Questions on proving, manipulating and applying inequalities.

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0
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0answers
76 views

An interesting trigonometric inequality on the 2-simplex.

Consider real parameters $-\pi<\alpha<0<\beta<\pi$. Prove that if $\alpha+\beta<0$, then \begin{align*} ...
4
votes
0answers
122 views

How to prove this inequality (already verified by numerical simulation)?

I have a conjecture which has been verified extensively by simulation. The conjecture is as follows: $\forall t \in [0, 1], \alpha \in [0,1]$, and positive real sequences $\{p\}_{i:1,\dots,n}, $, ...
0
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3answers
62 views

Prove that $|\frac{a-b}{1-\bar ab}|=1$ if $|a|=1$ or $|b|=1$ [duplicate]

Prove that $$|\frac{a-b}{1-\bar ab}|=1$$ if $|a|=1$ or $|b|=1$ I assumed $|a|=1$. Then tried to show that our statement holds. I wrote $a=a_1+ia_2$ and $b=b_1+ib_2$ and $\bar a=a_1-ia_2$ Also ...
0
votes
1answer
60 views

trace inequality $|tr(XY)| \leq tr(|XY|)$

Why does $|tr(XY)| \leq tr(|XY|)$ hold for any complex matrices where $|XY|$ denotes $\sqrt{Y^*X^*XY}$? would following proof be correct? So the trace of a matrix $A$ is the sum of its eigenvalues ...
2
votes
3answers
44 views

Prove that $ax+by+cz+2\sqrt{(xy+yz+xz)(ab+bc+ca)}\le{a+b+c}$

Let $a,b,c,x,y,z$ be positive real numbers such that $x+y+z=1$. Prove that $$ax+by+cz+2\sqrt{(xy+yz+xz)(ab+bc+ca)}\le{a+b+c}$$. my try: $2\sqrt{(xy+yz+xz)(ab+bc+ca)}\le{\frac{2(a+b+c)}{3}}$ But ...
2
votes
1answer
35 views

Prove that $\sqrt{x+y+z}\ge{\sqrt{x-1}+\sqrt{z-1}}$.

Let $x,y,z>1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that $$\sqrt{x+y+z}\ge{\sqrt{x-1}+\sqrt{z-1}}$$. I took $x=\sec^2{a}$, $y=\sec^2{b}$, $z=\sec^2{c}$ but it was not useful. ...
0
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2answers
49 views

Prove that $\frac{1}{a+ab}+\frac{1}{b+bc}+\frac{1}{c+ca} \geq \frac{3}{2}.$

Let $a,b,$ and $c$ be positive real numbers such that $abc = 1$. Prove that $$\dfrac{1}{a+ab}+\dfrac{1}{b+bc}+\dfrac{1}{c+ca} \geq \dfrac{3}{2}.$$ I thought about substituting in $abc = 1$ to ...
1
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0answers
45 views

Trace inequality $tr(|XY|) \leq \|X\| tr(|Y|) $

Why does one have $tr(|XY|) \leq \|X\| tr(|Y|) $ for any complex matrices? I do know that Cauchy Schwarz establishes $|tr(X^*Y)|\leq \|X\| \|Y\|$. Ok so far I have $\langle u,X^*Xu \rangle=\langle ...
0
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1answer
18 views

How to visualize a 3D region plot of an inequality easily?

I can't find the right way to think about the region plot of an inequality. Considering $A=\big\{ (x,y) \in \mathbb{R^2} \mid y<x+1 \big \}$, almost automatically I say: the points "under" the ...
0
votes
1answer
34 views

The norm of real powers of strictly positive bounded linear operators

Why does one have $\|A^x\|=\|A\|^x$ if $A$ is a positive, linear, bounded operator and $x$ is a real number? By spectral theorem I would deduce $$\|A^x \|=\| {U^*}^x D^x U^x ...
0
votes
1answer
65 views

Why in a triangle 2cos((A+B)/2)cos((A-B)/2)<2cos(90° -C/2)?

If $A,B,C$ angles of a triangle aka $A+B+C=180°$, and I need to evaluate $$\cos(A)+\cos(B)+\cos(C)$$ I'm out to get that at most it is $$\cos(A)+\cos(B)+\cos(C)<=3/2$$ Since: $A+B+C=180°$, then ...
5
votes
7answers
95 views

Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds

I aim to show that $$(n+1)^n<n^{(n+1)}$$ for all $n \geq 3$. I tried induction, but it didn't work. What should I do?
0
votes
1answer
14 views

Norm of pointwise product of Lp functions

Does the following inequality hold in $L_p$ spaces? $\|fg\|_p\leq\|f\|_p\|g\|_p$ How would I go about proving this? Do I need to apply Cauchy Schwarz?
0
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1answer
84 views

Naive Question Related to Inequality for Comparing a Fixed Real Number and 1/$\infty$

Given any real number $a>0$, can I say $$ a>1/b $$ for $b=\infty$? Here is my thinking: let $a>0$ be fixed, I see $$a>\lim_{b \to \infty}1/b=0,$$ hence, $a>1/b$ for $b=\infty$. ...
0
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1answer
28 views

Let $a,b$, and $c$ be real numbers. Suppose for every $c$ with $b < c$, we have $a\leq c$. Prove that $a \leq b$.

Let $a,b$ and $c$ be real numbers. Suppose for every $c$ with $b < c$, we have $a \leq c$. Prove that $a \leq b$. This is annoying me and I am stuck. Here is my approach: Given any $a,b$ ...
-2
votes
2answers
49 views

solve for x using inequalities

Solve $$\frac{x}{x-2} < \frac{x}{x-1}$$ I know for inequality you have to multiply by the denominator square but I'm not sure if this applies to this one since this contains two denominators.
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1answer
34 views

How to find the corners of a shape given 4 inequalities?

I'm trying to display the feasible space of four 2-variable linear inequalities as a quadrilateral shape. I have a simple solution so far but it makes a few key assumptions I want to remove: There ...
2
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3answers
48 views

Proving the inequality $ \frac {x+y}{x^2+y^2}\leq \frac 12 \left(\frac {1}{x}+\frac{1}{y}\right)$

Let x and y be definitely two positive numbers : Prove that : $$ \left( \frac {x+y}{x^2+y^2}\right) \leq \frac 12 \left(\frac {1}{x}+\frac{1}{y}\right)$$ I answered this one by squaring the two ...
0
votes
1answer
24 views

proving inequalities with 3 terms [duplicate]

how do you prove $9(a^3+b^3+c^3)$ $\ge$ $(a+b+c)^3$ I tried to expand by multinomial expansion the right side and got a long string so what do i do next?
0
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1answer
26 views

Find the minimum of $\sum_{cyc} {\sqrt{\frac a{2(b+c)}}}$ with $a,b,c \gt 0$

As said in the title, I have to find the minimum of the following: $$\sum_{cyc} {\sqrt{\frac a{2(b+c)}}} $$ with $a+b+c>0$ In my very last attempt, I tried to work it out using AM-GM: Since ...
1
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2answers
37 views

ceiling functions inequality

Please, help me in solving this ceiling function inequality. $ \lceil n/4 \rceil \ge 3$ I know the formal definiton of the ceiling functions: $\lceil x \rceil = n$ iff $n-1< x \le n $ ...
1
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2answers
19 views

Existence of an $L$, such that $\big|\lvert y_1\rvert^\alpha-\lvert y_2\rvert^\alpha\big|\le L\lvert y_1-y_2\rvert$, when $|y|\le b$ and $\alpha>1$

I'm trying to prove the uniqueness of solution in following IVP problem $$\frac{dy}{dx}=\lvert y\rvert^\alpha,\,\,\,y(0)=0 ~~~~~(\alpha>1)$$ One possible way here is to apply the Picard's ...
2
votes
1answer
30 views

How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$

If the set $\{f_1, ..., f_n\}$ is an orthonormal subset of inner product space $E$ and $f\in E$ then how can I show that: $$\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2.$$ How ...
1
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6answers
85 views

Is my solution for the proof of $x^2+xy+y^2 > 0$ correct? [duplicate]

The problem requires you to prove: $x^2 + xy + y^2 > 0$ assuming that $x$ and $y$ are not both zero I made use of a property proven earlier ($x^3 - y^3) = (x-y)(x^2 + xy + y^2)$) and rewrote $x^2 ...
1
vote
1answer
93 views

Show $\left( \int_1^e f(x) \; dx \right)^2 \leq \int_1^e x\,f(x)^2 \; dx$

Given that $f: [1,e] \to \mathbb{R}$ is a continuous function, show $$ \left( \int_1^e f(x) \; dx \right)^2 \leq \int_1^e x\,f(x)^2 \; dx $$ My Attempt: At first it looked rather like a ...
0
votes
0answers
21 views

Bounding simple series involving binomial coefficient

Let $r \ge 1$. What is a simple argument to show the following two inequalities: \begin{align*} \sum_{m=1}^n 2^m \binom{n}{m}^2 \Big( \frac{en}{m}\Big)^{-5rm} &\le n^{-r} \\ \sum_{m=1}^n ...
2
votes
5answers
375 views

Proving $\pi^3 \gt 31$

$$\large \pi^3 \gt 31$$ Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand. It is extremely easy with the use of any calculator, so I was wondering ...
0
votes
2answers
53 views

How do I show that this fraction is smaller than 1?

I have $a>b>0$ and $z = \frac{a-\sqrt{a^2-b^2}}{b}$ As $b\to a$ we have $z \to 1$ and as $b\to 0$ we have $z\to 0$. Is this sufficient to show that $z\lt 1$? If not how can I do it? ...
1
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1answer
30 views

Convexity of the natural exponential fuction - directly from the definition

Without using the Second Derivative Test, can the convexity of the natural exponential function be shown directly from the definition of convexity? The expression \begin{equation*} e^{t} = ...
0
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4answers
42 views

Find upper limit of summation inequality

How is it possible, with correct calculus, to find the upper limit of a summation in an equation, this could for instance be: $\sum_{n=1}^x\frac{1}{2^n}\tag{displayed}>0.99$ How would i go about ...
1
vote
2answers
63 views

Upper bound of $\int_{-\infty}^{\infty}\sin(x)dx$. [duplicate]

From another question, improprer integral $$\int_{-\infty}^{\infty}\sin(x)dx$$ is not $$\lim_{a \to \infty} \int_{-a}^a \sin x \, d x$$ but $$\lim_{a \to \infty}\lim_{b \to \infty}\int_{-a}^b \sin x ...
0
votes
3answers
41 views

Nested absolute-value inequality

I try to solve a problem in two ways, but the results are not the same. Method 1. $$\lvert \lvert x \rvert + x \rvert \le 2$$ For $x < 0$, we have $\lvert x \rvert = -x$. Therefore: $$\lvert ...
1
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1answer
29 views

$\inf_{a\in\mathbb{C}}\|f-a\|_{L^{\infty}(I)}\le|I|\|f'\|_{L^{\infty}(I)}$

Let $I\subset\mathbb{R}$ be an interval of finite length and $f:\mathbb{R}\to\mathbb{C}$ a function that is differentiable on a neighborhood of $I$. I tried to prove:$$ ...
2
votes
1answer
61 views

Given $a+b+c=3$ .Prove $ \sum _{cyc} \frac {1}{a^2+b^2+2} \le \frac 34$

Yesterday I found this on the Internet: Give 3 non-negative numbers $a,b,c$ that $a+b+c=3$. Prove $$ \sum _{cyc} \frac {1}{a^2+b^2+2} \le \frac 34 $$ I have tried to solve this using AM-GM: From ...
1
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3answers
80 views

Extreme of $\cos(A)\cos(B)\cos(C)$ in a triangle without calculus.

If $A,B,C$ angles of a triangle, show extreme value of $$\cos(A)\cos(B)\cos(C)$$ I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing helps. On this topic ...
3
votes
1answer
73 views

An inequality on three constrained positive numbers

Assume $a,b,c$ are all positive numbers, and $2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2$. Prove that: $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2\ge(ab+bc+ca)^2$$
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0answers
17 views

Degree of Majorization of Vectors

Given vectors $\textbf{x}$, $\textbf{y}$, $\textbf{z}$, such that each of $\textbf{x}$ and $\textbf{y}$ are majorized by $\textbf{z}$, i.e. $\textbf{x}\prec\textbf{z}$ and ...
3
votes
0answers
64 views

Is this inequality trivial?

Let $z = (z_1,z_2)\in \mathbb C^2$, $|z| \leq 1$. Then $$ \left( |z_1|^2 - |z_2|^2 \right)^2 \leq \left( |z|^2 + (z_1 \bar z_2 -\bar z_1 z_2 )^2 \right) \left( |z|^2 - (\bar z_1 z_2 - z_1 \bar ...
4
votes
1answer
64 views

Prove that $\frac{1}{1+a_1+a_1a_2}+\frac{1}{1+a_2+a_2a_3}+\cdots+\frac{1}{1+a_{n-1}+a_{n-1}a_n}+\frac{1}{1+a_n+a_na_1}>1.$

If $n > 3$ and $a_1,a_2,\ldots,a_n$ are positive real numbers with $a_1a_2\cdots a_n = 1$, prove that ...
2
votes
1answer
58 views

An application of Bernoulli's Inequality - relevant to an expression for $e^{x}$

I saw on another post the following proposition. \begin{equation*} 1 + x + \frac{x^{2}}{2!} + ... + \frac{x^{n}}{n!} ≤ \left( 1 − \frac{x}{n} \right)^{−n} \end{equation*} for every real number $0 ...
2
votes
1answer
55 views

Prove that at a wedding reception you don't need more than $20 \sqrt{mn}$ of ribbon to adornate the cakes.

At a wedding reception,$n$ guests have assembled into $m$ groups to converse.(The groups are not necessarily equal sized.)The host is preparing $m$ square cakes,each with an ornate ribbon ...
4
votes
1answer
77 views

Prove inequality $ a^4+b^4 \leq 2 a^2 b^2 +1. $

Let $a,b$ be such that $|a|\leq 1$, $|b|\leq 1$. Prove that then $$ a^4+b^4 \leq 2 a^2 b^2 +1. $$
1
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0answers
13 views

Arithmetic of sampling

INTRO: Suppose that a large set $Z$ is divided to two subsets $Z^A$ and $Z^B$, such that each element of $Z$ has a probability of 1/2 to be in each subset, independently of the others. By the law of ...
2
votes
3answers
57 views

Trace inequality on positive matrices

Let $A,B,C\geq 0$ be self-adjoint matrices. Assume $A\leq B$. Is it true that $$\mathrm{tr}(ACAC) \leq \mathrm{tr}(BCBC)?$$ How to prove this?
1
vote
0answers
50 views

Proof of the inequality $e^x \le x+ e^{x^2}$ [duplicate]

$e^x \le x+ e^{x^2}$ for $x\in \mathbb{R}$. I'm struggling to show this inequality. I've tried differentiation but to no avail. How can I show this? I would greatly appreciate some help.
0
votes
1answer
33 views

How to prove Clarkson's inequality?

I do not know how to prove one of the four Clarkson's inequalities: let $u,v \in L^p(\Omega)$, if $1 < p < 2$, then $$ \bigg\lVert \frac{u+v}{2} \bigg\rVert_p^p + \bigg\lVert \frac{u-v}{2} ...
2
votes
2answers
28 views

Prove the inequality $\tan{\frac{\pi\sin{x}}{4\sin{\alpha}}}+\tan{\frac{\pi\cos{x}}{4\cos{\alpha}}} > 1$

Prove the inequality $$\tan{\dfrac{\pi\sin{x}}{4\sin{\alpha}}}+\tan{\dfrac{\pi\cos{x}}{4\cos{\alpha}}} > 1$$ for any $x, \alpha$ with $0 \leq x \leq \dfrac{\pi}{2}$ and $\dfrac{\pi}{6} < ...
2
votes
1answer
77 views

Prove that if $a,b,$ and $c$ are positive real numbers, then $\frac{a^3}{a^3+2b^3}+\frac{b^3}{b^3+2c^3}+\frac{c^3}{c^3+2a^3} \geq 1.$

Prove that if $a,b,$ and $c$ are positive real numbers, then $$\dfrac{a^3}{a^3+2b^3}+\dfrac{b^3}{b^3+2c^3}+\dfrac{c^3}{c^3+2a^3} \geq 1.$$ This question seems hard since we aren't given any other ...
2
votes
0answers
29 views

Doob's $L^p$ inequality - Case $p = 1$

I have found in the wikipedia page following generalisation of Doob's so-called $L^p$ inequality, for general nonnegative submartinagles $X_s$: $$E[\sup_{0 \le s \le T} X_s] \le \frac{e(1 + ...
0
votes
2answers
27 views

A gambler bet half his money at each toss of coin.If at the end of the game the wins and loses of the gambler are equal did he gain?

A gambler played the following game with a friend The gambler bet half the money in his pocket on the toss of a coin;he won on heads and lost on tails.The coin was tossed and the money handed over.The ...