# Questions tagged [cauchy-schwarz-inequality]

Problems with using C-S (Cauchy-Schwarz inequality)

579 questions
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### Triangle Inequality of Tensor Products

If $$\|A - x\|_1 \le \epsilon$$ and $$\|B - y\|_1 \le \epsilon$$ where $A, B, x, y \in Herm(H_A)$, where $Herm(H_A)$ are the set of Hermitian matrices in a Hilbert space $H_A$, then can we say, by ...
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### prove the following inequality using am gm hm or weirstrass etc

For $a_0,a_1,a_2,......,a_n \in R, \,\, a_0<a_1<a_2<....<a_n$ show that $$\frac n{a_1-a_0}+\frac {n-1}{a_2-a_1}+....+\frac 1{a_n-a_{n-1}} \ge \sum_{k=1}^n \frac {k^2}{a_k}$$ i recently ...
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### prove the inequality using inequalities like AM GM HM OR CAUCHY or WEIRSTRASS ETC.

The inequality to be proven is $$2^n \gt 1 + n\cdot \sqrt{2^{n-1}} for\ all\ n>2$$ using any inequalities like am gm hm cauchy schwarz tchebychev etc I recently studied inequalities came across ...
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### Some Cauchy-Schwarz Inequalities [closed]

I am trying to learn how to deal with inequalities to prepare for a Math Olympiad and right now I am working on Cauchy-Schwarz. However, I am not that good at seeing the relationships and I don't have ...
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### Cauchy-Schwarz Inequality troubles

I have to prove the following inequality using the Cauchy-Schwarz inequality: $$\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\ge2$$ where a, b, c and d are positive real numbers. But ...
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### A direct result from the definition of operator norm

Although Wikipedia says this result comes from the definition of Operator Norm directly, I am not quite sure how to understand it: Let $||\cdot||$ denote Euclidean norm. Given a $n\times n$ matrix $A$...
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### Schwarz-Pick on a disc of radius R > 0…

Let $R, C > 0$. Let f be a holomorphic function defined on $D(0, R)$ and such that f is bounded above by $C$. Prove that $$|f'(z)| \leq \frac{R}{C}\cdot \frac{C^2 - |f(z)|^2}{R^2 - |z|^2}.$$ I am ...
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### Cauchy-Schwarz inequality for $L^2$-norm on periodic functions space

I have proven something that is definitely not true (Lemma 2), which is why I am intersted where I err. Definition Let $C(\mathbb{R}/\mathbb{Z},\mathbb{C})$ be the set of all continuous $\mathbb{Z}$-...
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### Find the smallest value of $f(x) := \left({1\over9}+{32\over \sin(x)}\right)\left({1\over32}+{9\over \cos(x)}\right)$ on the interval $(0,\pi/2)$

There's a function defined as: $$f(x) := \left({1\over9}+{32\over \sin(x)}\right)\left({1\over32}+{9\over \cos(x)}\right)$$ In interval $$\left(0,\frac{\pi}{2}\right)$$ Find the smallest value (...
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### Prove $l^2$ norm obeys the triangle inequality

I'm trying to work through Exercise 3 from this blog post, which is essentially a proof of the validity of the $l^2$ norm: Exercise 3: Let $(\mathcal{V},\left<\cdot,\cdot\right>)$ be an inner ...
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### How do I find the distance from a point to a plane?

I am trying to find the distance from point $(8, 0, -6)$ and plane $x+y+z = 6$. I tried solving it but I am still getting it wrong. Can anyone help me on this? Any help I would very much appreciate. ...
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### Does $\sum\limits_{i=1}^n x_i = 1$ imply $\sum\limits_{i=1}^n x_i^2 \geq \frac{1}{n}$?

Suppose we have real numbers $x_1, ..., x_n$ which satisfy $x_1 + ... + x_n = 1$. Do we have the lower bound $x_1^2 + ... + x_n^2 \geq \frac{1}{n}$? It seems intuitive that we can minimize this by ...
Assuming $p_k > 0$, $1 \leq k \leq n$ and $p_1 + p_2 + \cdots + p_n = 1$, show that: $$\sum_{k=1}^n \left( p_k + \frac{1}{p_k} \right)^2 \geq n^3 + 2n + 1/n$$ and determine necessary and ...