Questions on proving, manipulating and applying inequalities.

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2
votes
2answers
38 views

How do I find the domain/range of functions algebraically?

I've been having trouble when trying to find the domain/range of functions algebraically. Here is an example: $P(x)=\frac{1}{3+\sqrt{x+1}}$ Finding the domain: $x+1\ge0$ $x\ge-1$ Therefore, $x \...
7
votes
1answer
132 views

Inequality problem, for positive $a,b,c$, if $abc=1$, then $\frac{1}{1+a+b^2}+\frac{1}{1+b+c^2}+\frac{1}{1+c+a^2}\leq1$

I need help or guidance in solving this inequality that I am battling for 3 days now. I have tried everything that comes to mind, but I am stuck. The inequality is as $$\sum_\textrm{cyc}\frac{1}{1+a+...
2
votes
4answers
55 views

Inequality: powers of small numbers

If $\epsilon \approx 0 $ then which one is greater $1-\epsilon^{k}$ or $(1-\epsilon)^{k}$ where $k \in \mathbb{Z}_{> 0}$ is a positive integer.
0
votes
3answers
87 views

Find the minimum of the value $n$ such that $(1-0.03)^n<0.03$

How can I find the smallest positive integer $n$ such that $$(1-0.03)^n<0.03$$ without the help of a computer?
3
votes
1answer
29 views

For positive self adjoint $T$, show $|\langle Tx,y\rangle|^2 \le \langle Tx,x\rangle \langle Ty,y\rangle$

As in title, $T$ is a positive self adjoint, bounded linear operator on a Hilbert Space $X$ and I'd like to show $$|\langle Tx,y\rangle|^2 \le \langle Tx,x\rangle \langle Ty,y\rangle$$ Self adjoint ...
-1
votes
1answer
44 views

Reverse Holder continuity

Consider a function $f(x)$ with a point-wise Holder exponent $\beta \leq 1$. Definition of point-wise Holder exponent: $$ \beta_x: = \sup \left\lbrace \beta: \limsup_{h \rightarrow 0^+} \left|\frac{...
0
votes
1answer
65 views

Prove inequality: $\frac13\left(x^3+y^3+z^3\right)\ge xyz+\frac34|(x-y)(y-z)(z-x)|$

For any nonnegative numbers $x,y,z$ prove inequality: $$\frac13\left(x^3+y^3+z^3\right)\ge xyz+\frac34|(x-y)(y-z)(z-x)|$$ My work so far: I used formulas $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-...
3
votes
3answers
84 views

Why does $n \geq 2$ imply that $\frac n 2 < n$?

It has been a while since I did math proof in school, and I just can't figure out why $$n \geq 2 \text{ implies that } \frac n 2 < n$$ Anything would help! Thanks.
3
votes
4answers
90 views

Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers

Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers It is in the exercises of the AM-GM inequality chapter of a book, and that is why I believe it will be solved by ...
0
votes
3answers
30 views

Need help showing $(a^p + b^p) \le (a^2 + b^2)^{p/2}$, where $p \ge 2$, and $a,b \ge 0$.

I've been so far trying to show: $(\frac{a^2}{a^2 + b^2})^{p/2} + (\frac{b^2}{a^2 + b^2})^{p/2} \le 1.$ Also, it holds true that $\frac{a^2}{a^2 + b^2} \le 1$ and $\frac{b^2}{a^2+b^2}\le 1.$ I'm ...
3
votes
1answer
82 views

How to prove this inequality? $a^{2}+b^{2}+c^{2}\leq 3$

How to prove this inequality? If $a^{2}+b^{2}+c^{2}\leq 3$ and $a,b,c\in \Bbb R^+$, then $$\left( a+b+c\right) \left( a+b+c-abc\right)\geq 2\left( a^{2}b+b^{2}c+c^{2}a\right) $$ I tried AM>GM but ...
0
votes
3answers
61 views

To prove $\sup B \leq \sup A$

Assume $A$ and $B$ are non empty and bounded above and satisfy $B \subseteq A$. Show that $\sup B \leq \sup A$ I am thinking of proving using contradiction, but I am getting nowhere. Someone please ...
-1
votes
1answer
89 views

Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ [closed]

I'm not sure how to go about this. Does it involve using MVT? I got as far as saying $e = \frac{e^x - e^y}{x-y}$.
0
votes
1answer
19 views

Determine all intervals of numbers $x$ satisfying the following inequalities.

i) $(x-5)^2 (x+10)\leq 0$ ii) $(x-5)^4 (x+10) \leq 0$ My answer : i) $(-10)\leq x \leq (5)$. ii) $(-10)\leq x \leq (5)$. Can you check my answer?
0
votes
0answers
38 views

What is the minimum growing function here?

What is the minimal growth of $a$ as a function of $N$ for which in $${x}{a^x}>\frac{\log N^{}}{c\log\log N}$$ $x=O(1)$ holds for a fixed $c>0$? Clearly $a=O\big(\big(\frac{\log N}{c\log\log N}\...
1
vote
2answers
66 views

Prove inequality for $xyz=1$

I am having trouble to prove an intermediate step $(xy+xz+yz)^2 \geq 3xyz(x+y+z)$ for $x,y,z\geq0$.
1
vote
1answer
75 views

Maybe this inequality holds? $x!-y!>x^n$?

Let $x,y,n$ be postive integers such that $x\ge 2y,y>n,n>3$ I conjectured that $$\color{red}{x!-y!\ge x^n}$$ Now, I claim that $$\color{red}{x!-y!=y![x(x-1)(x-2)\cdots(y+1)-1]\ge (n+1)![x(x-1)(...
1
vote
4answers
60 views

If $9 ≥ 4x + 1$, which inequality represents the possible range of values of $12x + 3?$

If $9 ≥ 4x + 1$, which inequality represents the possible range of values of $12x + 3$? I've been trying to do SAT prep, and I came across this question. It allowed me to show an explanation and it ...
1
vote
1answer
50 views

Are equality and non-equality mutually dependent?

Is there any type of objects or ideas for which asking about their equality makes sense, but asking about non-equality doesn't? (or vice versa) Intuitively, "not equal" is a negation of "equal", so ...
1
vote
3answers
30 views

Show that $\max{\{|a|+|b|,|c|+|d|\}} \leq \max{\{|a|,|c|\}}+\max{\{|b|,|d|\}}.$

Show that $\max{\{|a|+|b|,|c|+|d|\}} \leq \max{\{|a|,|c|\}}+\max{\{|b|,|d|\}}.$ I wanted to show that $d(p,q)=\max{\{|x_1-x_2|,|y_1-y_2|\}}$ where $p=(x_1,y_1),q=(x_2,y_2)$ is a metric on $\mathbb{R^...
1
vote
0answers
30 views

Integration in an inequality

Does integrating on both the sides of inequality with the same upper and lower limits with respect to same variable somehow affect the inequality. I saw an example lets say, Sin x < x ,x>0 ...
1
vote
1answer
43 views

Prove inequality for $a, b , c, d$ for $a,b, c, d\ge 0$

How does one prove the following inequality: $$ \frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{c^2+1}+\frac{d}{d^2+1}\le\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{d^2+1}+\frac{d}{a^2+1}$$ without much ...
0
votes
0answers
9 views

How to prove this inequality for $h\leq h_0$?

Let $f(x,t)$ is a function, $f:\left[0,1\right]\times\left[0,T\right] \to \mathbb{R}_{\geq 0}$ with $T>0$ and $\left| f\left(x,t\right)\right|\leq C >0$ Let a partition on $x$; $\left\{x_0,\...
1
vote
1answer
33 views

Find the range of a

If the point $P(a^2,a)$ lies in the region corresponding to the acute angles between the lines $2y=x$ and $4y=x$ then range of a is... This would have been easy if the lines had constants and I ...
4
votes
1answer
59 views

Prove that $a_1+\frac{a_2^2}{a_1+a_2}+\frac{a_3^2}{a_1+a_2+a_3}>b_1+\frac{b_2^2}{b_1+b_2}+\frac{b_3^2}{b_1+b_2+b_3}.$

Suppose $a_1>a_2>a_3>0$ and $b_1>b_2>b_3>0$ and $a_1>b_1,a_2>b_2,a_3>b_3$. I want to prove that $$a_1+\frac{a_2^2}{a_1+a_2}+\frac{a_3^2}{a_1+a_2+a_3}>b_1+\frac{b_2^2}{...
7
votes
3answers
559 views

Which is larger, $70^{71}$ or $71^{70}$? [duplicate]

Yet another question of which is larger: $70^{71}$ or $71^{70}$. I solved it by observing that $f(x)=\frac{\ln(x)}{x}$ is decreasing for all $x>e$ since $f'(x)=\frac{1-\ln(x)}{x^2}<0$ for all $x&...
-5
votes
1answer
36 views

Graph the function [closed]

Graph the function $$ f(x)=\begin{cases} -x-2, & -2<x\le -1 \\ -x^2, & -1<x\le 1 \\ x+2, & 1<x\le 2 \end{cases} $$
1
vote
1answer
45 views

Finding the smallest value $c$ such that $x^n < c(1+nx)$

Can we find the smallest value $c$ such that $x^n < c(1+nx)$ for all positive real numbers $x$ and for all positive integers $n$, if the value exists? I always thought quadratic functions always ...
0
votes
2answers
119 views
-1
votes
1answer
47 views

Iterate through integers solutions of linear inqualities [closed]

Say we have a set of integers value $x_1,\ldots x_n$ such that $$ \left\{ \begin{array}{l} a_{1,1} x_1 + \ldots a_{1,n}x_n \leq b_1 \\ \vdots \\ a_{m,1} x_1 + \ldots a_{m,n} x_n \leq b_m \\ x_1, \...
0
votes
1answer
21 views

Upper bound for sum of powers

I have a sequence of positive real numbers $\{x_i\}_{i=1}^{N}$ and $k \in \mathbb{N}$. I was wondering if one can find an upper bound of the type $$ \sum_{i=1}^{N}{x_i^k} \leq f\left(\sum_{i=1}^{N}{...
0
votes
0answers
33 views

maximum value of $6$ variable expression. [duplicate]

If $a,b,c,d,e,f\geq 0$ and $a+b+c+d+e+f=1\;,$ Then $\max(ab+bc+cd+de+ef)$ $\bf{My\;Try::}$ Using Lagrange Multiplier Method:: Let $$F=f(a,b,c,d,e,f) = ab+bc+cd+de+ef-\lambda(a+b+c+d+e+f-1)$$ $$\...
0
votes
0answers
26 views

Inequality between norm of function and it's derivative

There is a theorem: Let $f$ be a continuously differentiable, $2\pi$-periodic function. Given $\int_{-\pi}^{\pi} f(x) dx = 0$, I need to prove that $$||f|| \le \frac{\pi}{2} \cdot ||f'||.$$ Where ...
0
votes
0answers
65 views

Why $y \le x\log_2(y)$ means $y < 2x \log_2(2x)$ when $x>2$ and $y\ge 1$

This simple inequality is very useful when estimating the VC dimension of certain functional class. Although starting from the result, I can show that it is correct after some ''ugly'' calculations, ...
3
votes
3answers
77 views

Prove $\sum_{i=1}^{n}\frac{a_{i}^{2}}{b_{i}} \geq \frac{(\sum_{i=1}^{n}a_i)^2}{\sum_{i=1}^{n}b_i}$

So I have the following problem, which I'm having trouble solving: Let $a_1$ , $a_2$ , ... , $a_n$ be real numbers. Let $b_1$ , $b_2$ , ... , $b_n$ be positive real numbers. Prove $$ \frac{a_{1}^{2}...
1
vote
1answer
46 views

If $x,y,z,t\geq 0,$ Then $\max(xy+yz+zt)$

If $x,y,z,t\geq 0$ and $x+y+z+t=1\;,$ Then $\max$ value of $(xy+yz+zt)$ $\bf{My\; Try::}$ We can write $x+y+z+t=(x+z)+(y+t)\geq 2(x+z)(y+t)$ and equality hold when $x+z=y+t$. So we get $\...
2
votes
3answers
27 views

why sometimes when solving inequalities, variables of both sides cancel?

Case in point: $|2x|\ge 2x+1$ can equivalent to $2x\ge 2x+1$ and $-2x\ge 2x+1$. The second resulting inequality yields you $x\le -1/4$ and is fine. But the first resulting inequality gives you $0\...
0
votes
2answers
85 views

$(\sin^{-1} x)+ (\cos^{-1} x)^3$

How do I find the least and maximum value of $(\sin^{-1} x)+ (\cos^{-1} x)^3$ ? I have tried the formula $(a+b)^3=a^3 + b^3 +3ab(a+b)$ , but seem to reach nowhere near ?
2
votes
1answer
53 views

Functional Inequality given $s\cdot f(t)+t\cdot f(s)\leq 2$

Question: For all continuous $f:\mathbb R\to \mathbb R\ $ and $\forall s,t \in [0,1]$ that satisfies: $$s\cdot f(t)+t\cdot f(s) \le 2$$ a) Prove that: $$\int_0^1 f(x) \,dx \le {\pi/2}.$$ b) How many ...
-2
votes
2answers
196 views

Show $\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2} \geq \sqrt{a^2+ac+c^2}$ [closed]

Can someone help me with this problem: Let $a$, $b$, $c$ be positive numbers. Show that: $\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2} \geq \sqrt{a^2+ac+c^2}$
0
votes
0answers
46 views

Linearize bilinear matrix expression

Given the matrix inequality in the unknowns $X,Y$: $\begin{bmatrix} X & (X-XAY)^T\\ X-XAY & X \end{bmatrix}>0$, I would like to linearize the bilinear matrix product $XAY$ using a change ...
1
vote
2answers
71 views

Equality and inequality with infinitesimal number

Is the following statement true? Suppose $a$ and $b$ are real numbers. If $0\leq a-b < \epsilon$ for every $\epsilon>0$, then $a=b$. What if I replace '$<\epsilon$' by '$\leq \epsilon$' and ...
0
votes
4answers
85 views

Proof that $\frac{2\theta}{\pi} < \sin \theta < \theta$

The following inequality hold: if $\theta $ is in radians and $ 0 < \theta <\pi/2$, then $\frac{2\theta}{\pi} < \sin \theta < \theta$ How can be proved this inequality?
5
votes
1answer
90 views

USA $2011$ contest inequality problem, proving $\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3$, under given condition.

If $a^2+b^2+c^2+(a+b+c)^2\le4$, then $$\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3.$$ My attempt: From the given criteria, one can easily obtain that $$(a+b)^2+(b+c)^2+(c+a)^...
1
vote
3answers
68 views

Proof by math induction with inequality example, why is “replacement” allowed?

I have trouble with the understanding of mathematical induction concerning inequalities. For example, the question is: Prove by mathematical induction that $ n ^ 2 <2 ^ n $ if $ \forall n \in {N}$ ...
0
votes
1answer
44 views

For which values of $\gamma$ does this inequality hold?

Edited: Just realised my first post was somewhat misleading and not precise. Thanks to the two commetators that pointed it out. I am working on an article and ended up wondering for which values of $\...
1
vote
2answers
39 views

Proof of inequality using AM-GM inequality generalisation

$(a+b)(a+c)(b+c) \geq 8abc$ I need a full proof and an explanation. I've tried it, but I don't even know how to begin.
0
votes
4answers
91 views

Is $\arctan(y/x) > xy/(x^{2}+y^{2})$ true for positive $x,y$?

I am trying to prove the following: $$\arctan\frac{y}{x}>\frac{xy}{x^{2}+y^{2}},\quad\forall x,y>0$$ Is the statement true, and if so how do you prove it?
1
vote
3answers
80 views

Solving the Inequality $\dfrac{x-1}{\lfloor x \rfloor}\ge 0$

Find all solutions of $$\dfrac{x-1}{\lfloor x \rfloor}\ge 0$$ $$$$ I know how to solve the Inequality $\dfrac{x-1}{ x }\ge 0$ using the Wavy-Curve/Method of Intervals technique. However I don't know ...
1
vote
1answer
56 views

Triangle Inequality for $\|x\|_{\infty}$

I have to show the triangle inequality for $\|x\|_{\infty}$. I'm not sure, if estimate is correct. To show: $\|x+y\|_{\infty} \le \|x\|_{\infty}+\|y\|_{\infty}$ Let $x \in \mathbb{R}^n$ and $\|x\|_{\...