Questions on proving and manipulating inequalities.

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2answers
63 views

solving inequalities with fractions on both sides

Solve this inequality: $(x^2 -2)/2 < (6x^2 -8x - 1)/(x+5)$. My solution: Multiply $(x+5)^2$ on both sides: $(x^2 -2)(x+5)^2 /2 = (x^4 + 10x^3 + 23x^2 -20x -50)/2$ $(6x^2 -8x - 1)(x+5) = 6x^3 ...
0
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4answers
59 views

solving a power 4 inequality

Solve this inequality: $x^4 - 3x^3 - 4x^2 \leq 0$ My answer (wrong) : $x^2 (x-4)(x+1)\leq0$ Case 1 $x^2 \geq 0$ , $x\geq4$ , $x\leq-1$ NO Case 2 $x^2 \geq 0$, $x\leq4$ , $x \geq -1$ ... so ...
2
votes
3answers
45 views

Solving a cubic inequality

$4x^3 - 3x^2 + x \ge 0$ , solve the inequality: My solution for this question: $x(4x^2-3x+1) \ge 0 $ $4x^2 - 3x + 1$ is not solvable, assume $x \in \mathbb R$ So $x \ge 0$ and $x \in \mathbb R$ ...
1
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1answer
54 views

I have a question on how to prove an inequality holds true.

How do I prove that for $a < b< c$ and $n \geq 3$ then the inequality $$b^n+c^n > a^n$$ holds true. I tried setting $c =b+n$ but I don't know we're to go from here !
0
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3answers
44 views

Solving this Inequation

Show that $e^x \geq x^e$ for $ 0 \lt x \lt \infty $. I tried to apply the normal logarithm here, which yields $x \geq e\times \ln(x)$ Still, I am kind of stuck here, anyone mind giving me a hand?
4
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6answers
540 views

How can I compare numbers raised to a square root

For example: $3^\sqrt5$ versus $5^\sqrt3$ I tried to write numbers as this: $3^{5^{\frac{1}{2}}}$ and then as $3^{\frac{1}{2}^5}$ But this method gives the wrong answer because $a^{(b^c)} \ne ...
0
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0answers
26 views

Decreasing of power function.

Show that $\frac{3}{4}(x-2)x^{-\frac{5}{2}}-(x+1)^{-\frac{3}{2}}<0$, wherer $x\ge0$. I tried by taking differentiation but then expression become more complicated. I also tried by checking the ...
3
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3answers
113 views

Absolute inequality derivation

I have been trying to prove an inequality that I am not even sure if it is even true or not. However I am experiencing great difficulties with this proof. I have an intuition that it is true and have ...
1
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3answers
67 views

How do I solve this rational inequality?

I'm having difficulty finding any help online for this equation: $$\frac{3x^2+2x-5}{2x^2+5x}\le0$$ I need to solve the inequality, and then put it in interval notation. So far I have taken the ...
1
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1answer
43 views

Chernoff-like bound for small intervals in tail distribution

I am searching for a Chernoff-like bound that controls the probability of small intervals in the tail distribution. More specifically, let $X_1, \ldots, X_n$ be independent random variables with ...
2
votes
3answers
62 views

$3x + 2 > 8$ solved not using order of operation?

So im basically relearning algebra, using a site to teach myself the basics. I read about bodmas then shortly after about inequalities. In the practice questions i got a bunch of questions where the ...
0
votes
1answer
86 views

How to prove the inequality $ (1+a+ab)(1+b+bc)(1+c+ca) \leq (1+a+a^2)(1+b+b^2)(1+c+c^2)?$

For $a,b,c>0$ prove the inequality $$ (1+a+ab)(1+b+bc)(1+c+ca) \leq (1+a+a^2)(1+b+b^2)(1+c+c^2). $$ I know that I should use the multiplicative rearrangement inequality but I am not sure how ...
1
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2answers
42 views

Why if $B = \{x : |x+1| ≤ 3 \}$ then $B$ equals $[ -4, \infty )$?

I really don't understand why $B$ is from $-4$ to infinity because $x+1 ≤ 3$ $x ≤ 2$ and $-3 ≤ x+1$ $-4 ≤ x$ Shouldn't it be $B = [-4, 2]$?
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0answers
30 views

Inequalities involving ordered variables

For given real numbers $\lambda_1\geq\lambda_2\geq\lambda_3\geq\lambda_4\geq\lambda_5\geq\lambda_6$ and the set identity $\{i,j,k,l,m,n\}=\{1,2,3,4,5,6\}$, show that the inequality \begin{equation} ...
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votes
4answers
68 views

A.M.>G.M. of four numbers [duplicate]

Prove that arithmetic mean of 4 numbers is greater than geometric mean of the same 4 numbers, i.e. prove that $$\dfrac{a+b+c+d}{4} > (abcd)^{\frac1{4}}$$
5
votes
5answers
189 views

A unusual inequality about function $\ln$

These day,I met a unusual inequality when I solve a difficult problem, and proving the inequality means I have done the work! Could you show me how to prove it or deny it? By the way, I believe that ...
5
votes
2answers
123 views

How to prove the inequality $(1+a+a^2)(1+b+b^2)(1+c+c^2) \leq (1+a+b^2)(1+b+c^2)(1+c+a^2)?$

For $a,b,c>0$ prove the inequality $$ (1+a+a^2)(1+b+b^2)(1+c+c^2) \leq (1+a+b^2)(1+b+c^2)(1+c+a^2). $$ Seems the rearrangement inequality must help but I can't do it. Any ideas?
5
votes
3answers
86 views

How can I prove this inequality $\frac{|xy|}{2x^2+y^2}\le1 $?

How can I prove this $\frac{|xy|}{2x^2+y^2}\le1 $ ? I was thinking about considering the left term a function and maybe show that 1 is the extreme point x,y can be any real number but not 0
3
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1answer
51 views

I need to show that the following function is positive.

I need to show that the following function is positive. $H(x)=2(7)^x+2(4)^x-2(3)^x+2(d+1)^x+2((d-2)(d+4))^x-2(d+2)^x-2((d-1)(d+5))^x$ Where $d=3,4,5 $ and $x\in[-1,0)$ From graph for different ...
6
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1answer
121 views

with inequality $\frac{1}{3a+5b+7c}+\frac{1}{3b+5c+7a}+\frac{1}{3c+5a+7b}\le\frac{\sqrt{3}}{4}$

let $a,b,c>0$, such $ab+bc+ac=1$,show that $$\dfrac{1}{3a+5b+7c}+\dfrac{1}{3b+5c+7a}+\dfrac{1}{3c+5a+7b}\le\dfrac{\sqrt{3}}{4}$$ by Macavity C-S:with inequality ...
0
votes
0answers
32 views

following system of inequalities

Graph the following system of inequalities. $$x+2y\leq 12$$ $$2x+y\leq 12$$ $$x\geq 0 , y \geq 0$$ I just need to know how Can show the augmented matrix that describes the situation
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1answer
44 views

Vectorial sequence space and its inner product

For $k\in \mathbb{N}$, let $\lambda_k \in \mathbb{R}^{d\times d}$ be a symetric positive definite matrix, and $\lambda_{kij}$ be it's coordinates. Suppose we have $c_k \in \mathbb{R}^d$, $c_{ki}$ its ...
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2answers
27 views

The shaded solution region in the following graph represents the inequality

The shaded solution region in the following graph represents the inequality please help I don't know how determine that from the graph
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2answers
30 views

Graph the following system of inequalities

Graph the following system of inequalities. Show (by shading in) the feasible region. $$x+2y\leq 12$$ $$2x+y\leq 12$$ $$x\geq 0 , y \geq 0$$ I would like to know how to graph these ...
0
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2answers
41 views

Why does this inequality stand?

I stand that $\log n=O(n^{\epsilon})$ for any $\epsilon >0$. At a previous example we have shown that $$e^{n^{\epsilon}} \geq \frac{n^{\epsilon d}}{d!}$$ where $d=\lfloor ...
4
votes
2answers
121 views

Prove the following inequality without using induction: $\frac{1}{2^k-1}\leq \sin^{2k}\theta+\cos^{2k}\theta\leq 1$

How to prove the following inequality (without using induction)? $$\frac{1}{2^k-1}\leq \sin^{2k}\theta+\cos^{2k}\theta\leq 1,\quad k\in\Bbb N.$$
1
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1answer
27 views

Proving triangular inequality in general

I want some advice on approaching proofs of the triangular inequality for various metrics. I have a suspicion the answer is simply 'improve mathematical maturity' but hopefully there is a general ...
2
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1answer
56 views

Upper bound for a interesting function.

Prove $$\frac{1}{2}^x+\frac{1}{2}^\frac {1}{x}\leq 1$$, where $x $ is a positive real number. This problem is from my friend. Here is my approach. It is sufficient to show that $$0\leq2^x-1-2^{x-\frac ...
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0answers
10 views

Lower Bound on Real Part of product?

For complex vectors $u,z,a\in\mathbb{C}^n$, are there any interesting lower bounds on the quantity $$ \Re\left[(a^*z)^2(u^*a)^2\right] $$ in terms of $|a^*z|^2, |u^*a|^2$ that are tighter than $$ ...
0
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1answer
41 views

Prove $x^{2n} + x^{2n-1}y + x^{2n-2}y^2 + \ldots + y^{2n} \geq 0$

This is trivial if there are binomial coefficients, but I don't know how to transform that case into this one. Obviously one can assume $xy < 0$ and then without loss of generality let $x$ be ...
2
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1answer
29 views

Show $\mathbb{E}{[\exp(-\sum_{n \mathop = 1}^{\infty}I_{E_{n}}})] \leq \exp[-(1-e^{-1})\sum_{n \mathop = 1}^{\infty}\mathbb{P}({E_{n}})]$

Let $E_{1}, E_{2}, \dots$ be independent events and $e^{-\infty} := 0$ Show that $\mathbb{E}{[\exp(-\sum_{n \mathop = 1}^{\infty}I_{E_{n}}})] \leq \exp[-(1-e^{-1})\sum_{n \mathop = ...
0
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1answer
65 views

prove that $A(n) : \left(\frac n3\right)^n\lt n!\lt \left(\frac{n}{2}\right)^n$ for all $n\ge 6$

prove that $A(n) : \left(\frac n3\right)^n\lt n!\lt \left(\frac{n}{2}\right)^n$ for all $n\ge 6$ first check $n=6$ : $2^6<6!<3^6$ ok then $n\gt 6$ assume $A(m)$ is true, then show ...
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1answer
26 views

(analysis) sequence a(n) is bounbed and sequence b(n) converges. Show that a(n)≤b(n) (∀ n ∈ ℕ) ⇒lim supa(n)≤lim b(n) [closed]

sequence a(n) is bounbed and sequence b(n) converges. Show that a(n)≤b(n) (∀ n ∈ ℕ) ⇒lim sup a(n)≤lim b(n) Since a(n) is bounded, a(n) has a convergent subsequence. let a1'(n) be a subsequence of ...
4
votes
2answers
104 views

Proving $1+\frac{1-\cos x}{x}>0$ for all $x \neq 0$

I proved that $$1+\frac{1-\cos x}{x}>0$$ for all $x>0$, but I fail to prove the same inequality for $x<0$. I just don't see a way of proving it. I tried proving it at least for ...
2
votes
1answer
10 views

Crossing of monotone function with linear function

Suppose $F(x)$ is a continuous monotone increasing function of an argument $x\in(0,x_M)$. To show that $F(x)<x\quad\forall x\in(0,x_M)$, is it enough to show that $F(0)<0$ and ...
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1answer
74 views

with inequality $\frac{y}{xy+2y+1}+\frac{z}{yz+2z+1}+\frac{x}{zx+2x+1}\le\frac{3}{4}$

Let $x,y,z\ge 0$, show that $$\dfrac{y}{xy+2y+1}+\dfrac{z}{yz+2z+1}+\dfrac{x}{zx+2x+1}\le\dfrac{3}{4}$$ I had solve $$\sum_{cyc}\dfrac{y}{xy+y+1}\le 1$$ becasuse After some simple computations, it is ...
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0answers
46 views

Application of Morse Inequalities

I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ...
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2answers
27 views

Explaining this inequality

In a proof I was working on today, I assumed this equation was true which lead to devastating results $$ \sqrt{\bar{x^2}} =\bar{\lvert x\rvert} $$ For instance, given the data 0 and 2, the left hand ...
5
votes
0answers
57 views

showing that an inequality holds

I am trying to figure out how to show that for $n\geq 3$, $$(2^n-1)^{\frac{n}{2(n-1)}}\geq (2^{n-1}-1)^{\frac{n-1}{2(n-2)}}+1.$$ I've tried basic algebra and induction, but the inductive hypothesis ...
0
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1answer
22 views

Dealing With Simple Inequalities

This is an odd question that may have to do with my lack of mathematical background prior to the study of the so-called "higher" mathematics. Say I have two pieces of information $$ a+b<1 \text{ ...
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5answers
167 views

with this inequality $\ln{x}\ln{(1-x)}<\sqrt{x(1-x)}$

If $0<x<1$, show that $$\ln{x}\ln{(1-x)}<\sqrt{x(1-x)}$$ use derivative it's not easy, such $$ f(x)=(\ln{x}\ln{(1-x)})^2-x(1-x), $$ $$ ...
3
votes
2answers
35 views

An inequality with $\sum_{k=2}^{n}\left(\frac{2}{k}+\frac{H_{k}-\frac{2}{k}}{2^{k-1}}\right)$

show that $$\sum_{k=2}^{n}\left(\dfrac{2}{k}+\dfrac{H_{k}-\frac{2}{k}}{2^{k-1}}\right)\le 1+2\ln{n}$$where $ n\ge 2,H_{k}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{k}$ Maybe this ...
0
votes
1answer
18 views

Schwarz inequality and uniform converges

Let $$f_n(x) = \frac{x}{1+nx^2}$$. Show that for $x\ne 0$ $f_n(x)$ converges uniformly to some $f$. So the solution suggests the Schwarz inequality, yielding: $$\left|f_n(x)\right| \le ...
0
votes
1answer
51 views

Jensen inequality for convex functions - infinite countable number of weights

Does Jensen inequality for convex functions hold if there is countable infinite number of weights? For example weights are given by sequence $(\frac{1}{2^n})_{n\in\mathbb{N}_1}$ ? If not, where is ...
1
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0answers
80 views

inequality about characteristic function

Let $X$ be a random variable with density $f(x)=|x|^{-3}1_{|x|\ge1}$ and $\phi_{X}(t)=E[e^{itX}]$. Show that $\forall t\in[-1,1] $ $$|\phi_{X}(t)-1-t^2log|t||\le3t^2$$ I noticed that $E[X]=0$, so ...
2
votes
1answer
39 views

Bounding an Integral with M-L Formula

I am stuck trying to understand Lemma 5.20 from Newman Bak's Complex Analysis. 5.20 Lemma Let s denote a root of the equation $z=g(z)$, for some analytic function g such that $g'(s)=0$. Suppose ...
2
votes
1answer
29 views

Extended Liouville Polynomial Inequality

I am trying to show that the noted inequality holds without using the Euclidean Algorithm. This appears in Complex Analysis by Newman and Bak on the top of page 63 (remark 1, my paraphrase): If ...
4
votes
1answer
63 views

Prove $(a^2+b^2+c^2+d^2)^2≥(a+b)(b+c)(c+d)(d+a)$

I've been unsuccessfully trying to solve this contest-style problem for a while. Tried different substitutions and the such, but nothing helped. I presume the solution is related to Cauchy-Schwarz? ...
0
votes
1answer
28 views

Number of grid points inside a triangle

I need to find a number of integer points inside a triangle with vertices $(0,0)$, $(A,0)$ and $(0,B)$, where $A$ and $B$ are positive integers. This problem can be reduced to the following ...
1
vote
2answers
53 views

How to prove the following inequality: $\frac{\sqrt{n + 1}}{\sqrt{n}} - 1 \leq \frac{1}{2n - 1}$

As a part of my practice for an upcoming mid-term, I managed to simplify the following inequality to what you see here: $$\frac{\sqrt{n + 1}}{\sqrt{n}} - 1 \leq \frac{1}{2n - 1}$$ And honestly I'm ...