Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

0
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1answer
38 views

Prove that $\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $

Prove that $$\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $$ I tried to apply Cauchy Schwarz but then I obtain Nesbitt's inequality with the wrong sign.
3
votes
0answers
50 views

Given two positive numbers $b,\,c$. Prove $\left ( \frac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \frac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .

Given two positive numbers $b,\,c$. Prove $\left ( \dfrac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \dfrac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ . My problem is given a solution by user ...
1
vote
1answer
19 views

$\sqrt{5}BA \leq PA +PB+\sqrt{2}PC$

Let $ABC$ be an isosceles right triangle ($\angle B=90^o$) and a point $P$ in its plane. Prove the inequality $\sqrt{5}BA \leq PA +PB+\sqrt{2}PC$. Find all poins $X$ for which the equality holds. It ...
2
votes
2answers
54 views

$ 4x -5y + 24z = 4A $, $ 2x - 2y + 2z = 10$. What is the largest possible value of $A$?

If $x,y,z$ integers that satisfy $$ 4x -5y + 24z = 4A $$ $$ 2x - 2y + 2z = 10$$ with $y < 2x$ and $y-20z< 0$, what is the largest possible value of $A$? Attempt: We can rewrite the equations ...
2
votes
1answer
38 views

Prove the inequality $b\le \sqrt{a^2 + b^2}$

How do i prove $b\le \sqrt{a^2 + b^2}$ ? What i think is : $\begin{align} a^2& \gt 0,\, b^2 \gt 0 \\ b &= \sqrt{b^2}\\ &\le\sqrt{a^2+b^2} \\ \therefore b &\le \sqrt{a^2 + b^2} \end{...
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2answers
26 views

Simplifying inequality raised to the power of 2

I was trying to solve a problem related to inequality and I came across the following: 1 - x^2 ≥ 0 x^2 ≤ 1 -1 ≤ x ≤ 1 <= how is this possible?! I do not ...
2
votes
2answers
86 views

Given three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+ b^{2}+ c^{2}= 5$. Prove $a^{3}b+ b^{3}c+ c^{3}a\leqq 8$ .

Problem. Given three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+ b^{2}+ c^{2}= 5$. Prove: $$a^{3}b+ b^{3}c+ c^{3}a\leqq 8$$ My solution in M&Y: (and I'm looking forward to seeing a ...
4
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0answers
61 views

Prove that $\sum\limits_{cyc}\sqrt{a+11bc+6}\geq9\sqrt2.$

Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc+abc=4.$ Prove that: $$\sqrt{a+11bc+6}+\sqrt{b+11ac+6}+\sqrt{c+11ab+6}\geq 9\sqrt2.$$ The equality occurs for $(a,b,c)=(1,1,1)$ and ...
1
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1answer
51 views

When does $ (1+a)(1+b)(1+c) \leq 4+4abc ~ $ hold? [on hold]

When I use $a = 0.1 , b = 1 , c = 10~ $ the following ineqaulity, $$ (1+a)(1+b)(1+c) \leq 4+4abc ~~~ ....~~~ (*) $$ does not hold. However, it holds for several other positive numbers $a,b,c$. ...
0
votes
3answers
56 views

Solve in integers $(x^2+2)(y^2+3)(z^2+4)=60xyz$

Solve in integers the equation $$(x^2+2)(y^2+3)(z^2+4)=60xyz$$ My try: We have $x,y,z \ne 0$ So we can write the equation as: $$\left(x+\frac{2}{x}\right)\left(y+\frac{3}{y}\right)\left(z+\frac{4}{...
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1answer
21 views

Nonstandard inequality with parameter and absolute value

The given inequality is $|x^2-ax+1|<3(x^2+x+1)$. The question is: For which values of $a$, every $x$ is a solution? I am trying to solve it by making the graphics of the two sides of the ...
2
votes
1answer
58 views

How to prove inequality $a^{\frac{1}{b}}+b^{\frac{1}{a}}\leq \frac {1}{2}$ if $0 <a,b < 1$ and $a+b=1$?

Suppose that $0 <a,b < 1$ and $a+b=1$ Today, I did some investigation of the expression $$a^{\frac{1}{b}}+b^{\frac{1}{a}}$$ and it seems that the maximum is at $a=b=\frac{1}{2}$ where the ...
1
vote
1answer
25 views

Showing $2(\|f\|_{L_p}^p+\|g\|_{L_p}^p)\le\|f-g\|_{L_p}^p+\|f+g\|_{L_p}^p$

I have the following question from a past qualifying exam: Given $2\le p<\infty$. Show that for any real-valued functions $f,g\in L_p(\mathbb{R})$, it holds that $$2\left(\left\|\frac{f}{2}\...
1
vote
1answer
36 views

Using triangle inequality to find $\lim_{(x,y)\to(0,0)}\frac{x^3-x^2y}{x^2+y^2+xy}$

This is an exercise from my textbook where the problem is to find the limit of the function $\frac{x^3-x^2y}{x^2+y^2+xy}$ when $(x,y) \to (0,0)$. So after changing to polar coordinates and ...
5
votes
1answer
81 views

An $\arcsin$ inequality

Show that if $0<|x|,|y|<1$, then $$\arcsin |x| +\arcsin |y| > \arcsin\left|\frac{x+y}{1+xy}\right|.$$ I found a proof (see below). Is there a different way (hopefully simpler) to show that ...
0
votes
1answer
58 views

Solution To Inequality: $q>\frac{x-y}{(1+x)(2+y)}$

I am studying the following inequality: $$q>\frac{x-y}{(1+x)(2+y)}$$ where $1>q>0$ and $x>y>1$. Wolframalpha provides the following solutions: $1>q\geq\frac{1}{3}$ with $x>y&...
1
vote
1answer
87 views

If $x,\,y$ be integers and suppose that $x^{2}- y^{3}= 17$, prove each following proposition true.

Problem. If $x,\,y$ be integers and suppose that $x^{2}- y^{3}= 17$, prove each following proposition true : $y^{2}+ 2\,x+ 2$ is a prime number if and only if $\{\!x= 4,\,y= -\,1\!\}$ $x^{2}+ y^{2}+ ...
2
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2answers
35 views

Can we expect find constant $C>0$ so that $(x+y)^r\geq C( x^{r}+y^r)$?

Let $x, y >0, r>0.$ Assume that $r\notin \mathbb N.$ Can we say that $(x+y)^r\geq x^{r}+y^r$? Edit: In view of the answer below: I'm modifying my question: Can we expect to find a ...
0
votes
1answer
28 views

Is $f\mapsto\int f^{-1}d\mu$ a weak-* continuous functional on $\{f\in L^\infty(\mu):\alpha\le f\le \beta\}$?

Problem Setting. Let $\mu$ be a probability measure on some space $\Omega$. Consider the subset of $L^\infty(\mu)$ defined by $S=\{f\in L^\infty(\mu):\alpha\le f\le \beta\}$, where $\alpha\le\beta$ ...
0
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0answers
16 views

weighted young inequality

Define weight $w(\theta) = (1+c \cos^2 \theta)^{s},$ $s>0$ $c$ some constant. Define $\|f\|_{L^p_w(\mathbb T)}^p=\int_{\mathbb T}|f(\theta)|^p w^{p}(\theta) d\theta $ Can we expect $\|f\ast ...
3
votes
1answer
65 views

Exploring an inequality between $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} $ and $\frac{3}{1+(abc)^{1/3}}$ if $a,b, c>0$

An AM-HM inequality for three positive numbers leads to $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \ge \frac{3}{{1+\frac{a+b+c}{3}}}. ~~~~(1)$$ Next, the well known AM-GM inequality $$\frac{a+b+c}{...
2
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2answers
25 views

Solving a inequality over the reals

I have the following inequality that I need to solve for real $z$: $$3\cdot\left(-d\left(d^3+4z^3-168\right)\right)\ge0\tag1$$ Where $d$ and $z$ are element of the real numbers (so they can be ...
3
votes
1answer
37 views

An integral inequality for a real-valued differentiable monotone function on $[0,1]$

I have the following question from a past analysis qualifying exam: Let $f$ be a real-valued differentiable monotone function on $[0,1]$. Define $$g(x)=\frac12[f(x)+f(1-x)]$$ for $0\le x\le1$. ...
0
votes
3answers
84 views

Prove $(xyz)^{2}(x^2+y^{2}+z^{2})≤(xy)^{2}+(yz)^{2}+(xz)^{2}$

Question: prove that if $x,y,z\in \Bbb R$ and $x+y+z=3$ then $$(xyz)^{2}(x^2+y^{2}+z^{2})≤(xy)^{2}+(yz)^{2}+(xz)^{2}.$$ I don't have any ideas to prove that.
0
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2answers
51 views

An inequality involving Euler’s $\varphi$-function

Let $\varphi$ be Euler’s phi-function. I have seen it claimed that $$\varphi(n)/n = O(\log \log n).$$ Could someone either give a proof of this fact or tell me a reference where I can find this proof?...
0
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3answers
25 views

distances of the end points of two orthogonal line segments

I have two line segments (with the same length) which are positioned on two orthogonal lines (see figure 1). For an application I would like to show that $(b-a)-d<0.5d$. It's easy to see that its ...
2
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1answer
62 views

Prove that $abc(a^2+b^2+c^2-2abc-2a-2b-2c)+ab(a+b-2)+bc(b+c-2)+ca(c+a-2)+a+b+c \le 2$

Prove that if the real numbers $a,b,c$ lie in the interval $[0,1]$, then: \begin{align*} abc(a^2+b^2+c^2-2abc-2a-2b-2c)+ab(a+b-2)+bc(b+c-2)+ca(c+a-2)+a+b+c \le 2. \end{align*} I've tried this: \...
1
vote
0answers
31 views

An inequality involving exponentials

Let $0< t \le 0.1$, $1<C<2$ and $0< x,y < 1/2$. Prove that $$ (1 - e^{-\frac{x}{t}})(1 - e^{-\frac{y}{t}}) \ge C (e^{\frac{xy}{t}} -1)(e^{-\frac{x}{t}} + e^{-\frac{y}{t}}) $$ It is ...
3
votes
1answer
75 views

Given $a,b,c,d\in\mathbb{Z}_{+}$ with $a\geqq b>c>d\geqq 0$ and $ac+bd=(\!b+d+a-c\!)(\!b+d-a+c\!)$. Prove $ab+cd$ is not prime.

Given positve integers $a, b, c, d$. For $a\geqq\!b>\!c>\!d\geqq\!0$ and $ac+\!bd= (\!b+ d+ a- c\!)(\!b+ d- a+ c\!)$ Prove $ab+ cd$ is not a prime number. I have a solution, and I'm looking ...
0
votes
1answer
24 views

Proof of Hardy Integral Inequality in N Dimensions

This comes from a recent lecture I've had. I have questions about on specific step in the short proof. This inequality is noted as the "Subcritical Hardy Inequality on the Whole Space". Statement: ...
1
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4answers
60 views

Find Maximum Value of Expression [duplicate]

I'm trying to learn about using the AM-GM inequality. I understand the basics of it so far, but I'm having a difficult time in terms of applying it to solve problems. I've encountered the problem ...
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4answers
51 views

Given $a>0$, $b> 0$, $a+b<10$, how to prove that $ab <25$ [on hold]

As titled, Given $a>0$, $b> 0$, $a+b<10$, how to prove that $ab <25$
2
votes
2answers
72 views

Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ .

Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ . My own problem is given a solution, and I'm looking forward to seeing a ...
0
votes
1answer
32 views

Why am I getting opposite?

The question asks to prove that $ab+bc+ca \geq\frac1 3$ given that $a,b,c $are positive real numbers such that $a+b+c=1$ I solved in this way. $(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$ $\implies1=a^2+b^2+...
1
vote
1answer
33 views

Prove $\frac{8}{3}\leqq c< 4$ for $a\geqq b> 0$ and $3\,a+ 2\,b- 6= ac+ 4\,b- 8= 0$ .

Given positive real numbers $a,\,b$ such that $a\geqq b$ and $3\,a+ 2\,b- 6= ac+ 4\,b- 8= 0$. Prove $$\frac{8}{3}\leqq c< 4$$ I have a solution, and I hope to see a nicer one(s), thanks a real lot! ...
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3answers
97 views

Olympiad inequality :$2\sqrt{(x^xy^y)^{\frac{1}{x+y}}\sqrt{xy}}\leq \frac{x+y}{2}$ for $x,y>0$

Hello I'm interested by the following problem : Let $x,y>0$ then we have : $$2\sqrt{(x^xy^y)^{\frac{1}{x+y}}\sqrt{xy}}\leq x+y$$ My try : Since the inequality is homogeneous we can put $x=1$ ...
1
vote
1answer
34 views

comparing length of the sides of a triangle

I have a triangle, with $\alpha\ge90^\circ$ (see figure). Now for an application I want to get a lower bound for $f+h-g$ (i.e the cost of the "detour" over $A$). But I don't know if there is a ...
4
votes
4answers
87 views

Prove $\log|e^z-z|\leq |z|+1$

Prove that $\log|e^z-z|\leq |z|+1$ where $z\in\mathbb{C}$ with $|z|\geq e$. Background: This is from a proof that $e^z-z$ has infinitely many zeroes. The present stage is that we assumed in ...
0
votes
5answers
60 views

Squaring inequalities

How do I square $-3<x<3$? Logically, $0\le x^2<9$ so what is the rule for squaring inequalities that allows someone to go from $-3<x<3$ to $0\le x^2<9$? It doesn't seem like there is ...
2
votes
4answers
64 views

How to correctly solve $\sqrt{1+x}+\sqrt{1-x}>1$?

$$\sqrt{1+x}+\sqrt{1-x}>1, x\geq-1 \wedge x\leq1$$ $$\sqrt{1+x}>1-\sqrt{1-x}$$ $$1+x>1-2\sqrt{1-x}+1-x$$ $$0>-2\sqrt{1-x}+1-2x$$ $$2\sqrt{1-x}>1-2x$$ Lets say both sides of inequality ...
-2
votes
1answer
45 views

What does c mean in Calculus (Absolute Value Inequality)? [on hold]

I am graphing Absolute Value Inequalities and I came across this problem. |x-c| < 0.1. I'm not sure what c represents in this inequality. Thanks for your help!
0
votes
1answer
44 views

Prove/disprove that $a \le b$ where $4x \le y$, $a = \sqrt{-8x + 2\sqrt x + 1}$, $b = \sqrt[3]{-48y + 3b\sqrt[3] y + 1}$ and $a, b, x, y \ge 1$.

Given that $a, b\ge 1$ and $x, y > 0$ satisfied $$\large 2\sqrt[4]{x} \le a = \sqrt{-8x + 2\sqrt x + 1}$$ and $$\large 2\sqrt[6]{y} \le b = \sqrt[3]{-48y + 3b\sqrt[3] y + 1}$$, prove/disprove that ...
0
votes
1answer
40 views

A little confusion about AM-GM proof

In Cauchy's forward-backward induction proof, why can we substitute $x_k=\frac{x_1+x_2+ ... +x_{k-1}}{k-1}$ without losing generality?
2
votes
0answers
60 views

New inequality involving exponent

I'm interested by the following problem : Let $x,y,z>0$ then we have : $$x+y+z\geq \sqrt{\Big((x^xy^y)^{\frac{1}{x+y}}+(y^yz^z)^{\frac{1}{y+z}}+(z^zx^x)^{\frac{1}{z+x}}\Big)\Big(\sqrt{xy}+\...
0
votes
0answers
26 views

Given $\boldsymbol{x} = \boldsymbol{y } \boldsymbol{A}$ with $ | \boldsymbol{x}|_{\infty} \le c $ , find a bound on $|\boldsymbol{y} | _{\infty}$

Problem Given \begin{equation}\label{4} \boldsymbol{x} = \boldsymbol{y } \boldsymbol{A} \quad \text{(i.e., Elementwise equality)} \end{equation} with $ | \boldsymbol{x}|_{\infty} \le c $, $ \...
1
vote
2answers
55 views

How can I prove these two inequalities?

For the first one I tried the inequality (x1-1)(x2-1)>=0 and I summed it up with the analougus inequalities but I didn't get what I needed. I also tried to prove it's smaller than (n-1)/2 instead of [...
1
vote
5answers
60 views

Why is $\tan(-\frac{\pi}{2})<\tan(x)<\tan(\frac{\pi}{2})$ equivalent to $-\infty<\tan(x)<\infty$?

Why is $\tan(-\frac{\pi}{2})<\tan(x)<\tan(\frac{\pi}{2})$ equivalent to $-\infty<\tan(x)<\infty$? I get that $\tan(-\frac{\pi}{2})$ and $\tan(\frac{\pi}{2})$ are both undefined but why ...
1
vote
1answer
69 views

Prove or disprove $\sum_{i=0}^\infty \frac{1}{i+j+1}\frac{1}{\sqrt{i+\frac{1}{2}}}<\frac{\pi}{\sqrt{j+1}}$

In Example 2.3.5; Functional Analysis book by S. Kesavan it was shown that for $j\gt -\frac{1}{2}$ $$s(j) := \sum_{i=0}^\infty \frac{1}{i+j+1}\frac{1}{\sqrt{i+\frac{1}{2}}}<\frac{\pi}{\sqrt{j+\...
1
vote
2answers
71 views

Where I made mistake

When I was proving the product rule of sequence limits, I find a fact that it seems to not allow us to do the substitution on $\epsilon$. (sorry my English is not good) $$\lim_{n\to \infty}a_n=a, \...
0
votes
2answers
65 views

how to prove this function is increasing?

i asked about thisfunction before but no one respond \begin{eqnarray*} \Gamma(x+1)>\frac{\frac{(900\gamma^2+73\pi^2)x}{900\gamma}+\frac{73}{100}}{\frac{\pi^2 x}{9\gamma}+1}\quad [1] \end{eqnarray*}...