Questions on proving, manipulating and applying inequalities.

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21 views

Inequalities with the Integral Test

a) Use the proof of the integral test to show that $\ln(n!)\ge n\ln(n)-n+1$ for $n>1$ b) Use part (a) to show that $\ln(n!)\ge n\ln(n)$ for $n\ge 10$ I was able to solve part a) but not ...
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23 views

One proof in Random graph Bela Bollobas about Hamiltonian Cycles

In Random Graph 2001 (Bela Bollobas), P209, the proof for lemma 8.7, it says that $\sum_{u=u_0-1}^{u_1}\sum_{w=1}^{\llcorner(\gamma - 1)u\lrcorner}(\log n)^w (\frac{e}{u})^u(\frac{eu}{w})^wn^{\gamma ...
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15 views

What is Hoeffding's inequality in Hilbert space?

Suppose I have random variables $X_1, X_2,...,X_n \in \mathcal{H}$, where $ \mathcal{H}$ is some Hilbert space. How can I bound the following term - $ P(\| \sum_{i = 1}^n X_i - E[X_i] ...
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42 views

Prove this inequality for sufficiently large $n$

Prove that the above inequality holds for sufficiently large $n$: $$\pi(2n) - \frac{3}{2} \pi(n) \ge O\left(\frac{\ln n}{(\ln \ln n )^2}\right)$$ $\ln n$ denotes to natural logarithm and $\pi(n)$ is ...
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27 views

Conditions for the satisfiability of a matrix inequality

Let $X\geq 0$ be a positive definite matrix and $A$ a square matrix, do there exist conditions either on $A$ or $X$ (or both) such that $$ X-AXA^\top\geq 0,\quad \quad (*) $$ holds, where $\top$ ...
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44 views

Maximum over Probabilistic Distribution Functions Space

Suppose $P$ is the set of functions where $p\in P: R^{+2}\to R^+$ and $p(t,s)$ is differentiable in $t$. $\forall t, p(t,\cdot)$ is a probability distribution on the positive axis $s\in [0,\infty)$, ...
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49 views

The derivative of piecewise linear function that vanishes at every other dyadic rational

Let $D_n = \{\frac{k}{2^n}:0 \leq k \leq 2^n\}$ Define function $F_n(x) = 0$ when $x \in D_{n-1}$ and it is piecewise linearly interpolated between consecutive points of $D_n$ (which means to define ...
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29 views

A tighter bound on Chebychev inequality

How can we prove that if $X$ is a random variable with mean $0$ and finite variance $\sigma^2$, then $$Pr[X \ge a] \le \frac{\sigma^2}{\sigma^2 + a^2}$$ for all $a>0$.
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34 views

If not $b<a$, then $a\leq b$

In trying to prove the following inequality $\neg(a\leq b)\Longrightarrow b<a$ i could produce the following indirect proof: Let $\neg(a\leq b)$. Let $\neg(b<a)$ But ...
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47 views

Cauchy-Schwarz for more than two vectors?

Given three or more vectors in an inner product space, $x,y,z, \ldots$, I am wondering whether there exist generalisations to the Cauchy-Schwarz inequality: \begin{equation} \left| \langle x,y \rangle ...
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27 views

Implicit function theorem: Lower bound on radius

Let $f: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be sufficiently smooth with $f(x_0,y_0) = 0$ and the Jacobian $J_yf(x_0,y_0)$ of $f$ with respect to the second variable $y$ is invertible at a ...
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40 views

A factorial related inequality

Given $n$ is there an explicit or asymptotic formula for least $m$ such that $$m!\geq n?$$ Essentially is there a good inverse to factorial?
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33 views

The least upper bound of $|\sum_{j=1}^{n}a_j^2|$

For $j=1,\cdots n$, let $a_j\in\mathbb{C}\setminus\{1\}$ and $|a_j|<n$. If $\sum_{j=1}^{n}a_j=n(n-1)$ and $\sum_{j=1}^{n}\frac{1}{1-a_j}=0$, what is the least upper bound of ...
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51 views

Archimedean integration of $x^3$?

I'm trying to integrate $x^3$ in the interval $[0,1]$ in a similar fashion to the famous integration of $x^2$ due to Archimedes. It's an exercise on Apostol's Calculus: I did all the ...
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15 views

Is $\int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 \cdots \int_{[(n+1)/2-\sum_{k=1}^{n-1} 1/x_k]^{-1}}^2 \prod_{k=1}^n f(x_{k-1},x_k) dx_1 \dots dx_n \geq cq^n$?

Let $f\colon (1,2)\times (1,2) \to \mathbb{R}$ be a Lebesgue measurable, bounded and non-negative function such that $$ \int_1^2 f(x,y) dy = 1, \qquad x \in (1,2). $$ Moreover, assume that for any ...
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24 views

How to upper-bound $\sum_{i=1}^n \frac{s_i}{i + \sqrt{s_i}}$ and $\sum_{i=1}^n \frac{s_i}{i + {s_i}}$?

Any ideas how to upper bound $A$, $B$, $C$ and/or $D$: $$ B = \sum_{i=1}^n \frac{s_i}{i + {s_i}} $$ $$ C = \sum_{i=1}^n \frac{1}{i + \sqrt{s_i}} $$ $$ D = \sum_{i=1}^n \frac{1}{i + {s_i}} $$ as a ...
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11 views

How can I prove the following inequality?

Let be $N_{n+1}(x)=\prod_{i=0}^n(x-x_i)$. Now I have to prove that $$||N_{n+1}(x)||_{\infty,[-5,5]}\leq n!\frac{h^{n+1}}{4},\qquad h:=\frac{5-(-5)}{n}=\frac{10}{n}.$$ I've started with ...
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32 views

inequality with radicals

Let $y\ge x\ge 1$. What is the best constant $k$ for which the following inequality holds \begin{equation*} \sqrt{x^2+y^2+xy}+2\sqrt{y^2+y+1}+3\sqrt{x^2+x+1}\le k(1+x+y)? \end{equation*} It is very ...
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28 views

Proof that $e^x\leq C_n(1+|x|)^n$ for every $n\in\mathbb N$?

It is well known $1+x\leq e^x$ for all $x\in\mathbb R$. See for example this post for several proofs of it. I wanted to show for every $n\in\mathbb N$ there is a constant $C_n>0$ such that ...
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31 views

Proof of a certain inequality in two-dimensional Euclidean space

Please think it easy because it is not an assignment. I'm trying to show the following problem. Show that the inequality $$ ...
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9 views

Convolutions of L^p functions

Denote by (fg) the convolution of f and g. If g is square integrable and (fg) is square integrable for every square integrable f can we conclude that g in integrable? This is a converse to the ...
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20 views

$\left|\frac{1}{A^2 B} - \frac{1}{C^2D} \right| \leq |A-C| \times constant $

Could you help me please, to make a majoration of this type : $$\left|\frac{1}{A^2 B} - \frac{1}{C^2D} \right| \leq |A-C| \times constant $$ My only hypothesis is : $$A,B,C,D \in ...
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45 views

Bound on the difference of infimums

Suppose we have to non-negative functions $f$ and $g$ and we want to bound the difference of their infimums \begin{align} \left|\inf_{x\in \mathbb{R}}f(x)-\inf_{x\in \mathbb{R}}g(x)\right| \end{align} ...
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42 views

How to formulate a linear programming model where the rates of production must be calculated?

A friend of mine is working on this problem for a course. I will post the problem in full and tell you what my/our thoughts are on it. The problem then asks for a model that can be used to develop ...
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31 views

Show that $|ax^2+bx+c|<1$ gives us $|c|<1$

$a,b,c \in \mathbb{R}$ and for all $-1<x<1$ Show that if $|ax^2+bx+c|=<1$ So : 1) $|c|=<1$ 2) $|a+c|=<1$ 3) $a^2+b^2+c^2=<5$ For the first one ; if I choose x=0 So |c|=<1 ...
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42 views

Compare absolute values of two expressions!

I have two sets of numbers as follows: $$X = \{x_1, x_2, ..., x_n\}\\ Y = \{y_1, y_2, ..., y_n\}$$ And a number $r$. Let $x^\ast$ and $y^\ast$ is average values of set $X$ and $Y$ respectively. ...
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26 views

Does the following inequality hold? $\sum\limits_{i=1}^a\sum\limits_{j=1}^b\sum\limits_{k=1}^c|A_iB_jC_k|\leq 1$ as $||A||=||B||=||C||=1$

Given three vectors $A\in \mathbb{R}^a$, $B\in \mathbb{R}^b$ and $C\in \mathbb{R}^c$, with $||A||=||B||=||C||=1$, where $||\cdot||$ denotes the $l_2$ norm. Then does the following inequality ...
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31 views

General inequality with powers

If $x,y\geq0$ and $b\geq1$. Does the inequality $(x+y)^b$ $\geq$ $x^b+y^b$ always hold? If so does this inequality happen to have a name?
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31 views

How to prove Bromwich inequality?( lower bound for inner product)

Consider Bromwhich inequality which is a generalization of the Abel's inequality: Theorem : For a given real n-tuple $p$ and given integer $v(1\le v\le n)$, define $H_1=h_1=0$, ...
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45 views

p - norms inequality

For $v \in \mathbb{R}^n$ denote by $||v||_p$ its p-norm. That is $(\sum_{i=1}^nv_i^p)^{\frac{1}{p}}$ where $v_i$ are the componenets of $v$. I'm looking for a way to bound the following expression: ...
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52 views

Any hints on how to solve this optimization problem?

I would like to solve the below optimization problem. Any hints is appreciated. I'm gussing the answer is 1. \begin{equation} \begin{aligned} & \min ...
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30 views

Upper bound for the $t$-th moment in terms of lower moments

Let $a_1, \ldots, a_n$ be positive integers. For positive integers $t$ and $m$ define the sum $$ M_t(m) = \dfrac{1}{n} \sum_{k=1}^n |a_k - m|^t. $$ I'm interested in upper bounds for $M_t(m)$ in ...
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34 views

A bound on powers of logarithms

I have come across the following ugly inequality, which I suspect holds for all integer $k \geq 1$: $$ h(k) := \sum_{n=1}^{\infty} \frac{(1-\log(k+1)^{-1}+\log(k+2)^{-1} )^n}{n^{1.5}} \leq \zeta(1.5) ...
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26 views

Inducing a map on a sub-polydisc

please forgive my title, I don't know what else to call this. In trying to prove a certain result, I've come across a technical part that I'm having trouble with.. Suppose that we call the unit ...
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30 views

strict inequality with norms $\frac{||y+p|(x+p)-|x+p|(y+p)+|x|y-|y|x|}{|(1+|y|)x-(1+|x|)y|}<2|p|$

We work in $R^n$ and $|x|=\sqrt{x_1^2+...+x_n^2}$. Let $p\in R^n$ different from $0$ and $x,y\in R^n$ with $x\neq y$ then \begin{equation*} ...
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25 views

How to determine the smallest N such that there are X (or more) prime numbers with exactly N bits?

I understand how I can use the prime number theorem to determine how many primes exist for a given bit length: $\pi(2^n)-\pi(2^{n-1})$. However, my specific problem is that I need to approach this ...
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45 views

Concave Upper Bound of Linear Combination of Exponential function

I have $f(x)=a_1e^{b_1 x}+...+a_Ne^{b_N x}$ where $ a_i\in \mathbb{R} \ \forall i$, $b_i\in \mathbb{R_+} \ \forall i$ and N is a finite integer. Is there any concave function that upper bound ...
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41 views

Hölder type inequality or similar or general inequality for integral of product

It is well known that for two given functions $f,g:\mathbb{R}^d \rightarrow \mathbb{R}^d$ such that $fg \in L^1(\mathbb{R}^d)$ and $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ with ...
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39 views

Expressing a set of discrete inequalities as a continuous differential equation

I'm trying to work out the solution to a problem of sequential inequalities. I believe the solution collapses to a set of differential equations, but I'm having trouble organizing things and I think I ...
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78 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
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48 views

finding the shortest distance of a hermitian matrix to a set of hermitian matricies with specific eigenvalues 2-norm

The title is more general, and all that I require is to show an inequality that I already have verified using random matrices in matlab. Let $\lambda_1 \leq ... \leq \lambda$ and $\mu_1 \leq ... \leq ...
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37 views

What is the name of this inequality?

What is the name of this inequality? If $a>0, b>0, c>0$ and $a<b+c$ then $\sqrt{a}<\sqrt{b}+\sqrt{c}$
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92 views

Is this proof that $\lim_{n \to \infty} (1+1/n)^n$ exists (1) new, (2) interesting?

I was playing around, and decided to see if I could come up with a proof that $$\lim_{n \to \infty} (1+1/n)^n$$ exists that was as elementary as possible in that it uses only Bernoulli's inequality. ...
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65 views

On the Chernoff bound

Recently, I saw the Chernoff bound written as follows. Let $X_1,X_2,\ldots,X_n$ be drawn i.i.d. on alphabet $\mathcal{X}$ and let $f:\mathcal{X}\to [0,1]$ be any function. Let $\mathbb{E}f(X_1) = ...
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32 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
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26 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 ...
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16 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 ...
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20 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 ...
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108 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 ...
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60 views

How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $

Let $a\in\mathbb N$, and $b\in\mathbb R, b\geq 1$ How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $