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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

1
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1answer
115 views
+50

Showing that two curves do not intersect

I want to show that $$x+1 \neq (x^3(x+2))^{1/4} + \sqrt{x+1-\sqrt{x^2+2x}}$$ for any real $x>0$. There are two approaches I've taken: showing they are equal and arriving at a contradiction (but ...
4
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2answers
87 views

$ \sum_{1 \le i < j \le n} a_{i} a_{j} \ge n(n-1)/2 $ , prove that $a_{1} + … + a_{n} \ge n$ for $n \ge 2$ using AM-QM

Let $a_{1}, a_{2}, ...$ be a sequence of positive real numbers. Let the following relation holds: $$ a_{k+1} \ge \frac{k a_{k}}{a_{k}^{2} + (k-1)}, \:\: k \ge 1$$ Prove that $ S_{n} = a_{1} + a_{2} + ...
0
votes
0answers
57 views

Inequality in a triangle with tan

Again, a construction Given a triangle $ABC$ with all its angles smaller than $90^\circ$, using standard notations ($\tau$ is the semiperimetr, $r$ the radius of the incircle.), prove that : $\...
3
votes
2answers
105 views

Area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$

Find the area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$ (using definite integration). I cannot understand how to find this area. I have graphed the lines and found ...
3
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2answers
235 views

Inequality in a Polynomial

This is a problem I made. Let $x^3-ax^2+bx-c$ be a polynomial with real coefficients and three real roots, all greater than $1$. Prove, that $b+c \geq 3a-5$. Due to the discussion made (see the ...
1
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2answers
72 views

Calculate the maximum value of $\sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}$ where $x + 2y + 3z = 2$

$x$, $y$ and $z$ are positives such that $x + 2y + 3z = 2$. Calculate the maximum value of $$ \sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}$$ This problem is ...
0
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1answer
52 views

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2})\ge 4$

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2} )\ge 4$, preferably with SOS. My approach: $\sum_{\mathrm{cyc}} \frac{a^2(b^2+c^2)+3abc(b+c)}{3bc(b^2+c^2)} \ge 4$. Let $f(a,b,c)=\...
3
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1answer
87 views

elementary but irritating algebra problem

The question itself is very short and sweet, and requires no background. Find a solution to the following system: $$ \left\{\begin{array}{l} a,b,c,d,e,f>0\\\\ a+b+c+d+e+f=1 \\\\\displaystyle\frac{...
1
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0answers
42 views

Prove that $\frac{d}{2^m + 1} + \frac{m}{2}\frac{2^m}{2^m+1} \geq \frac{\log_2 d}{4}$

I am reading Tau and Vu's book Additive Combinatorics, and I came across a step in a proof that I am not able to verify. On page 252, in the last line of the proof of Theorem 6.4, it is stated that ...
2
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2answers
54 views

How to prove that for all non-negative $\forall x\in\mathbb R: x\ge \frac{\ln^2(1+x+\sqrt{2x})}{2}$?

I'm trying to prove that for all non-negative $\forall x\in\mathbb R:$ $$x\ge \frac{\ln^2(1+x+\sqrt{2x})}{2}.$$ You can think of it as a tighter inequality than the useful $x\ge \ln(1+x)$ or $e^x\ge ...
0
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1answer
26 views

What are Contra Harmonic Mean and Inverse Contra Harmonic Mean? [on hold]

Are they related to Inequality? Like the $$ AM \times HM = GM^2 $$
1
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1answer
37 views

Tough substitution inequality

Prove that if $x, y, z >0$ and $xyz=x+y+z+2$, then $$ \sqrt{x}+\sqrt{y}+\sqrt{z} \leq \frac{3}{2}\sqrt{xyz}. $$ By the way, the first equation implies the existence of positive $a, b, c$ such ...
2
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2answers
63 views

Bound on $x$ for $x-\ln x\ge 1+\epsilon$

I'm looking for a function $x(\cdot)$ where the domain is $\mathbb{R}^+$, that satisfies the following: For all $\epsilon > 0$, the inequality $x-\ln x\ge 1+\epsilon$ is satisfied for all $x \geq ...
0
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1answer
23 views

Show that $\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|} $

We must show that $$\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|}. $$ Here is my attempt, however I wondered if there is a one-trick wonder ...
1
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3answers
34 views

Simplification of proof for $\exists c>0$ such that $|x+y|\leq c(1+\sqrt{x^2+y^2})$

Question: Consider function $f(x,y)=|x+y|$ for $x,y\in \mathbb{R}$. Show that $\exists c>0$ such that $|x+y|\leq c(1+\sqrt{x^2+y^2})$. My proof: i) if $|x|,|y|\leq 1$, then $|x+y|\leq 2\leq 2(1+\...
1
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1answer
30 views

Subset of $[n]$ without chain of leangth $5$ is of size $\leq \mathcal 2\Biggr(\binom{n}{(n-1)/2}+\binom{n}{(n-3)/2}\Biggl)$

Suppose $n$ is odd. Let $\mathcal P([n])$ denote the power set of $[n]$, that is, the $2^n$ subsets of $\{1,...,n\}$. We say that a family of sets $\mathcal F\subseteq \mathcal P([n])$ is nice if $\...
0
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3answers
48 views

Calculate the minimum value of $\frac{x^2 + 4}{y^2 + 1}$ where $1 \le y \le 2$ and $2y \le xy + 2$.

$x$ and $y$ are reals such that $1 \le y \le 2$ and $2y \le xy + 2$. Calculate the minimum value of $$\large \frac{x^2 + 4}{y^2 + 1}$$ This problem is adapted from a recent competition... It really ...
2
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1answer
49 views

Prove this generalization of Cauchy-Schwarz: $|(a.b)c+(b.c)a-(c.a)b|\le|a||b||c|$.

If you take $c$ unit perpendicular to $a,b$, Cauchy-Schwarz follows. I can prove the inequality in a roundabout way. First normalize to unit vectors. Then show that equality holds for 2-dimensional ...
0
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0answers
12 views

Question regarding Inequation with multivariable functions

In order to simplify the notation consider: $$ x=[x_1,x_2,...x_n] $$ Consider the following inequation: $$ s(x)(u(x)+A(x))<0 $$ My goal is to choose the function u(x) such that the inequation ...
3
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0answers
88 views
+50

Prove that inequality Hardy inequality

Suppose $n$ is a positive integer, $2n$ reals $x_i, y_i (1\le i \le n) $ satisfy $$\sum_{i=1}^n x_i^2 = \sum_{i=1}^n y_i^2 = 1.$$ positive reals $0 < \lambda_1 \le \lambda_2 \le ... \le \...
0
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3answers
63 views

AM-GM Inequality Involving Squares and Proof

Prove: $$(a^2 + b^2 + c^2)/3 \geq ((a + b + c)/3)^2$$ OR $$(a^2 + b^2 + c^2)/3 \leq ((a + b + c)/3)^2$$ for all $a, b, c \geq 0.$ The problem wants me to find which inequality is correct and then ...
2
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3answers
44 views

Prove $(2a + b + c)(a + 2b + c)(a + b + 2c) ≥ 64abc$ using the AM-GM method and establishing when inequality holds

Prove: $$(2a + b + c)(a + 2b + c)(a + b + 2c) ≥ 64abc$$ for all $a, b, c ≥ 0$. Also, calculate/prove when equality holds. To prove this, the first thing that came to mind was the Arithmetic Mean -...
1
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1answer
62 views

Prove that $|\frac{a+b}{1+ab}| < 1$ given that $|a|<1$ and $|b|<1$

Can I please get a hint/solution for why, if true, $$\bigg{|}\frac{a+b}{1+ab}\bigg{|}<1$$ given that $|a|<1$ and $|b|<1$, where $a,b \in \mathbb{R}$. I've tried the usual things like ...
2
votes
0answers
33 views

Proof of Khintchine's inequality on $\mathbb{C}$?

I'm trying to understand this proof of Khintchine's inequality for the complex case ($a_n\in\mathbb{C}$). The author claims that the complex case follows from the real case by taking absolute values ...
1
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1answer
21 views

Taking assigning an unknown variable $\theta = 0.5$ to solve an inequality

In order to solve the following inequality for $n$, where $0 < \theta < 1$, they use a trick that I do not understand $n \geq (\frac{1.96}{0.03})^2 \theta(1 - \theta)$ Since $\theta$ is ...
2
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1answer
28 views

The volume of a cover of an interval is greater than the volume of the interval

Let $A \subset \mathbb{R}^n$ be a $n-$dimensional interval, i.e. $A = [a_1,b_1] \times [a_2,b_2] \times \cdots \times [a_n, b_n]$, where $[a_i, b_i]$ is an interval in $\mathbb{R}$, for every $i \in \{...
1
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1answer
56 views

Solve the following Algebraic Logarithmic inequalities [closed]

Solve the following logarithmic inequality with all log base $10$. $$\left(\frac12\right)^{(\log x^2)} + 2 > 3\times2^{(-\log(-x))}$$ I have done many logarithmic inequalities but ...
1
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1answer
55 views

$\sum_{k=1}^n \Bigl(\frac{a_1+\dots+a_k}k\Bigr)^p\le\frac{p}{p-1}\sum_{k=1}^n\Bigl(\frac{a_1+\dots+a_k}k\Bigr)^{p-1}a_k$ for nonnegative $(a_k)$

I am struggling with the following task: Let$$1<p<\infty , n\in \mathbb{N}, a_1\geqslant 0,\dots,a_n\geqslant 0.$$ Prove that $$\sum_{k=1}^n \Bigl(\frac{a_1+\dots+a_k}{k}\Bigr)^p \le \...
-1
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1answer
37 views

proof of inequality with absolute-value and function

Prove that if $|f'(x)| \leq M$, for all $x \in [a,b]$, then $$f(a)-M(b-a)\leq f(b) \leq f(a)+M(b-a)$$ What I tried so far: Proof. Let $x \in [a,b]$ Assume $|f'(x)|\leq M$ Show $f(a)-(Mb-Ma)\leq f(...
1
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1answer
46 views

Given $ a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$, prove $ S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2 $

Suppose a sequence of positive real numbers with $$ a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$$ prove that $$ S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2 $$ Solution: I will ...
4
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1answer
106 views

Prob. 5, Sec. 6.2, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: How to show this function is strictly decreasing using derivative

Here is Prob. 5, Sec. 6.2, in the book Introduction To Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition: Let $a > b > 0$ and let $n \in \mathbb{N}$ satisfy $n \geq 2$. ...
2
votes
1answer
98 views

Nice olympiad inequality

Let's go for an olympiad inequality : let $a,b,c>0$ then we have : $$\sum_{cyc}\frac{ab}{a+b}\geq \frac{3\sqrt{3}}{2}\sqrt{\frac{abc}{a+b+c}}$$ My proof : $$\sum_{cyc}\frac{ab}{a+b}=\frac{(a^...
2
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2answers
122 views

Why AM-GM inequality showing different results?

I was given to find the minimum of $$1+a_1+a_2+a_3+...a_n$$. It was given that $$a_1\times{a_2}\times{a_3}...a_n =c$$ My approach Using AM-GM inequality $1+a_1+a_2+...a_n\ge (n+1)c^{\frac{1}{n+1}}$...
1
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2answers
43 views

Put the values of $a , b$ and $c$ in the equation $b^2 - 4ac \geq 0$

In the equation $a = (y - 1) , b = 0 , c = y$. Put the values in the following equation $$b^2 - 4ac \geq 0$$ Just tell me what answer you guys got. The answer should be $0 \leq y \leq 1$ but my ...
0
votes
0answers
42 views

Prove that the function $f(x)$ is increasing for $x<0$ and decreasing for $x>0$

I'm interested by the following problem : Let $a,b,c>0$ and $x>0$ then the following function is decreasing : $$f(x)=\sqrt{\frac{(abc)^x}{a^x+b^x+c^x}}\Big(\frac{1}{7a^x+b^x}+\frac{1}{7b^x+...
0
votes
1answer
26 views

Series with exponential upper bound

Let $K > 0$. Is it then true that there is some constant $C$ independent of $K$ such that $$\sum_{n=0}^\infty e^{-2^n K} \leq C e^{-K/C}$$ Thanks for the help!
1
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2answers
46 views

Show that $\frac{j(j-1)}{2n}> \frac{j^2}{4nr}$

In Lemmas 8.5 and 8.6 in book Irrational Numbers by Ivan M. Niven it uses the following : $$\frac{j(j-1)}{2n}> \dfrac{j^2}{4nr}$$ $n \ge 2$, $2 \le j \le n$ and $r \ge 2$, that's it! How the ...
3
votes
3answers
111 views

Prove that $(x + y + z)^3 + 9xyz \ge 4(x + y + z)(xy + yz + zx)$ where $x, y, z \ge 0$.

Prove that $(x + y + z)^3 + 9xyz \ge 4(x + y + z)(xy + yz + zx)$ where $x, y, z \ge 0$. This has become the norm now... This problem is adapted from a recent competition. We have that $6(x^2y + xy^2 ...
2
votes
1answer
61 views

Can we use QM-AM inequality to solve this?

There are two sequences (${a_1,a_2,a_3,....,a_n })$ and $( {b_1,b_2,b_3,....,b_n})$ such that $$\sum_{i=1}^n a_i = \sum_{i=1}^n b_i$$ Prove that: $$\sum_{i=1}^n \frac{a_i^2}{a_i+b_i} \ge \...
2
votes
2answers
76 views

Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

$x$, $y$ and $z$ are positives such that $x^2 + y^2 + z^2 = 3xyz$. Calculate the maximum value of $$\large \frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$$ This is (obviously) ...
12
votes
5answers
338 views

Interesting inequality $\frac{x^{m+1}+1}{x^m+1} \ge \sqrt[2m+1]{\frac{1+x^{2m+1}}{2}}$

Prove that $$\frac{x^{m+1}+1}{x^m+1} \ge \sqrt[k]{\frac{1+x^k}{2}}$$ for all real $x \ge 1$ and for all positive integers $m$ and $k \le 2m+1$. My work. If $k \le 2m+1$ then $$\sqrt[k]{\frac{1+x^k}{...
0
votes
1answer
31 views

A Euler summation like inequality.

Let $f: \mathbb{R} \to \mathbb{C} $ be a continuously differentiable function. Let $n$ be an integer. Why do we have: $$\int_n^{n+1} f(t) dt = f(n)+O\left(\int_n^{n+1} |f’(t)| dt\right)?$$
1
vote
3answers
56 views

$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

$x$, $y$ and $z$ are positives such that $xyz = \dfrac{1}{2}$. Prove that $$ \frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$$ Before you complain, this problem ...
-3
votes
0answers
51 views

is the last inequality in this page true?if yes how? [closed]

enter image description herein the middle of the page its trying to integrate the expression using complex numbers integration and Res...this page has a inequality in the bottom(for the cas e goes to ...
-4
votes
2answers
58 views

Prove the following inequality $\frac{1}{1+x}\leq\ln(1+x)\leq x$ [closed]

I need help proving the following inequality: $\frac{1}{1+x}\leq \ln(1+x)\leq x$ if $x > 0 $ Thanks
1
vote
1answer
28 views

Bound $\|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\|$ in terms of commutator $\|AB-BA\|$

For positive definite matrices $A$ and $B$, can $$ \|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\| $$ be bounded in terms of $\|AB-BA\|$? Note that if the matrices commute, then both norms ...
-1
votes
1answer
59 views

Nice refinement of an inequality by Michael Rozenberg

It's related to this If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$: In fact we have this refinement (wich I think much easier) : Let $a,b,c>0$ then we have : $$\...
0
votes
2answers
29 views

Doubt in basics of inequalities

While solving a problem I encountered with this $ 4m(4m-1) \leq 0 $ Solving this one gets the solution $ 0 \leq m \leq 1/4 $ Now, if the inequation is multiplied with -1 on both sides we get $ 4m(1-4m)...
0
votes
2answers
40 views

Induction with nth root of n

I am trying to prove by induction that $\sqrt[n]{n}<2-\frac{1}{n}$ where $n\ge2$. It seemed simple at first, but I am stuck with $log(2n-1)$ in the RHS. I am in an elementary undergraduate Maths ...
0
votes
1answer
41 views

Should $x=2$ be included in the solution?

Consider an inequation $\frac{(x-2)^6 (x-3)^3 (x-1)}{(x-2)^5 (x-4)^3}>0$. $(x-2)^5$ can be cancelled from both the numerator and denominator with the condition that $x\neq2$ leaving $(x-2)$ in the ...