Questions on proving, manipulating and applying inequalities.

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142 views

Nonnegative solution of a linear system

Given three collections of parameters $\epsilon_1 > ... > \epsilon_N$, $(a_1,...,a_{N-1})$ and $(b_1,...,b_N)$ that satisfy the following conditions $\forall i, a_i \geq 0, ...
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27 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
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36 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
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43 views

Why is the bold text true??

Ok I have stared at this for nearly 30 minutes now, and can't figure out why the bold text is true. Problem: If $z \in \mathbb{C}$ and $\mathrm{Re}(z^n) \ge 0$ for $n \in \mathbb{N}$, show that $z ...
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24 views

Finding feasible solution of inequalities in math software

I have a Math problem where I have some true statements, and I want to know if there is a feasible solution to an equation. I would like to know how to do that in either Matlab or Mathematica. The ...
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43 views

Truth of an inequality involving differentials

Is the following inequality true? $$ s\frac{\partial \frac{\partial f(s,t)}{\partial s}}{\partial t}-\frac{\partial f(s,t)}{\partial t}>0 $$ Given that $f(s,t)$ is a monotonically-decreasing ...
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76 views

Bound for variance of maximum of normal random variables

Suppose that $(X_1,\ldots,X_n)=\mathbf{X}\sim N(\mathbf{0},\Sigma)$ is an $n$-dimensional normal random vector. I want to show the bound $$ \text{Var}\left(\max_{i\leq n} X_i\right)\leq \max_{i\leq n} ...
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52 views

Showing $\frac{1+c}{a+b}\leq \varphi$, when $\frac{1}{1+b}\leq a\leq 1$, $\frac{c^2}{a+c}\leq b\leq 1$ and $0\leq c\leq 1$

The title pretty much says it all. Let $\varphi\triangleq\frac{1+\sqrt 5}{2}$ be the golden ratio. Let $a,b,c$ be some non-negative numbers such that: $\frac{1}{1+b}\leq a\leq 1$ ...
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70 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and ...
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119 views

Proving the inequality $\frac{\mathrm{arccot} 2\sqrt{2}}{\pi\log\zeta(3)}-\frac{\log^2(1+e^{-\pi})}{\pi}>\frac{131e^2+422e-1151}{222e^2+279e-757},$

I have come across the following inequality in my studies $$\frac{\text{arccot}2\sqrt{2}}{\pi\log\zeta(3)}-\frac{\log^2(1+e^{-\pi})}{\pi}>\frac{131e^2+422e-1151}{222e^2+279e-757},$$ where ...
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40 views

Norm Inequality (Vinogradov Notation)

I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality: $$ ...
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48 views

How prove $ \frac{2}{\sqrt3}F \geq s-1 $ for convex quadrilateral?

Let $Q$ be any convex quadrilateral of area $F$ and semiperimeter $s$. Suppose that length of any diagonal of $Q$ $ \geq$ length of any side of $Q$ $\geq 1$ How prove $ \frac{2}{\sqrt3}F \geq s-1 ...
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47 views

Reference for an inequality between $|x+y|^p$, $|x|^p$, $|y|^p$, and $|x-y|^p$

I am interested in an inequality for real numbers $x,y$ and $1<p<\infty$ that it should say something like $$ |x+y|^p -|x|^p - |y|^p \leq (1-2^{1-p})|x-y|^p. $$ Is this inequality (or ...
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62 views

Two-leg games in Elo rating for football teams

Do you know Elo rating for association football? It is a numerical estimation of strength of football clubs using simple mathematical formula based past results allowing predictions for the future. ...
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83 views

Looking for an existing proof for a property of triangles

In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote. Lemma 1: Let the nodes ...
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37 views

An upper bounded for partial Fourier sum

Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, ...
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57 views

On the average length of the Steiner net for $n$ randomly chosen points in the unit square

$n$ points are randomly chosen in the unit square with respect to the uniform measure. What is the average length $L$ of the associated Steiner net (tree of minimum length through each of the $n$ ...
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55 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
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98 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...
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99 views

Jensen's inequality: proof by using linear functions

Here's an extract from Stochastic Calculus for Finance Volume 1 by Shreve. I don't understand the statement that says a convex function is the maximum of all linear functions that lie below ...
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65 views

“Massaging” inequalities to prove them (esp. in contest math like the IMO/Putnam)?

What's the contest inequality solving technique where you do something like representing each side as the function of some sequence and replacing the max/min terms of the sequence with their average, ...
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132 views

Proving probability inequality (how to return to Chebychev?)

Supposing $X$ is a random variable, $X>0$, $E[X^2]<+\infty$, $\lambda \in (0,1)$, I have to prove the following inequality. $$P[X>\lambda E[X]] \geq (1-\lambda)^2 \frac{E[X]^2}{E[X^2]}$$ ...
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40 views

Find Max : $P=\frac{1}{x^2+1}+\frac{4}{y^2+4}+\frac{3z}{9+z^2}$

Let $x,y,z>0$ and satisfying $3xy+yz+2zx=6$ Find Maximum of this expression: $P=\frac{1}{x^2+1}+\frac{4}{y^2+4}+\frac{3z}{9+z^2}$
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83 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
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42 views

Inequality in matroid theory

Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct ...
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55 views

Miklos Schweitzer 2013, Strong lower bound on sumset $|A+qA|$,

Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: $$|A+qA|\ge (q+1)|A|-C_q.$$ This is a problem from ...
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45 views

Proving a quantity negative.

For $j\in{1,2,3}$ let $x_j,y_j \in R$ be nonzero and let $v_j=x_j+y_j$. Suppose that following holds: $$x_1x_2x_3=−y_1y_2y_3 \quad \text{and} \quad x^2_1+x^2_2+x^2_3=y^2_1+y^2_2+y^2_3$$ nd that ...
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94 views

Prove the inequality for composite numbers

Is it true that $c_m+c_n$ $>$ $c_{m+n}$ for all $m$, $n$ $\in$ $\mathbb{N}$? Though the result seems true, I can't get a solution. Even the bounds on $c_n$ obtained from Prime Number Theorem ...
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53 views

How prove this inequality $\sum_{cyc}\frac{1}{(ka+(k+1)b+(k+2)c)^2}\le \frac{1}{(k+1)^2(ab+bc+ac)}$

Inequality: Let $a,b,c\ge 0$. and $k\ge\dfrac{\sqrt{21}-3}{3}$ show that $$\sum_{cyc}\dfrac{1}{(ka+(k+1)b+(k+2)c)^2}\le \dfrac{1}{(k+1)^2(ab+bc+ac)}$$ This problem is from: ...
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62 views

How to prove or disprove this statement?

For $n\geq1$, $$\int_{0}^{\pi/2}(\theta \sin\theta)^{n+1}d\theta>\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$$ . It is hard to find $\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$ so I have no ...
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91 views

Minimum value of an algebraic expression

What is the minimum value of ($x,y,z$ are positive): $$\sum_{cyc} \frac{(x+y)^2}{x^2+6 x y+5 y^2+4 y z}$$ With the values I have tried it does seem to be smallest has been $\frac34$. However, I can't ...
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119 views

Proof of integral inequality

How does one prove (without making use of any approximations whatsoever) the following inequality: $$\int_1^2 \left(\ln(x)\right)^{2013}dx\leq\dfrac{1}{2^{2013}}.$$
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Proving AM-GM via $n \cdot (a_1^n + a_2^n + \dots + a_n^n) \ge (a_1^{n-1} + a_2^{n-1} + \dots + a_n^{n-1}) \cdot (a_1 + a_2 + \dots + a_n)$

I want to prove the arithmetic–geometric mean inequality. To prove that, I need the following inequality: Suppose that $n$ is an integer which is greater than or equal to $1$ and $a_1, a_2, \dots, ...
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676 views

Normal sequences and Montel's Theorem

I am currently stuck on an exam question involving normal sequences and Montel's theorem: Give two examples of non-constant normal sequences one in the $(a)$ unit disk $\mathbb{D}$ and one in $(b)$ ...
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33 views

Inequality similar to Hoeffding

I have a coin with heads probability $p_1$. I toss it $n_1$ times. Let $\hat{p}_1$ be the empirical heads probability. Then we know from Hoeffding that $$P\left( \left|\hat{p}_1-p_1 \right| \geq ...
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58 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
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37 views

Muirhead's Inequality (software?)

I just started learning about inequalities: Schur's, Karamata's, Muirhead's, etc... They are beautiful but it seems that in the case of more than two variables, some of the computations become a ...
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41 views

Solving an inequality involving sum of floors

Suppose I want to find $t_{critical}(u)$, the least $t\in\mathbb{R}^+$ for a given $u\in\left(0\ldots\dfrac{1}{s}\right]$ satisfying $$f(t)=\lfloor rt\rfloor x+\lfloor s (t-u)\rfloor y + y > h$$ ...
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40 views

When this inequality true?

If $a$ and $b$ are non-negative integers and $c$ and $d$ are non-negative real numbers, for what values is the following inequality true? $\log((a+b)!) - \log(a!b!) \ge(a+b) \log(c+d) - (a \log(c) ...
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94 views

Showing that a logarithmic inequality holds

Given $0 < x_1 < x_2 < x_3 < x_4 < 1$, how can I show that the following inequality holds: $$ \frac{1}{R(x_1, x_3)}+\frac{1}{R(x_2, x_4)}<\frac{1}{R(x_1, x_2)}+\frac{1}{R(x_3, x_4)} ...
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64 views

Help me find a less tedious way to prove this theorem about inequalities

I'm currently working through Spivak's Calculus and am currently in Chapter 1. So far, the book has covered commutativity, associativity, existence of an inverse and existence of an identity for ...
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95 views

Apostol Limit Proof

This is an interesting proof of the product limit law. I can see the squeeze theorem but how do you work out the step when applying the triangle-CS inequality with norms? Will this work for any $n$ ...
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90 views

How find this maximum and minimum of this $\sum_{k=1}^{n-1}(x_{k}-a)(x_{k+1}-a)$

let $a$ is give positive numbers,and $n$ is give positive integer number,and such $$x_{1}+x_{2}+\cdots+x_{n}=na,x_{i}\ge 0,i=1,2,\cdots,n$$ Find this function maximum and minimum ...
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70 views

Name of theorem?

I am trying to understand a proof which uses the following statement without further explanation, so I am wondering if this is a well known theorem? For the unit ball $B$ with radius $r>0$ and the ...
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94 views

Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
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30 views

Integral Inequality Question: Effect of Exponents

Is it true that, if $\alpha q < (1-\alpha)q$ then $\int|k(x,y)|^{\alpha q}dy \leq \int|k(x,y)|^{(1-\alpha)q}dy$, where $k(x,y)$ is a positive measurable function?
2
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193 views

Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
2
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101 views

necessary and sufficient conditions for the following inequality to be hold

Let $p=(p_n)$ is a sequence of nonnegative numbers with $p_0>0$ s.t $P_n= \sum_{k=0}^{n}p_k \rightarrow \infty$ as $n \rightarrow \infty$ and let $t\in(0,1)$. find necessary and sufficient ...
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86 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
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57 views

A probability inequality

Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume $X_1,X_2,Y_1,Y_2$ are four random variables. Assume $X_1$ and $X_2$ are independent. Is it necessarily true that ...