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 ...
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 ...
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} ...
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}+ ...
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.
1
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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 ...
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 $y=3$ ...
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
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
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.
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 $x, y \in \mathbb{R}.$
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 $x>1$ $((1 + 4 \sqrt{x} + x) \log x)'=\textstyle 1 + 4 x^{-1/2} + x^{-1} + (2 x^{-1/2} + 1) \log (x)$
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
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
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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? [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 $. ...
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
44 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
35 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
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 ...
2
votes
1answer
61 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
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 ...
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
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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
53 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
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0answers
34 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 ...
0
votes
1answer
26 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
2
votes
2answers
63 views

Cauchy-schwarz inequality problem

Can someone give me hint for the two problems. They are all based on cauchy-schwarz inequality. Just hint.
7
votes
0answers
113 views

why this polynomials is non-negative? [on hold]

show that this polynomials is non-negative $$f(x,y)=x^2(x^2-1)^2+y^2(y^2-1)^2-(x^2-1)(y^2-1)(x^2+y^2-1)\ge 0,\forall x,y\in R$$
3
votes
1answer
71 views

Minimize Value of Function with Constrain

Let $x$ and $y$ be real number with $xy\neq-1$ and $$\frac{x^7y+xy^7}{1+x^5y^5}=4$$ What is the minimum value of $x^2+y^2?$ I've been trying to solve it by Lagrange Multiplier but it's getting ...
1
vote
3answers
38 views

Is this problem wrongly built? Or is there a solution which I don't know how to arrive at?

I was solving a Cauchy-Schwarz's inequality based problem. Given that $x^2+y^2+z^2=1$ I am supposed to show that $x+y+z \le 6$. After struggling for a while I realised that I could solve this ...
0
votes
1answer
66 views

Vector inequation problem

$${A_i} = \left( {\begin{array}{*{20}{c}}{{A_{i1}}}\\{{A_{i2}}}\\ \vdots \\{{A_{in}}}\end{array}} \right),{B_i} = \left( {\begin{array}{*{20}{c}}{{B_{i1}}}\\{{B_{i2}}}\\ \vdots ...
-1
votes
2answers
66 views

How to show $\frac12\cdot\frac34\cdot\frac56\cdots\frac{99}{100}<\frac{1}{12}$? [closed]

How can I show that $$\frac12\cdot\frac34\cdot\frac56\cdots\frac{99}{100}<\frac{1}{12}?$$
0
votes
1answer
47 views

In Maple, how can I partially factor a lengthy symbolic expression (23 terms in 6 variables)?

I need to show that the following expression, $$a^3b-a^3c+a^3z+a^3x+a^3y-a^2bx+a^2by+a^2cx-a^2cy-a^2zx+a^2zy-a^2x^2+a^2y^2-abcz-abcx-aczx-acx^2+b^2c^2+2bc^2x+c^2x^2-b^2c-2bcx-cx^2,$$ is positive ...
0
votes
2answers
35 views

I want to know that $r^k \le C(1+r)^k$ holds.

When doe the inequality $$r^k \le C(1+r)^k$$ hold for $r>0$? I want to know the possible value of the real number $k$ so that the inequality holds. Here $C$ is independent of $r$.
0
votes
2answers
48 views

A tricky diophantine equation with factorials

I am being unable to solve this diophantine equation. Does anyone have any suggestions. Let $n$ and $m$ both be non-negative integers. Find all solutions to $$n(nm - 2)! = (n!)^m$$ How would one ...
3
votes
0answers
29 views

Fractional Part Inequality

How can I show that the following inequality holds when $x$ and $y$ are coprime positive integers greater than 2, and $r$ is an arbitrary rational number greater than or equal to $2$? ...
3
votes
1answer
96 views

if $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$ then minimum integer value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = ?$

Given $a,b,c,d > 0$, how do we find the minimum integer value of $n=\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}$ such that $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$.
4
votes
1answer
84 views

How to prove $a^{ka}+b^{kb} \geq a^{kb}+b^{ka}$

Let $0 \leq a \leq b, 0 \leq k \leq e$ $$ a^{ka}+b^{kb} \geq a^{kb}+b^{ka} $$ It's relatively easy to prove when $b \geq 1$(every non-negative $k$ satisfies this inequality), I can't prove the other ...