Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

2
votes
0answers
35 views

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$ [on hold]

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as ...
5
votes
1answer
72 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|]$$ This question is a re-posting of An expectation inequality. I can ...
1
vote
2answers
32 views

Inequality $\left|z+w\right|\geq\left||z|-|w|\right|$ when $|w|\leq A|z|$.

Let $z,w\in\mathbb C$ and $|w|\leq A|z|$ for $A>0$. I want estimate from below $|z+w|$. I proceeded as follows. Since $$\left|z+w\right|\geq\left||z|-|w|\right|$$ and $-|w|\geq -A|z|$, I write ...
-3
votes
1answer
36 views

The integer part of $x+1$ is the integer part of $x$ plus $1$ [on hold]

How do you solve the proof: If $x$ is a real number, then: $[x+1] = [x] + 1$. For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please ...
2
votes
0answers
14 views

$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$

Let $c_i\in\mathbb R$, $a_i\geq0$ with $\sum_{i=1}^n a_i=1$, prove $$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$$ This inequality comes from there, when $X$ is ...
1
vote
2answers
35 views

Solve for real value of $x$: $|x^2 -2x -3| > |x^2 +7x -13|$

Here I have a question: Solve for real value of $x$: $$|x^2 -2x -3| > |x^2 +7x -13|$$ I got the answer as $x = (-\infty, \frac{1}{4}(-5-3\sqrt{17}))$ and ...
2
votes
5answers
102 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
0
votes
1answer
34 views

About inequality $\sum_{k=1}^n |a_k|^2 \lessgtr \sum_{k\neq s} |a_k| |a_s|$

Let $a_k$ a sequence of complex number. We have $$\left(\sum_{k=1}^n |a_k|\right)^2 \geq \sum_{k=1}^n |a_k|^2$$ It is a basic fact because $$\left(\sum_{k=1}^n |a_k|\right)^2 = \sum_{k=1}^n |a_k|^2 + ...
2
votes
1answer
33 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
0
votes
0answers
15 views

Solving nonlinear inequality that involves norm2 operator

I have an equation of the form $$ \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \norm{\mathbf{Z} \left[ \sum_{n = 0}^{N - 1} (-1)^n \psi^n \mathbf{C}^n \right] \mathbf{q} }^2 \leq |p|^2, $$ where ...
5
votes
2answers
72 views

Proving that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$ using derivatives

Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$ This isn't hard problem. I have already solved it in following way: Let ...
1
vote
2answers
38 views

prove using Lagrange multipliers that for $x,y>0,\space n\in \mathbb N,\space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2 $

I have been asked to prove using Lagrange multipliers that for \begin{equation*} \space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2,~x,y>0,~n\in \mathbb {N} \end{equation*} I am familiar with the ...
1
vote
1answer
48 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
0
votes
1answer
27 views

Can the following inequality be directly infered?

If we have a condition as follows $$\log(1+\mathbf{h}_2^* \mathbf{\Sigma} \mathbf{h}_2) \leq \log(1+\mathbf{h}_1^* \mathbf{\Sigma} \mathbf{h}_1)$$ where $\Sigma$ is positive semi definite matrix ...
0
votes
3answers
35 views

Proving inequality involving real numbers [on hold]

$x, y$ are real positive numbers. Let $m$ be the smallest number among $x, y + \frac{1}{x}, \frac{1}{y}$. How to prove that $m \le \sqrt{2}$? I really don't know how to start.
0
votes
0answers
20 views

A smart way to bound this function and get rid of covariance matrix

I have the following function which I am trying to bound as follows $$A({\bf h},\Sigma)= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - \rho_1 \rho_2^* ...
1
vote
0answers
20 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
0
votes
1answer
25 views

How to show this is decreasing

I'd like to show $$\sum_{i=1}^n \frac{1}{i((n+1)-i)} $$ is decreasing for n>1, which is Cauchy product of $$\sum_{i=1}^n \frac{1}{i}$$ Numerical computation until n=50 shows it's decreasing but I ...
-1
votes
0answers
30 views

How to solve this inequality using AM-GM? [duplicate]

Let $a,b,c>0$ and $a+b+c=1$. Prove $$\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ca}{b+ca}}\le\frac32$$
0
votes
1answer
30 views

Chebiyshev Inequality

In proving the Chebyshev inequality in Probability theory an important step is to observe that: $P((|x-E(x)|≥a))=P(|x-E(x)|^2≥a^2)$. It is assumed that X has a moment of order 2. Can somebody help ...
1
vote
1answer
46 views

Proving $(n+1)c^{1/(n+1)} - nc^{1/n} \le 1$ from first principles

Is it possible to prove that \begin{align*} (n+1)c^{1/(n+1)} - nc^{1/n} \le 1 \qquad c \in \mathbb{R}_+, n \in \mathbb{N} \end{align*} using only elementary techniques? (No calculus, no appeasement to ...
1
vote
1answer
24 views

Upperbound a logarithmic expression that has a covariance matrix

Let $\Sigma$ be a $2\times 2$ covariance matrix and ${\bf h}$ a vector of complex values entries. $$A= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - ...
0
votes
3answers
42 views

The inequality $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$

How one can prove the following statement: $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$ Tried so far: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.
0
votes
1answer
31 views

Hilbert's inequality for $\left|\sum_{n,m}a_n \bar a_m\right|$.

We know that, an Hilbert's inequality states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. Then write an inequality ...
1
vote
1answer
52 views

Convex function inequality for Euclidean norm: $\|(f(x_1),\cdots,f(x_n))\|_2\leq f(\|x\|_2)$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a positive, convex, continuous function such that $f(0)=0$. (If you wish you can also suppose $f$ to be monotone increasing.) I would like to prove or to ...
0
votes
0answers
20 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
1
vote
1answer
46 views

Inequalities using AM-GM

Use the AM-GM inequality to prove $(5xy + 6y)^3$ ≥ $1215xy^3$ for all real numbers x, y > 0. Not sure if I was on the right track but so for my understanding is: since there is a power of 3 on the ...
3
votes
2answers
44 views

Proving $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$

Let a,b,x,y be positive reals. Prove $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$ I don't have any olympic background, so I may be missing some standard trick. The ...
1
vote
1answer
31 views

Maximum singular value of a matrix valued function

Let $f$ be an analytic matrix-valued function, $\Lambda(A)$ be the spectrum of $A$ and $\sigma_1(A)$ the maximum singular value of $A$. It is known that $$\Lambda(f(A)) = f(\Lambda(A)) := ...
1
vote
0answers
27 views

$ \| v\| \leq Ce^x\| w\| \quad \left(x\rightarrow 0 \right) $ if $\| w\|=0$

Let $v\in \mathbb{R}^J$ and $J\in \mathbb{N}$. I have the follow inequality: $$ \| v\| \leq \| w\| + x \quad \left(x\rightarrow 0 \right) $$ If $\|w\|\not=0$ we can find a $C$ positive constant with ...
3
votes
2answers
258 views

Absolute Value Equation

Please help me with this! $$x^3+|x| = 0$$ Now one solution is clearly $0.$ We have to find the other solution (i.e, $-1$) $$Solution:$$ CASE $1$: If $x<0,~|x| = -x$, we can write $x^3+|x| = 0$ as ...
13
votes
6answers
196 views

show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$

Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$) show that $$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$ if we use Closed-form expression ...
1
vote
4answers
54 views

A Stochastic Limiting Inequality Proof

Let $(X_p)_{p\ge 0}$ be a sequence of non-negative random variables with finite mean for each $p\ge 0$. Then $$\liminf_{p\to\infty} X_p^{\frac{1}{p}}\le \liminf_{p\to\infty}E(X_p)^{\frac{1}{p}}$$ ...
4
votes
1answer
47 views

Struggling with inequality involving a bunch of binomial coefficients

I want to find a lower bound on $n$, i.e. isolate $n$, or more realisticly, approximate $n$ that satisfies the following : $$ {n \choose k}\left( 1 - \frac{{n \choose \frac{n-1}{2} - k}}{{n \choose ...
0
votes
1answer
14 views

$y^tv<0,z^tv<0 \text{ unsolveable} \Leftrightarrow \exists \lambda\geq 0: y=-\lambda z$

I am trying to show: Let $y,z\in\mathbb{R}^n$ and $z\neq 0$. Then $y^tv<0,z^tv<0 \text{ unsolveable} \Leftrightarrow \exists \lambda\geq 0: y=-\lambda z$. '$\Leftarrow$' is trivial. ...
1
vote
4answers
30 views

Inequality for sides and height of right angle triangle

Someone recently posed the question to me for the above, is c+h or a+b greater, without originally the x and y lengths. I used this method: (mainly pythagorus) $a^2+b^2=c^2=(x+y)^2=x^2+y^2+2xy$ ...
2
votes
2answers
27 views

Find the minimum value of k $(k \in I)$ for which the equation $e^x =kx^2$ has exactly three real solution.

Problem : Find the minimum value of k $(k \in I)$ for which the equation $e^x =kx^2$ has exactly three real solution. My approach : We apply log on both sides $x=2\ln(k x^2)$ $\Rightarrow ...
1
vote
3answers
53 views

Is the following inequality true $(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$?

Let $a,b,c$ be all positive integers greater than $1$. If $$a>b^2$$ and $$a^3-b^6> 3c$$ Is this the following inequality true?: $$(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$$ I tried to ...
1
vote
1answer
7 views

When $f \mapsto \lambda\int_{\mu}^{x}f(t)dx$ is contration map

$$\Phi\colon C[a,b] \to C[a,b], f \mapsto \lambda\int_{\mu}^{x}f(t)dt$$ I want to find $\lambda,\mu$ such $\Phi$ is contraction map, so $$|\lambda\int_{\mu}^{x}f(t)dt| < q|f(x)|$$ on $[a,b]$ for ...
1
vote
3answers
48 views

Proof for Inequality

Can somebody tell me what is the name of the inequality: \begin{equation} \sum_{t=1}^T \frac{1}{\sqrt{t}} \leq 2\sqrt{T} \end{equation} or any hint/link how to prove above? Thanks.
6
votes
0answers
73 views
+50

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(B)$, where $B$ is the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any positive integer $k$, ...
1
vote
1answer
27 views

Integer division through multiplication by reciprocal

Please help me to understand (prove) why the following statement is true. For any natural number $w > 0$ and divisor $b \in \left[ 1, 2^w \right)$, if we define a natural number $inv(b)$ such that ...
1
vote
1answer
31 views

Prove that $\left|\sum_{r\neq s}u_r\overline u_s\csc\pi(x_r-x_s)\right|^2\leq\sum_{r}\left|\sum_s\overline u_s\csc\pi(x_r-x_s)\right|^2$

On a paper that I'm studying, it is written, without another: "By Cauchy's inequality $$\left|\sum_{r\neq s}u_r\overline u_s\csc\pi(x_r-x_s)\right|^2\leq\sum_{r}\left|\sum_s\overline ...
3
votes
1answer
58 views

Inequality of elementary symmetric polynomials

Let $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$ with $\lambda_i>0$ for $i=1,2,3,4$. Let $$\sigma_k(\lambda)=\sum_{1\leq ...
1
vote
2answers
33 views

A quick question on inequalities with floor function.

For any $x\in\mathbb{R}$, denote $\lfloor x\rfloor:=\max\{n\in\mathbb{Z}\mid n\leq x\}$, i.e. the floor function. Show that for any $x\in\mathbb{R}$ and $m,n\in\mathbb{N}$ with $m\leq n$ ...
0
votes
1answer
24 views

Solution range for an inequality

Given $a\in(\frac{1}2,1)$ such that $$0<(1-a)<\frac{1}2<1<\frac{3}2<(1+a)<2,$$ is there an $x\in\Bbb R$ such that ...
0
votes
0answers
37 views

To prove an inequality with square root term

now I have a formulation which has a square-root in the objective. I need to show the following inequality holds, to strength the Lagrangian relaxation: \[\sqrt{C-K_jp}-\sqrt{C} \leq p(\sqrt{A-K_j} ...
6
votes
7answers
200 views

Which term is bigger? $\sqrt[102]{101}$ or $\sqrt[100]{100}$

Which term is bigger? $\sqrt[102]{101}$ or $\sqrt[100]{100}$ I tried AM-GM but didn't succeed.
-1
votes
3answers
58 views

How to show simple inequality of fractions

If $$\frac {a}{a+b}<\frac{a'}{a'+b'}$$ then how can I show that $$\frac {a}{a+2b}<\frac{a'}{a'+2b'}\ \forall\ a,b,c>0$$ I tried puitting in a constant k so $$\frac ...
1
vote
3answers
85 views

Proof of inequality problem from Spivak, ch 1, 16 b)

If $$4x^2+8xy+4y^2 \ge 0$$ it follows that $$4x^2+6xy+4y^2 \gt 0 $$ unless $x=0$ and $y=0$. How can I prove that?