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Questions tagged [cauchy-schwarz-inequality]

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

Problems about Cauchy–Schwarz inequality.

  1. C-S inequality it's the following.

Let $a_1$, $a_2$,..., $a_n$, $b_1$, $b_2$,..., $b_n$ be real numbers. Prove that: $$(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n)^2\geq(a_1b_1+a_2b_2+...+a_nb_n)^2.$$

  1. C-S inequality in the Engel form it's the following.

Let $a_1$, $a_2$,..., $a_n$ be real numbers and $b_1$, $b_2$,..., $b_n$ be positive numbers. Prove that: $$\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+...+\frac{a_n^2}{b_n}\geq\frac{(a_1+a_2+...+a_n)^2}{b_1+b_2+...+b_n}$$

  1. C-S inequality in the integral form.

    Let $f$ and $g$ be integrable functions on $[a,b]$. Prove that: $$\int\limits_a^bf(x)^2dx\int\limits_a^bg(x)^2dx\geq\left(\int\limits_a^bf(x)g(x)dx\right)^2$$

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Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem…
HN_NH
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Intuition for the Cauchy-Schwarz inequality

I'm not looking for a mathematical proof; I'm looking for a visual one. I'm having trouble understanding (in my mind's eye) why the dot product of two vectors V and W produces a scalar that is less than the length of V multiplied by the length of…
Sydney Maples
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Prove this inequality with Cauchy-Schwarz inequality

Let $x_{1},x_{2},\cdots,x_{n}>0$, show that $$\left(\sum_{k=1}^{n}x_{k}\cos{k}\right)^2+\left(\sum_{k=1}^{n}x_{k}\sin{k}\right)^2\le \left(2+\dfrac{n}{4}\right)\sum_{k=1}^{n}x^2_{k}$$ I can prove it when $2+\dfrac{n}{4}$ takes the place of $n$, It…
math110
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Inequality with five variables

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$ Easy to show that…
Michael Rozenberg
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Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds. I spent a lot of time trying to solve this and, having consulted some books, I came to this: $$2x^2+2y^2+2z^2 \ge 2xy +…
ILoveChess
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Prove $\left|1+z_1\right| +\left|1+z_2 \right| + \left|1+z_1z_2\right|\geq 2$

I am struggling with this problem Let $z_1,z_2\in\mathbb{C}$ prove that $$\left|1+z_1\right| +\left|1+z_2 \right| + \left|1+z_1z_2\right|\geq 2$$ I know a similar question has been solved: if $|z_i|=1$ prove $|z_1+1|+|z_2+1|+|z_1z_2+1|\ge 2$…
Hackerman
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How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

let $x,y,z$ are postive numbers,show that $$\dfrac{x+y}{\sqrt{x^2+xy+y^2+yz}}+\dfrac{y+z}{\sqrt{y^2+yz+z^2+zx}}+\dfrac{z+x}{\sqrt{z^2+zx+x^2+xy}}\ge 2+\sqrt{\dfrac{xy+yz+xz}{x^2+y^2+z^2}}$$ My try: Without loss of generality，we assume that…
math110
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How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

let $a,b,c>0$,and such $a+b+c=3$, show that $$\dfrac{2}{(a+b)(4-ab)}+\dfrac{2}{(b+c)(4-bc)}+\dfrac{2}{(a+c)(4-ac)}\ge 1$$ I think this inequality use this $$ab\le\dfrac{(a+b)^2}{4}$$
math110
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Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that: $$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6.$$ Here's what I've tried: Using Cauchy-Schawrz I proved that: $$(3a + b^3)(3 + 1) \ge (3\sqrt{a} +…
Stefan4024
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Kantorovich inequality and Cauchy-Schwarz inequality

On the wikipedia site for the Kantorovich inequality, it is claimed ... the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality... Here, "Kantorovich inequality" refers to $$ (x^\top A \, x) \, (x^\top A^{-1} \, x) \le…
gerw
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Prove that $f(x, y) \le 3 $ for $x \ge 0, y > 0$

Let $x \ge 0, y>0$ and \begin{align*} f(x,y)&=\sqrt{\dfrac{y}{y+x^2}}+4\sqrt{\dfrac{y}{(y+(x+1)^2)(y+(x+3)^2)}}\\[6pt] &\qquad +4\sqrt{\dfrac{y}{(y+(x-1)^2)(y+(x-3)^2)}}. \end{align*} Prove that $f(x,y) \le 3$. I can prove when $x\ge 2, f(x,y) <…
chenbai
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If $x,y,z\in[-1,1]$ and $1+2xyz\geq x^2+y^2+z^2$, then can we infer $1+2(xyz)^n\geq x^{2n}+y^{2n}+z^{2n}$?

This problem was in IMC 2010. Assuming $x,y,z\in [-1,1]$, suppose that $$1+2xyz\geqslant x^2 + y^2 + z^2$$ Can we infer from this that $$1+2(xyz)^n\geqslant x^{2n} + y^{2n} + z^{2n}$$ for any positive integer $n$?
user60253
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Why does the Cauchy-Schwarz inequality hold in any inner product space?

I am working through linear algebra problems in Apostol's Calculus, and he has numerous problems that seem to imply that Cauchy-Schwarz holds no matter how the inner product is defined. Then, he has problems where the triangle inequality holds…
user59083
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Is the following generalization of Cauchy-Schwarz inequality true?

Let $\{a_{1,i}\}_{i=1}^k,\{a_{2,i}\}_{i=1}^k,\dots ,\{a_{n_,i}\}_{i=1}^k$ be real sequences. Does the following inequality hold $$(\sum_{i=1}^k a_{1,i}^2)\cdot(\sum_{i=1}^k a_{2,i}^2)\cdots(\sum_{i=1}^k a_{n,i}^2)\geq (\sum_{i=1}^k…
Oshawott
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Does Cauchy-Schwarz Inequality depend on positive definiteness?

Let $V$ be a vector space over $\mathbb{R}$. Suppose we have a product $\langle \cdot,\cdot\rangle:V^2\to \mathbb{R}$ that satisfies all the inner product axioms except the second part of positive-definiteness: $$\langle x,x\rangle=0\iff…
Riley
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