show that $E(\sqrt{X}) \leq \sqrt{E(X)}$ for a positive random variable $X$? I may be intended to use the Cauchy-Schwarz Inequality, $[E(XY)]^2 \leq E(X^2)E(Y^2)$, but I'm not sure how.
-1
votes
2answers
47 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 ...
0
votes
2answers
645 views

Combining inequalities into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
2
votes
1answer
224 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 ...
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 ...
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 ...
8
votes
4answers
409 views

How to prove inequality $\frac{a}{a+bc}+\frac{b}{b+cd}+\frac{c}{c+da}+\frac{d}{d+ab}\ge 2$

Question: Let $$a,b,c,d>0,a+b+c+d=4$$ show that $$\dfrac{a}{a+bc}+\dfrac{b}{b+cd}+\dfrac{c}{c+da}+\dfrac{d}{d+ab}\ge 2$$ when I solved this problem, I have see following three ...
5
votes
3answers
73 views

Inequality $\frac 1{\sqrt{1+xy}}+\frac 1{\sqrt{1+yz}}+\frac 1{\sqrt{1+zx}}\ge \frac 9{\sqrt {10}}$

Let $x,y,z$ be non-negative real numbers such that $x+y+z=1$ , then is the following true? $$\dfrac 1{\sqrt{1+xy}}+\dfrac 1{\sqrt{1+yz}}+\dfrac 1{\sqrt{1+zx}}\ge \dfrac 9{\sqrt {10}}$$
0
votes
1answer
58 views

Cauchy Schwarz inequality and absolute value

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$ Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?