Questions on proving and manipulating inequalities.

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0
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1answer
23 views

Proving a simple inequality with closure under multiplication

Suppose we have inequalities $0≤a<b$ and $0≤c<d$ prove $$ac<bd$$ using closure under multiplication, as if $x>0$ and $y>0$ then $xy>0$.
2
votes
0answers
108 views

Proving the inequality $\frac{\mathrm{arccot} 2\sqrt{2}}{\pi\log\zeta(3)}-\frac{\log^2(1+e^{-\pi})}{\pi}>\frac{131e^2+422e-1151}{222e^2+279e-757},$

I have come across the following inequality in my studies $$\frac{\text{arccot}2\sqrt{2}}{\pi\log\zeta(3)}-\frac{\log^2(1+e^{-\pi})}{\pi}>\frac{131e^2+422e-1151}{222e^2+279e-757},$$ where ...
2
votes
2answers
67 views

If $a,b,c$ are real numbers all less than or equal to $1$ such that $a+b+c=0$ , then is it true that $(1-a)(1-b)(1-c) \le 1$?

If $a,b,c$ are real numbers all less than or equal to $1$ such that $a+b+c=0$ , then is it true that $(1-a)(1-b)(1-c) \le 1$ ? I tried Weierstrass inequality but noticed that $a,b,c$ might all not be ...
4
votes
1answer
82 views

Inequality in 4 variables

I came across the following problem in a book. Four real numbers $p,q,r,s$ satisfy $p+q+r+s=9$ and $p^2+q^2+r^2+s^2=21$. Prove that there is a permutation $a,b,c,d$ of $p,q,r,s$ Such that $ab-cd\ge ...
5
votes
0answers
125 views

To prove triangle inequality for $d : \mathbb C \times \mathbb C \to \mathbb R$ ; $d(x,y):=\frac {|x-y|} {\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ [duplicate]

Is the function $d : \mathbb C \times \mathbb C \to \mathbb R$ defined by $d(x,y):=\dfrac {|x-y|} {\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ a metric ? I can easily prove it is symmetric and positive-definite ; ...
0
votes
3answers
102 views

A symmetric inequality with three variables

Here is an inequality I came across in a book that I was doing:- Prove that for all $a,b,c\gt 0$ $$\frac {a+b+c}{(abc)^{1/3}}+\frac {8abc}{(a+b)(b+c)(c+a)}\ge 4$$ I have no idea about how to approach ...
2
votes
4answers
73 views

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$

Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$ I am doing it by simple induction adding $1/\sqrt{x+1}$ to both sides, but I can't find a way to manipulate this expression and find ...
3
votes
2answers
49 views

How prove this inequality $\frac{e^x}{x+1}>\frac{\cos{x}}{\sin{x}+\sqrt{2}}$

Today,when I use wolf found this following inequality let $x>-1$, show that $$\dfrac{e^x}{x+1}>\dfrac{\cos{x}}{\sin{x}+\sqrt{2}}$$ I found this I want $$\Longleftrightarrow ...
0
votes
1answer
29 views

equivalence of any two polynomials of same degree

Let $P$ and $Q$ be two polynomials of same degree with real coefficients. Assume that $P$ and $Q$ have no real roots. It seems to me that $P$ and $Q$ are equivalent in the sense that there are some ...
1
vote
1answer
73 views

Proving the inequality $4(\ln 3)^2/33+187/(15\pi^{3/2}) < \sqrt{29}-3.$ without a calculator [closed]

I need help showing that $$\frac{4(\ln 3)^2}{33}+\frac{187}{15\pi^{3/2}} < \sqrt{29}-3.$$ I've tried many methods, however could not prove it.
-6
votes
1answer
295 views

proving that $3^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}}}}< 8$ [closed]

proving that $$3^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}}}}< 8$$ I don't have any idea about how to prove it.
1
vote
1answer
30 views

Gronwall's Lemma type problem

I have a function $X(t)\geq 0$, with initial condition $X(0)=X_0\geq 0$ and constants $\alpha < 0$, $\beta > 0$ and $\gamma <0$ such that $$\frac{d}{dt} X(t)^2 \leq \alpha X(t)^2 + \beta ...
1
vote
0answers
18 views

Hyperbolic trigonometric inequality

Is the following hyperbolic trigonometric inequality correct and if so, is there a simple derivation? $$\tanh A- \tanh B \geq (A-B)(\operatorname{sech}{^2}(A)), \qquad \forall A\ge B\ge 0.$$
0
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1answer
16 views

Sum involving the “distance to the nearest integer function”

I want to prove that if $||x||$ is the distance between $x$ and the nearest integer to $x$, $\{\alpha_1,\ldots, \alpha_N\}$ are points in $\mathbb{R}$/$\mathbb{Z}$ and we define $$S(y) = ...
-9
votes
1answer
57 views

Please help, i bet this is 2 simple 4 u [closed]

=IF 300=(X-2000)*-0,009 then what is X ??
1
vote
2answers
36 views

Represent if-else or OR condition in a linear equation (optimisation with simplex algorithm)

I would like to write some linear equations and inequations to state that the sum of all possive x - C is smaller than L. As my ...
2
votes
1answer
141 views

Reference or proof for an integral inequality

The following seems believable and quasi-intuitive to me, but it also doesn't quite seem trivial, and I'm not sure whether I've seen it stated before. Let $f$ be a complex-valued integrable function ...
0
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0answers
49 views

Bounded second derivative implies $|f(a)| \leq C(1+|a|^2)$

Let $f: \mathbb{R} \rightarrow [0, \infty)$ be a twice differentiable function such that $0 < A \leq f''(x) \leq B$ for all $x$. Then there exists a $C>0$ such that $|f(x)| \leq C(1+|x|^2)$. ...
1
vote
1answer
18 views

Inequality with absolute value of sum of terms

I am having the following inequality $$x>|A+B+C+D|$$ and I can prove that $|A|<a$, $|B|<b$, $|C|<c$ and $|D|<d$ Can I express the inequality in terms of $a,b,c,d$? Or am I following ...
3
votes
0answers
37 views

The relation between $\inf_{R\in \mathsf{U}_n} \left\Vert A - BR\right\Vert^2_F$ and $\left\Vert AA^*-BB^*\right\Vert$

Suppose that $A$ and $B$ are two arbitrary $m\times n$ matrices with $m>n$. Let $\mathsf{U}_n$ denote the set of $n\times n$ unitary matrices. I'd like find positive constants $c_1$ and $c_2$ such ...
21
votes
2answers
526 views

Proving that the triangle inequality holds for a metric on $\mathbb{C}$

Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as: $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} + \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$ I ...
0
votes
1answer
11 views

positive lower bound of $\tanh t -\tanh (t/(1+a)),0<a<1$

I am looking for a positive lower bound for $f(t)=\tanh t -\tanh (t/(1+a)),0<a<1,t\geq 0$. I expect to find that: $$f(t)\geq a g(t),\qquad g(t)>0\tag{1}$$. What does the $g(t)$ look like? ...
0
votes
1answer
63 views

Mathematical induction

The sequence of real numbers $a_1, a_2, a_3,...$ is such that $a_1=1$ and $$a_{n+1}=\left(a_n+\frac1{a_n}\right)^\lambda,$$ where $\lambda$ is a constant greater than $1$. Prove by mathematical ...
1
vote
2answers
43 views

Minkowski inequality of infinite sum

For $1\leq p <\infty,$ Given $\{f_n\}^{\infty}_{n=1}$ be a sequence of function in $L^{p}(\mathbb{R}).$ Show that $\left\Vert \sum\limits_{n=1}^\infty f_n\right\Vert_p \leq \sum\limits_{n=1}^\infty ...
0
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0answers
31 views

Markov/Chebyshev - Returning homework to students

Homework was returned to students in a random way. What can we say about the probability that more than 10 students will get back the homework that they gave in in the beginning? Try to find the ...
1
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0answers
33 views

A maximal inequality on distance to median, so called Lévy's inequality?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.3, ex6) Suppose $X_1,\dotsc,X_n$ are i.i.d. random variables, and $S_j:=X_1+\dotsb+X_j,S_j^0:=S_j-m_0(S_j)$, where $m_0(S_j)$ is a ...
1
vote
2answers
56 views

Inequality with 6 variables

I am trying to show the following, Let, $x,y,z,a,b,c$ be positive reals such that $x+y+z=1$ Then $ax + by + cz +2 \sqrt{(ab+bc+ca)(xy+yz+zx)} \leq a+b+c$ This is practice problem 6 in this document ...
0
votes
1answer
33 views

Are these regions subsets of one another?

Assume we have a region defined by a relationship between three variables as follows $$R_1: |x|^2 +2\big|\ xz-|y|^2\big| \leq |y|^2$$ Also assume we have another relationship as follows $$R_2: ...
4
votes
5answers
79 views

Solution to $\sqrt{x^2-5}+3>|x-1|$

I tried many ways to solve this but I just can't figure it out... $$\sqrt{x^2-5}+3>|x-1|$$
0
votes
1answer
22 views

Prove there is a unique solution for $\log x=cx$ when $c \le 0$

When $c=0$ it is clear, because $\log x=0 \iff x=e$. But when $c<0$, given $x$ has to be bigger than $0$ (otherwise $\log x$ is not defined), $0<x<1$ and then both $\log x$ and $cx$ are ...
2
votes
3answers
35 views

“Simple” rational inequality - How does one deduce this?

I am a little puzzled by a past exam question. Part (a) is very easy and part (b) is easy too. However, according to the question, and the mark scheme, I should be able to deduce the answer to part ...
0
votes
2answers
72 views

Inequality masquerading as equations

Find all triples $(x,y,z)$ with $x,y,z \in (0,1)$ satisfying $$(x+\frac 1{2x}-1)(y+\frac1 {2y}-1)(z+\frac 1{2z}-1)=(1-\frac{xy}z)(1-\frac{yz}{x})(1-\frac{zx}{y})$$I took the LCM and simplified the ...
1
vote
1answer
17 views

Is it an axiom that the inequalities are translation-invariant or can we prove it?

I was thinking about the inequalities on the set of real numbers. To me and everyone else, it's been taught that an inequality is translation-invariant, i.e.: $x < y \implies x + c < y + c ...
0
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1answer
33 views

Help proving that $\lim f(x,y)=|x|^{a}|y|^{b} / (|x|^{c} + |y|^{d}) = 0$ if $a/c + b/d >1$.

First, $a/c + b/d >1$ is the same as $a + b > c + d$. Next, my plan is to use squeeze theorem. Find $B(x,y)$ such that $|f(x,y)-0|<B(x,y)$, where $\lim B(x,y) = 0$. What I have so far: ...
0
votes
2answers
38 views

Lower bound an expression given a relationship between three variables

Given the following relationship between three variables $$ |x|^2 \leq 2|y|^2 \leq |z|^2$$ I would like to lower bound the function $F$ below and end up with function of $z$ only. I think this can ...
1
vote
3answers
80 views

Is there a trick to prove that $(x^4+y^4)^{1/2} \leq x^2+y^2$?

$(x^{4}+y^{4})^{1/2} \leq x^{2}+y^{2}$ I tried multiplying the original by 1 = $\displaystyle \frac{(x^{4}+y^{4})^{1/2}}{(x^{4}+y^{4})^{1/2}}$, but that just brings me back to the original. Ive ...
1
vote
1answer
19 views

Classic $c_r$-inequality in $l_r$ space: $E(|X+Y|^r)\leq c_r[E(|X|^r)+E(|Y|^r)]$

How does one prove: $$ E(|X+Y|^r)\leq c_r\bigg[E(|X|^r)+E(|Y|^r)\bigg] $$ where $c_r=1$ if $0<r\leq 1$ and $2^{r-1}$ if $r>1$? This is a classic result whose proof I once knew but have ...
0
votes
1answer
54 views

Prove $(x^n+1)^{\frac 1n}≤ x+1$. x>0 What is wrong with my proof?

This is what I did so far: I shall prove this by induction on $n$. Let $n=1$, then $x+1≤x+1$ which is correct. Let the inequality hold for $n$. Then, $(x^n+1)^{\frac 1n}≤x+1$, e.g, ...
0
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0answers
17 views

when does the equality hold for the matrix norm triangle inequality and product inequality

So here is the problem: When does the equality hold for the following two famous "matrix" norm inequalities: $\|A+B\|\leq \|A\|+\|B\|$ $\|CD\|\leq\|C\|\|D\|$ For any norm you prefer. But I'm ...
6
votes
2answers
86 views

Prove the following inequality without using differentiation

Let $a,b,c$ be real numbers that satisfy $0\le a,b,c\le 1$. Show that $$\frac a{b+c+1} + \frac b{a+c+1} + \frac c{a+b+1} + (1-a)(1-b)(1-c) \le 1.$$ I don't know where to start. Multiplying ...
2
votes
3answers
31 views

Hölder inequality in case $q=p=2$.

It should return the Cauchy-Schwarz inequality, but I'm having trouble with comparing the left sides of the inequalities: For example, if $x=(4,3)$ and $y=(3,-4)$, then $\sum_{v=1}^2 |x_vy_v| = ...
0
votes
2answers
83 views

Proving the inequality $e^{\sin(\sqrt{2}/{7})}<11/9$ without calculator

Proving the inequality $$e^{\sin(\sqrt{2}/{7})}<11/9$$ without calculator I tried to prove it by taking some terms of Taylor series to find a value which comparison with the $11/9$ but I ...
0
votes
1answer
32 views

How can Chebychev's inequality give this (from Random Forests by Breiman)?

I read Random Forests by Leo Breiman and found this on page 8: (the notation has been simplified): We have $s = E(mr)$ and $s\geq 0$, then $P(mr < 0) \leq P(\vert mr-s \vert \geq s) \leq ...
1
vote
1answer
32 views

Prove $\mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 ) $ for binomial variable $k$

Suppose we have a Binomial variable: $$ k \sim Bin(l,\alpha) $$ Is it possible to prove/disprove that: $$ \mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 ) $$ EDIT: it's been ...
-1
votes
1answer
40 views

Is this logarithmic inequality true?

Assume we have two complex variables $h_i$ and $h_d$ which satisfy the following relationship $$ 2\ |h_i|^2\leq \ |h_d|^2$$ can we say that $$\log\left( 1+ \frac{\big||h_d| - ...
0
votes
4answers
84 views

Is the inequality true? [closed]

Is the following relationship true $$\big| |x|-y\big|^2 \leq \big| |x|-|y|\big|^2$$ I am guessing we have to use the triangle inequality... Thanks
0
votes
3answers
63 views

Inequality $\arctan x ≥ x-x^3/3$

Can you help me prove $\arctan x ≥ x-x^3/3$? I have thought of taylor but I have not come up with a solution.
0
votes
0answers
26 views

Looking for proof of Young's inequality avoiding ( or only using very little) calculus [duplicate]

What are the different proof of Young's inequality that if $p,q $ are positive real numbers such that $\dfrac 1p+ \dfrac 1q=1$ then for positive real numbers $x,y$ , $\dfrac {x^p}{p}+\dfrac {y^q}q\ge ...
0
votes
2answers
46 views

what to do with: logarithmic, trigonometric and exponential inequalities with variable outside

After encountering this inequality: $$ e^{x/2}=2x+1 $$ that leads me to: $$ x=2\ln(2x+1) $$ I realized that I don't know how to solve it. But this lack of knowledge expands also to $\cos(x)=x$ or ...
4
votes
1answer
295 views

inequality $10<2^{2^{\frac {3}{\log_2 \log_2 10}}}$

While working on this question I ended up with $10<2^2{^{\frac {3}{\log_2 \log_2 10}}}$ I am looking for answers using methods similar to this or this or this or this. Alternative original ...