Questions on proving and manipulating inequalities.

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1answer
18 views

Is Hlawkas Inequality holds for sobolev space

im wondring is that inequality holds for any functionnal space such as sobolev space and if it's true how we can write it in that space /HlawkasInequality any help would be apperciated
2
votes
1answer
65 views

Inequality in tetrahedron

You are given a tetrahedron $ABCD$. $ACB = ADB = 90^\circ$. $AC = CD = DB$. Prove that $AB < 2 * CD$. I know that $AD = CB$ and $CBD = DCB = ADC = CAD$.
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0answers
47 views

Inequality in inverse Laplacian

I have the following problem, which is motivated by geometric diffusion on a directed graph. Conjecture. Let $A \in [0,1]^{n\times n}$ be strictly substochastic - i.e. $\forall i ~ \sum_j A_{i,j} ...
1
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1answer
18 views

Procedure in proving inequalities (or bounds) involving minus sign

Usually when we want to bound an expression involving sums, it is easy to proceed by bounding each term separately since we "do not lose" too much using the triangle inequality $|x+y|\leq |x|+|y|$ ...
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2answers
49 views

Find the minimum possible value of $x(1-z)+y(1-x)+z(1-y)$

It is given that $$xyz=(1-x)(1-y)(1-z)$$ and $$x, y, z \epsilon (0,1)$$ Find the minimum possible value of the expression: $$x(1-z)+y(1-x)+z(1-y)$$ Using the AM-GM inequality concepts, I can write ...
3
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3answers
278 views

Prove $|\cos^2(z)| + |\sin^2(z)| > 1$ for complex numbers $z$ with nonzero imaginary part

Prove $$|\cos^2(z)| + |\sin^2(z)| > 1$$ for $\operatorname{Im}(z) \ne 0$ I know from using the triangle inequality, $|x+y| \leq |x| + |y|$, that $|\cos^2(z)| + |\sin^2(z)| \geq 1$ but I don't ...
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1answer
39 views

Inequality which involves complex numbers and absolute values

How can I solve the following inequality: $|\frac{(1+(1-\theta)z)}{1-\theta z}| \leq 1$ ? $z$ is a complex number. I have to find the values of $\theta$ for which the inequality is satisfied.
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4answers
98 views

Proof that $n^n<(n!)^2$ for $n>2$

Prove that $n^n<(n!)^2$ for $n>2$ I tried math induction, but couldn't prove that $(k+1)^{k+1}<((k+1)!)^2$.
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0answers
35 views

A simple elementary inequality (without ABC Thm)

I want to solve this following inequality. $$\sum_{sym } x^5 - 7 \sum _{sym} x^4y + 7 \sum_{sym} x^3 y^2 + 10 \sum_{sym} x^3 yz - 11\sum_{sym} x^2y^2z \ge 0 $$ whenever $ x,y,z \ge 0 $ I do not want ...
0
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1answer
34 views

If $x\gt0$ prove that $\sqrt x\gt0$

I tried to prove by letting $x=y^2$ then $y^2\gt0$ and by taking the square root of both sides we have that $y\gt0$ and therefore $\sqrt x\gt0$. The problem mentions that a proof by contradiction can ...
2
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2answers
130 views

Miklos Schweitzer 2014 Problem 8: polynomial inequality

Look at problem 8 : Let $n\geq 1$ be a fixed integer. Calculate the distance: $$\inf_{p,f}\max_{x\in[0,1]}|f(x)-p(x)|$$ where $p$ runs over polynomials with degree less than $n$ with real ...
0
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1answer
17 views

Find constant K such that the inequality is satisfied

Find constant K s.t. $|tx| + |e^{t^2/2}| \leq K(|x|+1)$ $|tx-ty| \leq K(|x-y|)$ What K works, and how to come up with one? It doesn't look like there is any such K considering there is no upper ...
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1answer
32 views

how to solve absolute inequality functions

I have noticed in the past while solving inequality functions that when you want to change the inequality symbol you need to switch the $+$ or $-$ signs of the function itself. How do I solve this ...
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1answer
92 views

How to prove this integral-inequality.

Suppose $f$ is twice differentiable and satisfies $f(0)=0$. Prove the inequality. $$\int_0^1 |f(x)f'(x)| dx \le\ \frac{1}{2} \int_0^1 |f'(x)|^2 dx $$ This is a problem from undergraduate math ...
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2answers
76 views

Proof of a Landau-inequality

I have to prove or disprove the following: $$ 2xlog_{10}((x+2)^2) + (x+2)^2log_{10}(\frac x2) \in O(x^2log_{10}(x))$$ My approach (with $log$ is meant $log_{10}$): $4x log(x+2) + (x+2)^2log(x) - ...
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6answers
50 views

Prove that $n^2 > n+1 \quad\forall n \geq 2$ using mathematical induction

Prove $n^2 > n+1$ for $ n \geq 2$ using mathematical induction So I attempted to prove this, but I'm not sure if this is a valid proof. Base case, $n = 2$ $$ 2^2 > 2+ 1 $$ $n = k + 1$, ...
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2answers
93 views

If $a,b,c$ are positive real numbers, then $\frac{a}{ab+1}+\frac{b}{bc+1}+\frac{c}{ca+1}\geq \frac{3}{2}$ [closed]

If $a$, $b$, and $c$ are positive real numbers such that $abc=1$, then prove that $$\frac{a}{ab+1}+\frac{b}{bc+1}+\frac{c}{ca+1}\geq \frac{3}{2}$$ Progress I think the relevant concept would ...
1
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1answer
28 views

Check this inequality using induction

I would like to prove this inequality using induction $$\sum_{k=1}^r \frac{2^k}{k^2} \le 9 \frac{2^r}{r^2}$$ The base case is simple enough: for $r=1$, we have: Here's my attempt at the inductive ...
2
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1answer
47 views

How to use Cauchy-Scharwz inequality to prove differentiable?

I'm attempting to understand how to prove the function f such that $$f(x,y)=\frac{x^3y}{x^4+y^2}\;if\;(x,y)\neq (0,0)$$ $$f(x,y)=(0,0)\;if\;(x,y)=(0,0)$$ is continuous in $\mathbb R^2$. The solution ...
2
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3answers
65 views

Let $a, b, c$ be positive real numbers such that $a + 2b + 3c = 26$ and $a^2 + b^2 + c^2 = 52$.

Find the largest possible value of $a$. I used the Cauchy Schwarz inequality $(ax+by+cz)^2 \leq (a^2+b^2+c^2)(x^2+y^2+z^2)$ as follows: $a + 2b + 3c = 26$ is given; adding $a$ to both sides gives ...
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1answer
20 views

Upper bound for $\left|(a+\pi)^k e^{i \pi x }- (a-\pi)^k e^{-i \pi x }\right|$.

I want to find an upper bound for $$\left|(a+\pi)^k e^{i \pi x }- (a-\pi)^k e^{-i \pi x }\right|\leq ?$$ where $a,k\in\mathbb{N}, x\in \mathbb{R}$. thanks a lot
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0answers
18 views

Verbal explanation of meaning of biological equation

Now replacing each variable with its meaning looks like this: Can anyone explain to me in "layman's terms" how to verbally state what the left side of this inequality means? In context: I have ...
4
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1answer
86 views

How to prove that $\sqrt{15}-\sqrt{8}+\sqrt[3]{7}<3$?

How to prove that $\sqrt{15}-\sqrt{8}+\sqrt[3]{7}<3$ without using calculus/high school methods or routine calculation of the roots with the precision needed? The difference between them is close ...
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3answers
35 views

How to solve greater & less / inequal equation

Two farmers, Eric and Josef where talking. "How many sheeps do you have?" asked Eric. "If I divide my sheeps in $2, 3, 4, 5,$ or $6$ groups, there will always be 1 sheep left." Josef answered. How ...
1
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1answer
55 views

Problem with an easy inequality

Help me please with this inequality . I need to prove this inequality, but i don't have any idea how to do this! $$\frac{(1+\frac{x}{2})^n-1}{(1+x)^2-1}\leq\frac{1}{2}$$
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1answer
46 views

Variance of the minimum between a constant and a random variable

Let $X$ be a random variable, and $c$ a constant. How can we prove that: $var[min(X,c)]\le var[X]$?
2
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1answer
50 views

estimate of infinite norm by $(p,q)$ norms

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$ I think this is true. I tried to prove it using integration by ...
0
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1answer
30 views

Probability with inequality condition

Can someone explain how to solve this problem. Since I can get 7 integers from 1st inequality 5 integers from 2nd inequality I got total number of cases of 35. Then I counted the possibilities that ...
0
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1answer
46 views

A binomial inequality

I've tried both expanding the binomials as well as trying to deduce something from the hypergeometric distribution, but I don't see how to prove: $${N\choose n}^{-1}\sum_{i\geq j}{M\choose ...
2
votes
3answers
51 views

How to isolate $n$ in the inequality $ 3n + 7n^3\gt c(17 + 34n^2) $?

I have an equation $$ 3n + 7n^3\gt c\left(17 + 34n^2\right) $$ and I want to turn this inequality into something like $$ n \gt c(\mbox{something that does not have}\ n) $$ I don't know why but ...
5
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0answers
58 views

How big is a tetrahedron?

Let $T$ be a tetrahedron with volume $vol(T)$ and edge lengths $a,b,c,d,e,f$ and let $sum(T) = a^3 + b^3 + ... + f^3$. We wish to compare $vol(T)$ with $sum(T)$. [ IMO (1961 #2 ) handles the case of ...
5
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1answer
71 views

Lower estimate for $(\frac{\ln(1+2x)}{\ln(1+x)}-1)(1+2x)^{1/2}$ where $x>0$

I want to prove that: $$\left(\frac{\ln(1+2x)}{\ln(1+x)}-1\right)(1+2x)^\frac{1}{2}\geq 1$$ where $x>0$. Any help appreciated. Thanks!
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2answers
31 views

Prove least upper bound for piecewise inequality

Say $S = \left\{ x \in \mathbb{Q} \mid 0 < x < 1 \textrm{ or } 3 < x < 4 \textrm{ or } 6 < x < 8 \right\}$. I want to show that the least upper bound of this set is $8$, but I don't ...
1
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1answer
37 views

Inequality problem involving Cauchy Schwarz inequality

If $a+b+c=3$, prove that $\sum \frac{a^2}{b^2-2b+3} \geq 3/2$. How to prove it using Cauchy Schwartz inequality? Denote the expression with $P$. What I got was $P$ is greater than or equal to ...
1
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1answer
35 views

Proving the finiteness of an integral

Really stuck in this question: Let $f: [0, +\infty) \rightarrow \mathbb{R} $ and $g: [1, +\infty) \rightarrow [-\infty, +\infty] $ be non-negative continuous functions such that for all $\epsilon ...
0
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1answer
57 views

How prove this inequality $|x|+|y|+|z|\ge 2$ with $xy+yz+xz=-1$

Question: let $x,y,z\in \mathbb{R}$ such that $$xy+yz+xz+1=0$$ show that: $$|x|+|y|+|z|\ge 2$$ I know $$(|x|+|y|+|z|)^2\ge 3(|xy|+|yz|+|xz|)\ge 3(xy+yz+xz)$$ But this relust is not ...
0
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1answer
48 views

Finding the minimum value of $M$. [closed]

For $0<z<y<x$. Finding the minimum value of $M$, where: $$M=\frac{y}{x-y}+\frac{z}{y-z}+\frac{x^2}{8z(\sqrt{xz}-z)}.$$ Thank you very much!
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3answers
26 views

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$. Using $e^x=\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$. I have this inequality in the proof of ...
5
votes
2answers
59 views

Prove or disprove that $p_n > e^{p_n - p_{n-1}}$ for large enough $n$.

Let $p_n$ denote the $n$-th prime. Prove or disprove that for large enough $n$ we have $$p_n > e^{p_n - p_{n-1}}.$$ The inequality trivially holds for all the twin primes larger than $7$. With ...
1
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1answer
61 views

Prove that the product space is a metric space.

I have the following problem: Let $(S,d)$ and $(T,e)$ be two metric spaces. Their product space has underlying set $$S\times T=\{(s,t)|s\in S,t\in T\}$$ and metric ...
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1answer
47 views

$\cos\left(\frac{x+y+z}{3}\right)\geqslant \frac{\cos(x)+\cos(y)+\cos(z)}{3}$ for $-\frac{\pi}{2}<x,y,z<\frac{\pi}{2}$

I am trying to prove the inequality of the title, in vain so far. It is easy to prove that $\cos$ is concave downward on $(-\frac{\pi}{2},\frac{\pi}{2})$ using differentiation, and this would imply ...
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0answers
36 views

Meromorphic function with a simple pole and a simple zero, and satisfies an inequality. What can it be?

Describe all meromorphic functions f(z) in the complex plane with a simple pole at z=1, a simple zero at z=-1, and for which $$|f(z)|\le M|z|,$$ for $|z|\ge 2$ for some $M>0$. I know that, ...
2
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1answer
32 views

How can I show that these two problems have the same optimal solution?

How can I show that these two problems have the same optimal solution: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\}$$ $$\inf \{ x^TAx + b^Tx : 1-x^Tx = 0,\ x \in \mathbb R^n\}$$ when ...
0
votes
2answers
19 views

Absolute error in machine-precision terms.

I am trying to wrap my head around errors in floating point calculations. Let me denote absolute error as follows: $e = |x - \hat{x}|$, where $x$ is the exact number and $\hat{x}$ is its floating ...
0
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0answers
21 views

A consequence of the Lipschitz inequality

Is the following true? If $F : \mathbb R\to\mathbb R$ is Lipschitz then there exists $L'$ such that for every $x \in\mathbb R$ $$|F(x)| \le L'(1+|x|)$$
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3answers
42 views

Why does $-b < a < b \implies |a| < b$ (and also the converse)?

I don't have any intuitiom for this because it's just something I memorized. I only understand that $|a| = a$ if $a$ is already positive (or $0$), and $|a| = -a$ if $a$ is negative since we want to ...
-2
votes
1answer
37 views

How prove this inequality $n^{kn}\ge\left(\sum_{i=1}^{k}n^i\right)^{n-1}$

show that $$n^{kn}\ge\left(\sum_{i=1}^{k}n^i\right)^{n-1},\forall k,n\in \mathbb{N}^{+}$$ here give a induction mathlinks I think have integral methods? I know ...
0
votes
2answers
21 views

Prove proposition on real numbers and inverses.

Prove the following proposition Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$ So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$.
4
votes
3answers
52 views

Show: $\left(\sum_{k=0}^n a_k\right)^2\leqslant (n+1)\sum_{k=0}^n a_k^2$

Show: $\left(\sum_{k=0}^n a_k\right)^2\leqslant (n+1)\sum_{k=0}^n a_k^2$ for $n\geqslant 0$ and $a_k\in\mathbb{Z}_{\geq 0}$. Wanted to show this by induction: $n=0: a_0^2\leqslant a_0^2$ Assume it ...
3
votes
1answer
23 views

limsup of average smaller than limsup

I have read this solution, but I could not understand it. It has shown $$\sigma_n\leqslant \frac 1n\sum_{j=1}^ks_j+\sup_{l\geqslant k}s_l,$$ but how does it go to $$\sup(\sigma_n)\leqslant \frac ...