Questions tagged [inequality]
Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.
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Recurring A.M-G.M. in limiting case
I came across a question where a recurring AM-GM sequence converges in the limit, but I am not able to find point of convergence as a closed form expression.
Question: Let $f(x)$, $g(x)$ be defined as:...
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One Confusion in Proving Brezis-Lieb Lemma
Here is the Brezis-Lieb Lemma:
$(X,\mathfrak{A},\mu)$ is a measure space, consider $L^{p}(X):p \in(0,\infty)$ . Then $\left\{ f_{n} \right\}$ is a sequence of extended complex-values measurable ...
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Proving a the distance between Cauchy sequences converges [duplicate]
Assume we have two Cauchy sequences { $x_n$ } and {$y_n$} in the metric space $(X,d)$. Is it true that the sequence {$a_n$}$=d(x_n,y_n)$ is convergent in $\mathbb{R}$? Here is my try: $$$$
Since those ...
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A rational function dominated by a linear function on a certain interval [closed]
Considering $\alpha, \beta, \gamma, c > 0$, $\alpha - c > 0$, and $\mu_0 = \frac{\gamma (\alpha - c + \gamma)}{4 \beta}$, we have the following two functions:
$$ f(\mu) = \frac{(\alpha - c + \...
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Given that $x_i\ge0$ and $0<p<1$, find an upper bound for $\sum_{i=1}^n x_i^p$ as a function of $a:=\sum x_i$ [duplicate]
Suppose $\boldsymbol{x}$ is an $n$-dimensional vector and all elements are nonnegative with the condition that $\sum_{i=1}^nx_i= a$. I am wondering how it is possible to find an upper bound on $\sum_{...
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If $x > 0$ is a real number and $x+\frac{1}{x} \leq 7$ then the maximum value of the expression $x^2-\frac{1}{x^2}$ is...?
The problem
If $x > 0$ is a real number and $x+\frac{1}{x} \leq 7$ then the maximum value of the expression $x^2-\frac{1}{x^2}$ is...?
my idea
We know that $x^2-\frac{1}{x^2}= (x- \frac{1}{x})(x+ \...
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"Peeling Technique" in Probability
So I am reading "Bandit Algorithms" by Lattimore wherein for one of the proofs he uses a technique called as "Peeling Device" which he says is a widely used tool in probability. I ...
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Simple proof (avoiding Gamma function) for a Gaussian lower bound?
Let $g \sim N(0, I_n)$ be a standard multivariate Gaussian vector in $\mathbb{R}^n$.
It can be shown via use of Gamma function identities and inequalities that
$$
\sqrt{\frac{n}{n+1}} \leq \mathbb{E}\...
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An inequality for a bisected "shifted quadrant" under a continuous symmetric bivariate distribution?
Suppose a bivariate probability distribution is continuous and has circular symmetry about the origin; i.e., the lines of constant density are concentric circles centered on the origin.
Now consider ...
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Inequalities and averages
Dikshant writes down $2 k+1$ positive integers in a list where $k$ is a positive integer. The integers are not necessarily all distinct, but there are at least three distinct integers in the list. The ...
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Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$
The problem
Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$
My idea
$(\sqrt{a}+\sqrt{b})^2=a+b+ 2\cdot \sqrt{ab}= 101 + 2\cdot \sqrt{ab}$
...
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How to prove this simple inequality based on convexity of $e^{x}$?
Suppose $\theta > 0$ and $x>0$. I would like to show that
$$
e^{\theta(x+1)} - e^{\theta x} - \frac{ e^{\theta x}-1}{x} \geq \frac{e^{\theta x}-1}{x} - (1-e^{-\theta})
$$
Another way to put it: ...
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fractional power function inequality [closed]
I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result.
Let ...
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Differential inequation $f'(x) > f(x)/x$
I am interested in the following differential inequation, for $f : \mathbb{R} \to \mathbb{R}$:
$$f'(x) > \dfrac{f(x)}{x}$$
I only care about this being satisfied at infinity, ie for $x \gg 1$. ...
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Show that: $(\frac{MN}{MA})^2+ (\frac{MP}{MB})^2+ (\frac{MQ}{MC})^2+ (\frac{MR}{MD})^2 \geq \frac{4 }{9}$
The problem
Let $M$ be a point inside the tetrahedron $ABCD$. We denote by $N,P,Q,R$ the intersections of the lines $AM,BM,MC,DM$ with the planes $(BCD),(ACD),(ABD)$, respectively $(ABC)$. Show that:
$...
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