Questions on proving, manipulating and applying inequalities.

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1
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2answers
51 views

Prove that $ \sum_{cyc} \sqrt{\cot{A}+\cot{B}} \ge 2\sqrt2$

Let $ \triangle ABC$ be an acute-angled triangle. Prove that $ \sum_\text{cyc} \sqrt{\cot{A}+\cot{B}} \ge 2\sqrt2 $ Attempt Since $\triangle{ABC}$ is acute, we may say that $A,B,C \in (0, ...
1
vote
0answers
42 views

Prove that if $x,y,z$ are positive real numbers and $ xy+xz+yz = 1$ then $\sqrt{x}+\sqrt{y}+\sqrt{z} \geq 2$

Prove that if $x,y,z$ are positive real numbers and $ xy+xz+yz = 1$ then $\sqrt{x}+\sqrt{y}+\sqrt{z} > 2$. I am having a hard time relating the square roots in the inequality to the given ...
7
votes
3answers
82 views

If all entries of matrix $X$ are the same, then $\det (A+X)\det (A-X) \leq \det (A^2)$

I want to prove that $\det (A+X)\det (A-X) \leq \det (A^2)$ where $X $ is a matrix whose $n^2$ entries are all the same. I tried to write down the expressions involved but that didn't help me prove ...
2
votes
1answer
36 views

Number of solutions for inequality $3x + 7y + z \leq 198$ in the non-negative integers

Please explain me how can I calculate a number of possible solutions for such inequality and obtain those values: $$3x + 7y + z \leq 198$$ where $x,y,z$ are integers (non-negative).
2
votes
2answers
51 views

Determine (or evaluate) the sum of the series $\sum_{j=0}^\infty (-1)^j \frac{\frac{3}{2}}{\frac{3}{2}+j} \frac{x^{2j}}{(2j)!}, \ \ \ x\in\mathbb R$.

Determine (or evaluate) the sum of the series $$\sum_{j=0}^\infty (-1)^j \frac{\frac{3}{2}}{\frac{3}{2}+j} \frac{x^{2j}}{(2j)!}, \ \ \ x\in\mathbb R$$ and $$\sum_{j=0}^\infty (-1)^j ...
4
votes
3answers
108 views

Induction on inequalities (a sum less than a particular value) [duplicate]

I am trying to solve this inequality by induction. I just started to learn induction this week and all the inequalities we had been solved were like an equation less than another equation (e.g. $n! ...
3
votes
3answers
61 views

Prove that for all positive real numbers $a,b,$ and $c$ we have $a^5+b^5+c^5 \geq a^3bc+ab^3c+abc^3$

Prove that for all positive real numbers $a,b,$ and $c$ we have $a^5+b^5+c^5 \geq a^3bc+ab^3c+abc^3$. This question reminds me of rearrangement, but I can't really find two sequences that fit. ...
1
vote
2answers
42 views

Prove that $n^k<(1+a)^n$ for sufficiently large $n$

Prove using the binomial theorem, that for natural numbers $n$ and $k$ and real positive number $a$, $n^k<(1+a)^n$ for all $n>N$. Using the binomial theorem, ...
4
votes
1answer
86 views

Prove that $0 \leq ab + ac + bc - abc \leq 2.$

Let $a,b,$ and $c$ be nonnegative real numbers such that $a^2+b^2+c^2+abc = 4$. Prove that $$0 \leq ab + ac + bc - abc \leq 2.$$ I tried using rearrangement to get $a^2+b^2+c^2+abc = 4 \geq ...
0
votes
0answers
70 views

Upper bound on successive difference inequalities

I would like to know tight upper bounds of the following equations perhaps some might have telescopic sum which can result in very tight bound.In the following equations assume $B\geq a_i \geq ...
0
votes
1answer
43 views

Where is that half coming from?

A few lessons ago, my professor proved Poincaré inequality in the following form: Let $\Omega$ be a domain contained in $\mathbb{R}^{N-1}\times(0,a)$ for some $N\in\mathbb{N},a>0$. Then for all ...
1
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0answers
26 views

Hilbert's inequality for $\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{\left(n-m\right)^\lambda}\right|$.

We know that, the Hilbert's inequality for double series states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ for $a_n\in\mathbb C$. I'd like to know if inequalities ...
0
votes
4answers
39 views

Determine if there is a solution in inequality [closed]

Does this inequality have a solution? $$x^2 < 5x - 6$$ I try to solve but after I check, the solution is wrong.
3
votes
0answers
88 views

Simpler proof for $\frac{a^3b}{c}+\frac{b^3c}{d}+\frac{c^3d}{a}+\frac{d^3a}{b}\geq a^3+b^3+c^3+d^3$

Let $a\geq b\geq c\geq d>0$. Prove that: $$\frac{a^3b}{c}+\frac{b^3c}{d}+\frac{c^3d}{a}+\frac{d^3a}{b}\geq a^3+b^3+c^3+d^3$$ I have a proof, but my proof is very ugly: Let $c=d+u$, $b=d+u+v$ and ...
0
votes
0answers
21 views

Positive correlation with the sequence $\sqrt{ij}/2-\min(i,j)$

Is there a sequence of positive real numbers $x_1,\ldots,x_n$ for which $$ \sum_{1\leq i,j\leq n}\left[\frac{\sqrt{ij}}{2}-\min(i,j)\right]x_ix_j> 0? $$
2
votes
2answers
74 views

Find the minimum of $\sum_{k=1}^n \frac{x^k_k}{k}$

Let $n$ be a positive integer. Find the minimum of $\displaystyle \sum_{k=1}^n \dfrac{x^k_k}{k}$, where $x_1,x_2,\ldots,x_n$ are positive real numbers such that $\displaystyle \sum_{k=1}^n ...
2
votes
6answers
89 views

Solving the absolute value inequality $\big| \frac{x}{x + 4} \big| < 4$

I was given this question and asked to find $x$: $$\left| \frac{x}{x+4} \right|<4$$ I broke this into three pieces: $$ \left| \frac{x}{x+4} \right| = \left\{ \begin{array}{ll} ...
0
votes
1answer
27 views

Matrix and vector norms Inequality

I want to prove that $\frac{\left \| u - \tilde{u} \right \|}{\left \| u \right \|}\leq \frac{k(A)}{1-k(A)\frac{\left \| A - \tilde{A} \right \|}{\left \| A \right \|}} \frac{\left \| A - \tilde{A} ...
4
votes
3answers
113 views

Prove $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a} \geq \dfrac{a+b}{a+c}+\dfrac{b+c}{b+a}+\dfrac{c+a}{c+b}.$

Prove that for all positive real numbers $a,b,$ and $c$, we have $$\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a} \geq \dfrac{a+b}{a+c}+\dfrac{b+c}{b+a}+\dfrac{c+a}{c+b}.$$ What I tried is saying ...
0
votes
1answer
44 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
0
votes
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
123 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
votes
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
votes
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
vote
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
votes
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
66 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
votes
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
votes
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.
0
votes
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
votes
3answers
66 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
votes
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
vote
2answers
38 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
vote
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
vote
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 ...
1
vote
6answers
443 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
vote
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
votes
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 ...