1
vote
2answers
50 views

minimal value of $x^2+2y^2+5z^2$ with constraint.

$x,y,z>0$, and $xy+yz+zx=1$. I need to find the minimum value of $x^2+2y^2+5z^2$ In general what can we say about the minimal value of $\frac{ax^2+by^2+cz^2}{xy+xz+yz}$, over all positive numbers ...
1
vote
1answer
61 views

The expression $1 + x^2 +(-T_px+y)^2 +z^2$ is bounded below by a constant multiple of $(1+x^2+y^2+z^2)$

Suppose $T_p > 0$. Is there a simply way to show that $1 + x^2 +(-T_px+y)^2 +z^2 \geq C (1+x^2+y^2+z^2)$, for all $(x,y,z) \in \mathbb R^3$, where $C>0$.
2
votes
1answer
43 views

verifying extrema found by Lagrange multipliers

This question was inspired by reading this problem: Prove the inequality $\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74$ Suppose I have a function $f(x,y,z)$ with continuous ...
3
votes
2answers
148 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
0
votes
0answers
30 views

Taylor expansions, inequalities and more

(Part A) I have to find the Taylor expansion of order 2 around (0,0) of $$f: \mathbb{R}^{2}\rightarrow \mathbb{R}$$ $$(x,y)x \mapsto f(x,y) = x\log (1+y)+sin(x+y) $$ Furthermore I have to prove if ...
0
votes
2answers
16 views

Graph the region of a two variable inequation

I need help with $( x + y -1) ( x - y +1) y <= 0$ There is a known method to conclude what area of the XY plane satisfy that inequation?. I usually have the same problem with other inequations ...
1
vote
1answer
30 views

What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$?

What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$? I see that is should be $-\dfrac{\pi}{2}\leq \ln(x\,\arccos{y}) \leq \dfrac{\pi}{2}$ and then $e^{-\dfrac{\pi}{2}} \leq x*\arccos(y) \leq ...
0
votes
3answers
38 views

What is the domain of $z=\arcsin\dfrac{x}{y}$?

I get that it should be $|y|>|x|$ and in the Wolfram it looks like this. But when I graph it by hand is that it should be only the "upper" part of intersection and not the "bottom" part as well, ...
0
votes
1answer
31 views

Calculus-based proof that $ x_1^{p_1}\cdots x_n^{p_n}\le p_1x_1+\dots+p_nx_n$ when $\sum p_i=1$

Let $$g(x_1...x_n)=x_1^{p_1}\cdot...x_n^{p_n}$$ $$u(x_1...x_n)=p_1x_1+...p_nx_n$$ Where $\sum p_i = 1$. I have to show that $f(x)=g(x)-u(x)$ is always negative or $0$ over $\Bbb R_+^n$. I've ...
0
votes
1answer
37 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
2
votes
0answers
76 views

Apostol Limit Proof

This is an interesting proof of the product limit law. I can see the squeeze theorem but how do you work out the step when applying the triangle-CS inequality with norms? Will this work for any $n$ ...
3
votes
2answers
60 views

Minimum value of the function $\sqrt{(1+1/m)(1+1/n)}$

If $m, n$ are positive real variables whose sum is a constant $k$, then what is the minimum value of $$\sqrt{\bigg(1 + \frac{1}{m}\bigg)\bigg(1 + \frac{1}{n}\bigg)}$$
5
votes
4answers
160 views

How to find the minimum value of this function?

How to find the minimum value of $$\frac{x}{3y^2+3z^2+3yz+1}+\frac{y}{3x^2+3z^2+3xz+1}+\frac{z}{3x^2+3y^2+3xy+1}$$,where $x,y,z\geq 0$ and $x+y+z=1$. It seems to be hard if we use calculus methods. ...
2
votes
2answers
82 views

Directional derivative in a Sobolev-like inequality

I am trying to do the following problem: Let $\Omega \subset \subset \overline{\mathbb{R}_{+}^{n+1}} = \{ (x, x_{n+1}); \, x_{n+1}\geq 0\}$ (i.e., $\Omega$ is bounded inside the closed upper ...
0
votes
1answer
116 views

Help with Inequality involving absolute values of trig

I am trying to wrap my ahead around the following problem: Prove that for all $x,y$ in $\Bbb R$ $ |\sin(x) - \sin(y)| \leq |x-y|$ And prove that for $x,y$ in $R$ $|\cos(x) - \cos(y)| \leq |x - y|$ ...
1
vote
2answers
85 views

How find this maximum of $f(n)$

let $x_{i}\in (0,1),i=1,2,\cdots,n,x_{n+1}=x_{1}$,give for any positive integer numbets $n$, find $$f(n)=\max{\sum_{i=1}^{n}x_{i}(1-x_{i+1})}$$ find the $f(n)$ it is easy find when $n=1$, then ...
7
votes
2answers
203 views

How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...
1
vote
2answers
96 views

Does this hold for three numbers [duplicate]

If $a\ge b\ge c\ge0$, does it hold that $\sqrt[3]{\left(a-b+c\right)^{2}}\ge\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}+\sqrt[3]{c^{2}}$? Thanks for any help.
4
votes
1answer
86 views

If $a\ge b\ge-c\ge0$, is $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?

Let $a\ge b\ge-c\ge0$. Is it true that $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?
4
votes
1answer
139 views

Is this always true?

Suppose $\left|x_{1}\right|\ge\left|x_{2}\right|\ge\left|x_{3}\right|$, $\left|y_{1}\right|\ge\left|y_{2}\right|\ge\left|y_{3}\right|$, and ...
3
votes
1answer
98 views

An inequality involving multi-index

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this: For $x \in \mathbb{R}^{n}$ and $\alpha = ...
6
votes
1answer
275 views

Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$

Let $a$, $b$, $c$ and $d$ are non-negative numbers such that $abc+abd+acd+bcd=4.$ Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$ I simplified it and it turns out that ...
4
votes
4answers
116 views

Find minimum in a constrained two-variable inequation

I would appreciate if somebody could help me with the following problem: Q: find minimum $$9a^2+9b^2+c^2$$ where $a^2+b^2\leq 9, c=\sqrt{9-a^2}\sqrt{9-b^2}-2ab$
1
vote
1answer
826 views

Inequalities related to infimum and supremum

Let $f,g: A \rightarrow \mathbb{R}$ be integrable functions on a closed rectangle $A \subset \mathbb{R}^n$. Let $P$ be a partition of $A$ and $S \in P$ a sub-rectangle. Show that: $m_S(f+g) \geq ...
3
votes
2answers
288 views

How to prove $f(x,y) = \frac{2 \log(y) \sqrt{y ( x-y)}}{x \log(x)}$ is bounded by $1$ for $(x,y) \to (0,0) $?

Let $$f(x,y) = \frac{2 \log(y) \sqrt{y ( x-y)}}{x \log(x)} $$ with $(x,y) \in D = \{(x,y)\mid 0 < y \leq x \le 1 \}$. How would one show (or disprove) that $$ \forall \epsilon > 0\ \ \exists ...
3
votes
2answers
86 views

Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$

I have the folowing exercise (which I've been thinking quite a while and couldn't figure out): Show that $\forall (x,y)$ in the first quadrant: $$\frac {x^2+y^2}{4}\leq e^{x+y-2}$$ My idea was to ...
1
vote
1answer
42 views

Maximization of an integer input function

Maximize the value of the function $$ z=\frac{ab+c}{a+b+c}, $$ where $a,b,c$ are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, ...
1
vote
1answer
55 views

Is there a constant for this?

Suppose that $\sum_{i=1}^{n}\lambda_{i}=1$, where $\lambda_{i}>0$, and $\sum_{i=1}^{n}x_{i}^{2}=1$, where $x_{i}>0$. Does one have $n^{3/2}\min_{1\le i\le n}\lambda_{i}x_{i}\le B$ for some ...
1
vote
0answers
179 views

Does this hold?

Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $$ \min_{1\le i\ne j\le ...
6
votes
0answers
241 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
1
vote
1answer
211 views

Proving Schwarz inequality for complex numbers using calculus

I am working on this problem for real analysis and it's rather baffling. I've never really seen any of this kind of notation before which is part of it I think? But also it just seems utterly ...
0
votes
1answer
83 views

Class of functions with negative mixed partial derivatives

Lets say we define a class of functions $g: \mathbf{R}^2 \rightarrow \mathbf{R}$ by the requirement that $$ \frac{\partial^2 g}{\partial x_1 \partial x_2}(x_1,x_2) \le 0 $$ for all $x_1$ and $x_2$. ...
1
vote
1answer
226 views

Approximation of Integration by parts

I'm trying to approximate a integral of the form: $$\int_V{g({\bf x})f({\bf x})} \; d^3x$$ Where the functions $f(\bf{x})$ and $g(\bf{x})$ are positive functions, but where only $f(\bf{x})$ is known ...
1
vote
1answer
80 views

Looking for hints of this inequality

I think the following two inequalities are true. However, the proof may not be easy. Does anyone have any hints? Thank you very much! Fix $a>1$. there exists two constants $K_1$ and $K_2$, such ...
3
votes
6answers
335 views

Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$

If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$. How to solve this one? (Here's the source of inspiration for the problem.)
2
votes
1answer
1k views

Minkowski's Inequality

I am wondering how to prove inequality $$\left\| \int f(.,y)dy \right\|_{p}\leq \int \left\| f(.,y) \right\|_{p}dy~~~~?$$ Here, $f$ is an integrable function on $\mathbb{R}^n$ and $\displaystyle ...
3
votes
1answer
98 views

Bound on Derivatives

Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function with the following properties. $\phi(x) = 1$ if $|x| \leq 1$ $\phi(x) = 0$ if $|x| \geq 2$ $0 \leq \phi \leq 1$ $\phi$ is ...
2
votes
3answers
243 views

finding unknown variable in Gaussian Integral

Given values of d, p and $\sigma$, is it possible to calculate the value of $\mu$? $$1-\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{y-d}^{y+d}\exp\big(-{x^2}/{2\sigma^2}\big) ...