Questions on proving and manipulating inequalities.

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1answer
49 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 ...
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2answers
66 views
+50

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 ...
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3answers
36 views

Proving inequality involving real numbers [closed]

$x, y$ 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.
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0answers
23 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^* ...
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0answers
22 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 ...
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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 ...
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0answers
33 views

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

Let $a,b,c>0$ and $a+b+c=1$. Prove $$\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ca}{b+ca}}\le\frac32$$
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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
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1answer
48 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
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1answer
28 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 - ...
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3answers
48 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 $(n−1)^2−1$.
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1answer
32 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
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1answer
53 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 ...
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0answers
22 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 ...
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1answer
47 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
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2answers
45 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
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1answer
32 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)) := ...
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0answers
28 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
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2answers
260 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 ...
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6answers
204 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 ...
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4answers
66 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 $$\liminf_{p\to\infty} X_p^{\frac{1}{p}}\le \liminf_{p\to\infty}E(X_p)^{\frac{1}{p}}$$ ...
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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 ...
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1answer
15 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. ...
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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) $a^2+b^2=c^2=(x+y)^2=x^2+y^2+2xy$ ...
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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 ...
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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
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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 ...
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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.
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0answers
129 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 $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
1
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1answer
29 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
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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
61 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 ...
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2answers
39 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 $m\leq n$ ...
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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 ...
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0answers
38 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} ...
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7answers
205 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.
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3answers
59 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 ...
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3answers
87 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?
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2answers
27 views

Proving an inequality by induction.

Question: If $n\ge2$, prove that $n!/n^n\le(1/2)^k$, where k is the greatest integer $\le(n/2)$. My answer: It is clear that statement is true for $n=2$ Inductive step: $n\ge2 \Rightarrow ...
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3answers
113 views

Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$

Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $$a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$$ The inequality can be written in the condensed form ...
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1answer
49 views

Case of equality in Bernoulli's inequality

How can I prove that the following equality holds only for $x=0$? $$\binom{n}{2}x^2 +\cdots+ \binom{n}{n}x^n=0\text{ when }x\gt-1\text{ and }n\gt1$$
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2answers
31 views

Prove that $a-a\cdot\left(1-\frac{1}{a}\right)^{x} \leq x$ given $x,a\in[1,\infty)$ [closed]

How can be proven that: $$a-a\cdot\left(1-\frac{1}{a}\right)^{x} \leq x$$ while $a\geq 1$ and furthermore $x\geq 1$.
3
votes
1answer
61 views

Determining the sign of a term

I have a problem in proving the sign of a term. It is as follows: $$x=\dfrac{1-a}{b_1b_2-a}+1,\qquad y=\dfrac{1-a}{b_1-a}+\dfrac{1-a}{b_2-a},\qquad z=x-y$$ with $0<b_1<1,\quad ...
2
votes
3answers
102 views

How can I prove inequality? [closed]

Let $x$, $y$, $z$ be real nonegative numbers so that $x+y+z=1$. Prove that $$8 \leqslant (x + 1) (y + 2) (z + 3) \leqslant 12.$$ I posted here ...
3
votes
2answers
71 views

Inequality - L'Hôpital's rule, logarithms

Let's consider a function: (EDIT here: the number "2" was lacking) $\mathcal{B}(x) = \frac{2}{x^2} \big( (1+x) log(1+x) - x \big)$ I need to prove inequality: $ (1 + \frac{1}{3}x)\, \mathcal{B}(x) ...
4
votes
3answers
98 views

$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $

could anyone give me any hint how to prove this ? $$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $$ just came acroos this expression in my book.
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3answers
35 views

Solve the inequality

$2\left|x+\frac{1}{4}\right| < 9$ I keep trying to figure this out and I can't. I tried to split up the absolute value first. $2 (x+\frac{1}{4}) < 9$ $2+x+\frac{1}{4} < 9$ subtract $2$ ...
1
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1answer
43 views

Proof of inequalities

I am having trouble with the following exercise: Let $p$ and $n$ denote positive integers. Use the formula $$b^p - a^p = (b-a)(b^{p-1}+b^{p-2}a + \cdots +ba^{p-2} + a^{p-1})$$ to show that $$n^p ...
0
votes
0answers
8 views

Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$

I have been stuck at this problem for a while :( Given $\mathbf{A}\in\mathbb{S}^{p\times p}, \mathbf{A}\ge 0,\mathbf{A} \text{ symmetric}, \mathbf{b}\in\mathbb{R}^n,\mathbf{c}_i\in\mathbb{R}^p\forall ...
1
vote
1answer
58 views

$ A\sin(x + a) = B\sin( x + b)$ implies $a = b$?

Given $A,B,a,b$ are constants and x are variable. $A,B$ don't equal $0$. Does $A\sin(x + a) = B\sin(x + b)$ for all $x$ implies $a = b + 2n\pi$? I only managed to show that $a-b = n\pi$ by putting ...