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Questions tagged [cauchy-schwarz-inequality]

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

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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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Exercise on Schwarz inequality: $\left(\int_a^b |f(x)| dx \right)^2 \leq (b-a)\int_a^b|f(x)|^2 dx$

I'm trying this exercise: Suppose that $f(x)$ is a complex-valued function square integrable on the interval $]a,b[$. Show that $$\left(\int_a^b |f(x)| dx \right)^2 \leq (b-a)\int_a^b|f(x)|^2 dx$$ I'...
injo's user avatar
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Generalized Cauchy-Schwarz for fractional Sobolev space [closed]

Let $\Omega$ be a bounded domain with Lipschitz boundary $\partial\Omega$. Let $f\in H^{1/2}(\partial\Omega)$ and $g\in H^{-1/2}(\partial\Omega)$. It is possible to prove that $$ \bigg|\int_{\partial\...
Vinz's user avatar
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Cauchy Schwarz inequality with modulo

I can't understand why integral $\int_x^y{(y-t)dt}$ became $|\delta x|$. By Cauchy Schwarz inequality $$|\int_x^y{(y-t)f''(t)dt}| \leq (\int_x^y{(y-t)^2dt})^{0.5}(\int_x^y{f''(t)^2dt})^{0.5}$$ ...
Pyrettt Pyrettt's user avatar
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Is there a Cauchy-Schwarz inequality for $Tr[A B C D]$?

I am familiar with the Cauchy_Schwartz inequality $$|Tr[A^*B]|^2 \le |Tr[A^*A]| |Tr[B^*B]|$$ where $*$ denotes the conjugate-transpose operation. I am wondering if there is a similar inequality for $...
Mike's user avatar
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Show that $ \left( \int_1^\infty f \right)^2 \leq \int_1^\infty x^2(f(x))^2 dx $ [duplicate]

In Axler's Linear Algebra Done Right (4e), Chapter 6A (Inner Product Spaces), exercise 18 is as follows: (a) Suppose $f: [1, \infty) \to [0, \infty)$ is continuous. Show that $$ \left( \int_1^\infty ...
Kyle L's user avatar
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Matrix Norm Inequality: $c \sum_{k=1}^{n} \left| \sum_{l=1}^{n} \epsilon_{l} A_{k,l} \right| \geq \sum_{l=1}^{n} \sqrt{\sum_{k=1}^{n} A_{k,l}^{2}}$

Question Show that there is a real universal constant $c$ with the property that for all positive integers $n$, and all nonzero $n \times n$ real matrices $A$, there are signs $\epsilon_1, ..., \...
Saucitom's user avatar
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Trouble finding norm of $T: H_1 \to H_2$, which is defined by $T(x)=\sum_{i=1}^n\lambda_i \langle x, a_i \rangle b_i \quad \text{for each } x\in H_1$.

Let $H_1$ and $H_2$ be complex Hilbert spaces. Let $\lambda_1, \lambda_2, \ldots, \lambda_n$ be complex numbers, and let $\{a_1, a_2, \ldots, a_n\} \subset H_1$ and $\{b_1, b_2, \ldots, b_n\} \subset ...
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Operator inner product

Let $\mathcal S(H)$ be the set of linear self-adjoint operators on Hilbert space $H$ with inner product $\langle\cdot,\cdot\rangle$. For $A, B\in \mathcal S(H), v=\sum_{i,j}a_{i,j}u_i\otimes u_j\in H\...
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Maximizing with Cauchy-Schwarz inequality

I want maximize the function $f(x)=\cos(x)+\sin(x)\cdot\cos(x)$, with $x \in (0,\frac{\pi}{2})$. By derivation $f'(x)=0 \Rightarrow x=\frac{\pi}{6}$. But, if we write $f(x)=\cos(x)+\frac{1}{2}\sin(2x)$...
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$\sum_1^{+\infty} a_j^2 < \infty$ if and only if $\psi_n$ is a Cauchy sequence?

Let $(\xi_j)_j$ be a complete orthonormal basis of $L^2(\mathbb{R})$, I want to prove , Given that $\sum_1^{+\infty} a_j^2 < \infty$, the sequence $$\psi_n(x) = e^{-\frac{1}{2}\sum_{j=1}^n a_j\...
Houssem Ajili's user avatar
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Let $a,b,c$ be positive real numbers such that $(a+b)(b+c)(c+a)=1$. Find $\max(ab+bc+ca)$ [duplicate]

Let $a,b,c$ be positive real numbers such that: $(a+b)(b+c)(c+a)=1$. Find $\max(ab+bc+ca)$. I used AG and AH to get that $a+b+c≤3/2$ and $abc\leq 1/8$ but I don't know what else to do...
Ficooo's user avatar
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How to Prove $ E[X^2Y^2] \leq E[X^2]E[Y^2] $

I'm working on a problem involving the expectations and variances of random variables, and I encountered a step that I'm not sure how to prove rigorously. Specifically, I need to show that: $ E[X^2Y^2]...
matmi's user avatar
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Maximum of $P=\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}.$

Let $a,b,c\geq 1$ such that $a+b+c=9$. Find $\max$ and $\min$ of $P=\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}.$ My attempt By C-S, I have $P\geq\dfrac{9}{2(a+b+c)}=\dfrac{9}{2.9}=\dfrac{1}{2}$. ...
lee max's user avatar
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prove that $\left( \sum_{i=1}^{n} a_i \right) \left( \sum_{i=1}^{n} a_i^{n-1} \right) \leq n \prod_{i=1}^{n} a_i + (n-1) \sum_{i=1}^{n} a_i^n.$

question:Let $a_1, a_2, \ldots, a_n$ be nonnegative real numbers. Prove that $$\left( \sum_{i=1}^{n} a_i \right) \left( \sum_{i=1}^{n} a_i^{n-1} \right) \leq n \prod_{i=1}^{n} a_i + (n-1) \sum_{i=1}^{...
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Knowing $x,y,z\ge0$ prove $x^2+xy^2+xyz^2\ge4xyz-4$

Knowing $x,y,z\ge0$ prove $x^2+xy^2+xyz^2\ge4xyz-4$ I thought that I should rearrange this inequality to be somewhat of the form of Schur's Inequality and WLOG I assumed $x\ge y\ge z$. Trying this way ...
FabDust's user avatar
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Knowing $a,b,c>0$ and $abc\le1$, prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge1+\frac{6}{a+b+c}$

Knowing $a,b,c>0$ and $abc\le1$, prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge1+\frac{6}{a+b+c}$ I tried to AM-GM the inequality, which gave this: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3\...
FabDust's user avatar
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A symmetric inequality involving product of three variables

Let $a, b,c \ge 0, ab + bc + ca + abc = 4$. Find the minimum of $S = \sqrt{a}+\sqrt{b}+\sqrt{c}$. My guess is that $S$ attends its minimum at $b = c = 2, a = 0$ and the other permutation of $(2, 2, 0)$...
anonimo's user avatar
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If $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$

Problem. Prove that if $\sum_{n=1}^\infty a_n^2$ converges, then $\lim_{n\to \infty} \frac 1 {\sqrt n} \sum_{k=1}^n a_k =0$. The problem arises from the following question: Let $(e_i)_{i=1}^\infty $ ...
Robert's user avatar
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a Cauchy Schwarz application [duplicate]

Here an inequality that I feel that CS may prove it but I can't find the right way to use it : $\left (\sum_{i=1}^{n}a_{i}\right )^{2}+\left (\sum_{i=1}^{n}b_{i}\right )^{2}\leqslant \left (\sum_{i=1}^...
Loca's user avatar
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prove that $\frac{x^x}{|x-y|} + \frac{y^y}{|y-z|} + \frac{z^z}{|z-x|} \geqslant \frac{7}{2}.$ where $x \neq y \neq z$

Question Statement: I came across the following intriguing inequality problem involving positive real numbers $x$, $y$, and $z$, where $x \neq y \neq z$: $$\frac{x^x}{|x-y|} + \frac{y^y}{|y-z|} + \...
Mods And Staff Are Not Fair's user avatar
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The Promotion of Cauchy-Schwarz Inequality in Probability Theory

In probability theory, if we have $\xi$ and $\eta$ as random variables, the Cauchy Inequality can be designated as: $$ E(\xi \eta) \leqslant \sqrt{E(\xi^2)E(\eta^2)} $$ Then I have an idea to make an ...
Zequan Bear's user avatar
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Finding the value range using Cauchy-Schwarz inequality.

English is not my mother tongue so my English may sound weird. When real numbers $x,y$ satisfies $\sin x+3 \sin y=1$,let $z=\cos x+3 \cos y$. Find the range of $z$. I came across to this problem when ...
Russel0201's user avatar
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If $x, y, z$ are real numbers with $xy+yz+zx\geq0,$ then $\left(x^5 + y^5 + z^5\right)^2 \geq3xyz\left(x^7 +y^7 + z^7\right).$

Question: If $x, y, z$ are real numbers with $xy+yz+zx\geq0,$ then $$\left(x^5 + y^5 + z^5\right)^2 \geq3xyz\left(x^7 +y^7 + z^7\right).$$ Attempt: Let's consider the expression $$\left(x^5 + y^5 + z^...
Mods And Staff Are Not Fair's user avatar
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finding a min point of a function using Cauchy-Schwarz inequality

I would like to know why the next state holds true: $$ \min \| y \| +a^T y= \displaystyle \begin{cases} 0, & \text{if }\| a \| \leq 1 \\ - \infty, & \text{if }\| a \| > 1 \\ \...
Chen's user avatar
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Cauchy-Schwarz Inequality Proof in $\mathbb{R}^n$

Let $n \in \mathbb{N}$ and let $\sum_{i=1}^n x_i = 1$ where $x_i$ are positive real numbers. Use the Cauchy-Schwarz inequality to show that $$ \sum_{i=1}^n \frac{1}{x_i} \geq n^2 $$ The Cauchy-Schwarz ...
InvestingScientist's user avatar
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$\frac{\sum_{i=1}^n a_i^2}{\sum_{i=1}^n a_ib_i}-\frac{\sum_{i=1}^n a_ib_i}{\sum_{i=1}^n b_i^2}\le\left(\sqrt{\frac{A}{b}}-\sqrt{\frac{a}{B}}\right)^2$ [closed]

Let $a_i,b_i (i=\overline{1,n})$ be real numbers, set $A=\max a_i, a=\min a_i, B=\max b_i, b=\min b_i$. Prove that $$\dfrac{\displaystyle\sum_{i=1}^{n} a_i^2}{\displaystyle\sum_{i=1}^{n} a_ib_i}-\...
30 Anh Ti 711's user avatar
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Chi-squared divergence inequalities proof using Cauchy-Schwarz

I am trying to prove the two following inequalities for distributions $P$ and $Q$ with densities $p$ and $q$ respectively, using the Chi-squared divergence defined as: $$\mathcal{X}^2(p||q) = \int\...
nothatcreative5's user avatar
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1 answer
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Bounding a sequence involving minimizers of strictly convex quadratic functions

Suppose that $H\succ 0$ and we have a sequence $\{x_j\}_{j\ge 1}$ such that each one is the unique solution of the following problem: $$ x_{j} = \text{argmin}_{x\in \mathbb R^n}(x-x_{j-1})^T\nabla f(...
Sam's user avatar
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1 answer
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Integral inequality with exponents

Let $f(x):[0;1]\to\mathbb{R}$ be continuous function. Prove that $$\int_0^1e^{f(x)}dx \cdot \int_0^1e^{-f(x)}dx \geq 1+\int_0^1(f(x))^2dx-\Bigg(\int_0^1f(x)dx\Bigg)^2.$$ I tried to use Cauchy–Schwarz ...
perenqi's user avatar
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$\lambda:=\frac{\sum_{i=1}^na_ib_i}{\sum_{i=1}^n|b_i|^2}$ makes $\sum_{i=1}^n\left|a_i-\lambda\overline{b_i}\right|^2$ as small as possible. Ahlfors.

I am reading "Complex Analysis Third Edition" by Lars V. Ahlfors. Let $a_i,b_i\in\mathbb{C}$ for all $i\in\{1,\dots,n\}$. Then $$\left|\sum_{i=1}^n a_ib_i\right|^2\leq\sum_{i=1}^n |a_i|^2\...
佐武五郎's user avatar
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Equivalent to Cauchy-Schwarz inequality

This is a question form a sample paper whose solutions where not published: Let $a$ and $b$ be two nonzero vectors in $3$ dimensions. The orthogonal projection of $b$ onto $a$ is$$c=(e\cdot b)e,$$...
Ishant Dumane's user avatar
3 votes
1 answer
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Proving an inequality: $\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)} \ge \frac{1+abc}{2}$

The question is as follows: Let $a, b, c \ge 0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)} \ge \frac{1+abc}{2}$$ I started as follows: (1) Proving for one variable- $a$ $$\frac{1+a^2}{...
xoxo's user avatar
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Show the inequality $\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (c+a)^3}} \ge 1$

For all positive reals $a, b, c$, we wish to prove the inequality $$\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (c+a)^3}} \ge 1$$ My approach was Hölder: $$\left(\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (...
Martin Westin's user avatar
1 vote
2 answers
71 views

Proof of one version of Cauchy-Schwarz in $\mathbb{R}^n$

Show that for $a,b \in \mathbb{R}$ and $x,y > 0$ that $$\frac{(a+b)^2}{x+y} \le \frac{a^2}{x} + \frac{b^2}{y}$$ and generalize this result for $a_1, a_2, \dots, a_n \in \mathbb{R}$ and $x_1, x_2, \...
IcedTea's user avatar
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3 votes
2 answers
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Prove the inequality $(1+x^2)(1+y^2)(1+z^2) \ge 64$ if $xy + yz + zx = 9$

Given non-negative reals $x, y, z$ satisfying $xy + yz + zx = 9$, we're supposed to prove the inequality $(1+x^2)(1+y^2)(1+z^2) \ge 64$. My first approach was something like C-S or Hölder, since we're ...
Martin Westin's user avatar
2 votes
1 answer
41 views

Integration Inequality with Inner Products

Suppose $f: [1, \infty) \rightarrow [0, \infty)$ is continuous. Show that $(\int_{1}^{\infty}f)^{2} \leq \int_{1}^{\infty} x^{2}f(x)^{2}dx$ This problem is from Axler's Linear Algebra Done Right, 4e (...
pseudobulbose's user avatar
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Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$ What's the minimum value of $(a^2+b^2)$? [duplicate]

Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$ What's the minimum value of $(a^2+b^2)$? Ok so here I tried to divide by $x^2$ and get this : $$x^2+\frac{1}{x^2}+a\left(...
FabDust's user avatar
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1 answer
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Prove that $(a^\frac23 + b^\frac23)^\frac32 > (a^2 + b ^2)^\frac12$ for $a, b$ being positive real numbers.

I'm working on an optimization problem in which to make my argument rigorous, I find myself needing to prove that $(a^\frac23 + b^\frac23)^\frac32 > (a^2 + b ^2)^\frac12$ for $a, b$ being positive ...
ten_to_tenth's user avatar
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5 votes
1 answer
251 views

Showing inequality involving log

I want to show that for all $x_i > 0$: $$\sum_{i=1}^{n}\dfrac{x_i}{x_1 + \ldots+x_n}\;\log(x_i) \geq \log\left(\dfrac{x_1 + \ldots + x_n}{n}\right)$$ I thought of Jensen's inequality but since $\...
wheeler 's user avatar
4 votes
1 answer
132 views

A sort of extension of Cauchy-Schwarz inequality?

Let $a_{i}$ and $b_{i}$, $i = 1,\dotsb,n$ be real numbers. Denote $S_{pq}:=\sum_{i=1}^{n}a_{i}^{p}b_{i}^{q}$. I want to show that, $$ (S_{20}S_{02} - S_{11}^2)S_{00} - (S_{10}S_{02} - S_{01}S_{11})S_{...
Don Lee's user avatar
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1 answer
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$\log_a 10 + \log_b 10 +\ log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$

Prove the following inequality for $a,b,c, \in (1,\infty)$ , such that $abc = 10$ $$\log_a 10 + \log_b 10 + \log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$$ I will transform logarithms to ...
Unknowduck's user avatar
1 vote
1 answer
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$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$

Prove the following inequality $$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$$ where $x,y,z \ge 0$ and $x+y+z=3$ I don't really know how to approach such an ...
Unknowduck's user avatar
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2 answers
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Prove $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$, with $x, y, a, b \in \mathbb{R}$ and $a, b > 0$

Prove $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$, with $x, y, a, b \in \mathbb{R}$ and $a, b > 0$ Proof: $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$ $\iff (bx^2 + ...
ten_to_tenth's user avatar
  • 1,426
2 votes
1 answer
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An inequality with a "Cauchy-Schwarz" flavour

Let $x_1\leq x_2\leq ... \leq x_n$ and $y_1<y_2< ... <y_n$ be positive integers such that $x_1\geq 2$, $x_i < y_i$ and $y_i + 1<y_{i+1}$. Do we have that $$x_ny_n (\sum_{i=1}^n x_i)^2 \...
Juan Moreno's user avatar
  • 1,190
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0 answers
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Maximal mean value of a differentiable function with small variation

Let $B$ be a real valued function on $[0, 1]$ such that $\|B'\|_{L_2}\le 1$ and $B(0) = 0$. Denote \begin{gather}\label{small var condition} \varepsilon = \int_0^1 B^2(s)\, ds - \left(\int_0^1 B(s)...
Pavel Gubkin's user avatar
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3 votes
0 answers
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Inequality question(wish to confirm working)

Let $a_{i}$ be a set of positive real numbers such that $\sum_{i=1}^n a_{i}^3=3$ and $\sum_{i=1}^n a_{i}^5=5$. Prove that $\sum_{i=1}^n a_{i} > \frac{3}{2}$. My attempt: Using Titu's lemma, $$\frac{...
A shubh's user avatar
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1 answer
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Equality characterization of the reverse Cauchy-Schwarz inequality in a Lorentzian manifold

Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone(ie, both future-directed or past-...
some_math_guy's user avatar
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1 answer
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I can't seem to understand the proof of the Cauchy-Schwarz inequality

The proof said to use the quadratic polynomial: $$P(t) = \sum_{i=1}^n(a_it - b_i)^2$$ by which we notice that $P(t) \ge 0$ , but then this is the part which I didn't understand where we conclude that ...
Aymane's user avatar
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0 answers
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A modified version of log-sum inequality.

Suppose $a_i>0, b_i>0, c_i>0 \; \forall i = 1, 2, \dots, n$, and $$\sum_{i} a_i b_i \ln(\frac{c_i}{b_i}) \geq 0, $$ where $a_i, b_i, c_i$ are not constant over $i$. Moreover, $$\sum_{i}a_ib_i\...
entropy's user avatar
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