Questions on proving and manipulating inequalities.

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3
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0answers
64 views

Prove this inequality $\frac{a}{b+cd}+\frac{b}{c+da}+\frac{c}{d+ab}+\frac{d}{a+bc}≥\frac{16}{5}$

Show that if $a, b, c, d \in (0, \infty)$ and $a + b + c + d = 1$, then $$\frac{a}{b+cd}+\frac{b}{c+da}+\frac{c}{d+ab}+\frac{d}{a+bc}≥\frac{16}{5}$$
2
votes
6answers
91 views

Prove this inequality $25ab+25a+10b\le38$

let $a,b>0$,and such $a^2+b^2=1$,show that $$25ab+25a+10b\le38$$ Now I have found this inequality $"="$,if and only if $a=\dfrac{4}{5},b=\dfrac{3}{5}$ then How to prove this inequality by AM-GM ...
0
votes
2answers
25 views

Solution set of inequality

This is the question: $$\frac{1-2x-3x^2}{3x-x^2-5} \gt 0$$ What I did : I got the answer as $$\left(x-3\right)\left(x+1\right) \gt 0$$ giving me the solution set : $x \in (-\infty,-1 ...
-2
votes
2answers
39 views

Inequality of numbers.

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a\le0$. (May be Jensen's inequality help but need help how to apply.)
3
votes
0answers
35 views

How find this minimum

Help me! Let $x,y,z\ge0$ such that: $xy+yz+zx=1$. Find the minimum value of: $A=\dfrac{1}{x^2+y^2}+\dfrac{1}{y^2+z^2}+\dfrac{1}{z^2+x^2}+\dfrac{5}{2}(x+1)(y+1)(z+1)$ I found minimum value of $A$ ...
8
votes
1answer
63 views

If $u\in L^1(0,1)$ is nonnegative and $E_n = \int_0^1 x^n u(x) \, dx$, prove $E_{n-k} E_k \leq E_0 E_n$.

$\textbf{Question:}$ Let $ u \in L^1(0,1)$ be a nonnegative function. Define $$E_n := \int_0^1 x^n u(x) dx$$ Prove the following inequality, $\forall n \ge 0$, and $\forall k \in [0,n]$, we have $$ ...
1
vote
2answers
19 views

Cardinality of the union of two sets

I am having trouble attempting to prove the inequality $|X\cup Y| \le |X|+|Y|$. Here is my intuitive argument when we take the union of $X\cup Y$ if there are repeated elements then they are not ...
0
votes
1answer
37 views

Floor inequality with prime

If $a$ and $b$ are positive integers and $a\ge b$ and $b$ is an odd prime, show that: $$\left\lfloor \frac{6a-1}{b}\right\rfloor+\left\lfloor\frac{a}{b}\right\rfloor\ge \left\lfloor ...
1
vote
1answer
26 views

Floor inequality: $\lfloor \frac{6a-1}{b}\rfloor+\lfloor\frac{a}{b}\rfloor\ge \lfloor \frac{2a}{b}\rfloor+\lfloor \frac{3a-1}{b}\rfloor+\cdots$

If $a$ and $b$ are positive integers and $a\ge b$, show that: $$\left\lfloor \frac{6a-1}{b}\right\rfloor+\left\lfloor\frac{a}{b}\right\rfloor\ge \left\lfloor \frac{2a}{b}\right\rfloor+\left\lfloor ...
0
votes
1answer
31 views

Floor inequality: $\lfloor x+y\rfloor\ge \lfloor x\rfloor+\lfloor y\rfloor$

I remember seeing the inequality $\lfloor x+y\rfloor\ge \lfloor x\rfloor+\lfloor y\rfloor$ somewhere which is true for all reals. So I was wondering what's wrong with this proof? For all reals $a,b$ ...
0
votes
0answers
10 views

Difficulty understanding the source of an inequality solution.

I used Wolfram-Alpha to verify an answer that I got for simplifying an inequality, and it turns out I was only partially correct. Original equation: ...
1
vote
2answers
47 views

this sum inequality $\sum_{i=1}^{n}\frac{1}{4i(i+1)-1}<\frac{2}{7}$

show that $$\sum_{i=1}^{n}\dfrac{1}{4i(i+1)-1}<\dfrac{2}{7}\tag{1}$$ we have $$4i(i+1)-1>4i^2$$ But $$\sum_{i=1}^{n}\dfrac{1}{i^2}<\dfrac{8}{7}$$ it is clear not hold, because ...
0
votes
3answers
17 views

A store is offering a 20% discount on a certain item. The store’s sale of the item is subject to a 6% sales tax.

Quantity A : The item’s purchase price, if the discount is applied to the after-tax price Quantity B : The item’s purchase price, if the tax is applied to the discounted price Which of the ...
1
vote
0answers
26 views

How to show that $0<b'<b$?

If $a, b, q$ are natural numbers with $\frac{1}{q + 1} < \frac ba < \frac1q$, show that $$ \frac{b}{a} - \frac{1}{q+1} = \frac{b'}{a(q+1)} $$ where $0<b'<b$. I did the following: ...
2
votes
1answer
41 views

Proving an Inequality (terms won't cancel out)

Problem: If $x$ and $y$ are real numbers such that $y \geq 0$ and $y(y+1) \leq (x+1)^2$, prove that $y(y-1) \leq x^2$. This is what I tried: \begin{align} y(y+1) \leq (x+1)^2 &\implies y^2 + y ...
1
vote
2answers
19 views

How to simplify the inequality $5c * 3^{n-2} + 3 \le c3^n$

I'm stuck on trying to simplify the inequality: $5c * 3^{n-2} + 3 \le c3^n$ I'm looking for an expression such as $c \ge x$ without the n to show when this inequality holds true Edit: This ...
0
votes
3answers
50 views

Proof By Induction $2^n \ge n^2$ for $n\ge4$

I am trying to prove the following, and here is what I have done: Can somebody help to complete this? $2^n \ge n^2$ for $n\ge 4$ $n=4$, LHS: $2^4 = 16$, RHS: $4^2=16$, $16=16$ Therefore TRUE Assume ...
2
votes
2answers
32 views

Proof By Induction $n^2 > 3n$ where $n\ge 4$

I am trying to prove the following example, however I seem to be getting a little stuck: For $n\in\mathbb N$, $n\ge 4, n^2>3n$ What I have Done: Base Case:$ n=4$, LHS: $4^2 = 16$, RHS: $3\cdot 4 ...
1
vote
1answer
68 views

Is $a \sin x + b \sin y \leq \sin(ax + by)$ true?

Studying math essay exam, I saw the following strange formula $$ a \sin x + b \sin y \leq \sin(ax + by), $$ where $x, y$ are arbitrary angles and $a + b = 1.$ Is the above inequality true, and can it ...
0
votes
1answer
19 views

Inequality regarding modulus

I am trying to prove this limit to be true: $$\lim_{x\to a}(x^2)=(a^2)$$ using the Epsilon Delta Limit Definition. So far I can understand how it works but I got stumped on this inequality ...
2
votes
2answers
77 views

Inequality with condition $x+y+z=xy+yz+zx$

I'm trying to prove the following inequality: For $x,y,z\in\mathbb{R}$ with $x+y+z=xy+yz+zx$, prove that $$ \frac{x}{x^2+1}+\frac{y}{y^2+1}+\frac{z}{z^2+1}\ge-\frac{1}{2} $$ My approach: After ...
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vote
0answers
46 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
3
votes
4answers
414 views

Why is the Riemann sum less than the value of the integral?

Why is $ \frac{1}{n}\sum_{k=1}^n \frac{1}{1+\frac{k}{n}}\leq\int_0^1 \frac{dx}{1+x}=\log 2 $? Because I think: $$\int _0^1\frac{dx}{1+x}=\frac{1}{n}\sum _{k=1}^n\frac{1}{1+\frac{k}{n}}$$ Why is the ...
1
vote
2answers
53 views

How we can prove that: $\sum _{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot \log(2)$?

$f:\left[0,1\right]\rightarrow R,\:f(x)=\frac{1}{1+x}$ and we have to show that $\sum_{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot\log(2)$. What I know is just that: $n\cdot \log(2)=\int_0^1 ...
0
votes
4answers
41 views

How do we solve this inequality

I have the following inequation : $$\frac{1}{x-x^2-1}< 0$$ I know that the solution set will be all $x\in R$ but how do we find the answer? if we take the root of equation, we get imaginary ...
4
votes
3answers
101 views

Proving $\left(a+\frac{2}{a}\right)^2+\left(b+\frac{2}{b}\right)^2\ge \frac{81}{2}$ for all positive real $a,b$ such that $a+b=1$

I approached this problem in two different ways, but only one was successful. I'll post the latter as an answer, while here follows the first approach: I expanded the squares: ...
0
votes
1answer
33 views

The first assumption leads to the third one that looks inconsistent at a glance. Can you explain it better?

Background I am trying to solve the following problem: > Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a>0$. The objective is to find the range of $a$ such that ...
0
votes
0answers
23 views

The $2$-norm of a Hermitian matrix does not exceed its $1$-norm

How to prove that the $2$-norm of a Hermitian matrix does not exceed its $1$-norm? In wiki, I see $2$-norm of matrix $A$ is $\le \sqrt{\|A\|_1\|A\|_\infty}$. But I don't know how to prove that ...
5
votes
5answers
88 views

Prove by mathematical induction: $\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1$

Could anybody help me by checking this solution and maybe giving me a cleaner one. Prove by mathematical induction: $$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1; n\geq2$$. So after I check ...
2
votes
3answers
48 views

Inequalities and Differentiation

Having become so accustomed to differentiation and integration being applied just like normal algebraic operators, I was somewhat suprised yesterday when I realized that $f(x) \geq g(x)$ does not ...
0
votes
1answer
54 views

inequality with gaussian cdf and density involved

in my calculations I've arrived at the following inequality $$ |\frac{4\phi(x)(1-2\Phi(x))}{(1+(1-2\Phi(x))^2)^2}| \leq 0.5 $$ where $\phi$ is Gaussian density, and $\Phi$ Gaussian cdf, which can ...
0
votes
4answers
68 views

Proof by Induction $3^n > n^3$

I am trying to prove the following, however I'm stuck at the Induction hypothesis Prove by induction that, for all integers $n$, if $n\geq 5$, then $3^n>n^3$ What I have Done: Base Case: $n ...
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vote
2answers
37 views

Misunderstanding inequalities of integrals

We have to prove the following inequalities: 1) to show that $\frac{2x}{\pi }<sin\left(x\right)<x,\:and\:after\:1-e^{-\frac{\pi }{2}}\le \int _0^{\frac{\pi ...
0
votes
1answer
26 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b ...
5
votes
1answer
142 views

Prove a relationship involving floor functions

I am trying to prove that a particular expression is a lower bound for a very unusually-behaved function. The whole proof will be complete if I can just nail down the details of one technical lemma ...
1
vote
1answer
50 views

prove Inequalities for integrals

prove $\frac{\pi}{6}+\frac{1}{3}\leq \int_0^\frac{\pi}{2}\frac{1+\cos(x)}{2+\sin(x)}dx \leq \frac{\pi}{4}+\frac{1}{2}$ I got to the point where $\frac{1}{3} \leq f(x) \leq 1$, so $\frac{\pi}{6} ...
1
vote
3answers
54 views

Prove that $x^2 - 2013^2 \le y \le 2013^2 - x^2$ has an odd number of solutions

$x$ and $y$ are integers. $N$ is the number of solutions $(x, y)$ of this inequation $x^2 - 2013^2 \le y \le 2013^2 - x^2$. Prove that N is odd.
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votes
3answers
80 views

Proving that $2^n+1\leq 3^n$ by induction

I need to prove the following using mathematical induction: $$2^n+1\leq 3^n\qquad\forall n\in\Bbb{Z^+}$$ Been working on this problem for a while and cannot figure it out. Any guidance or help would ...
2
votes
2answers
71 views

Prove $(\alpha_1 + … + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + … + \alpha_n^2)$

For any real numbers $\alpha_1, \alpha_2, . . . . ., \alpha_n$, $$(\alpha_1 + ...... + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ..... + \alpha_n^2)$$ And when is the inequality strict?
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0answers
33 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
0
votes
1answer
23 views

Satisfying a set of inequalities

Having a set of conditions (1)(2) and (3) as follows 1-$$\beta < 1$$ 2-$$ \beta > \alpha$$ 3-$$\alpha < 1$$ Can I say that the following inequality is incorrect? $$1-\alpha -\beta ...
7
votes
4answers
104 views

How to evaluate $\log x$ to high precision “by hand”

I want to prove $$\log 2<\frac{253}{365}.$$ This evaluates to $0.693147\ldots<0.693151\ldots$, so it checks out. (The source of this otherwise obscure numerical problem is in the verification ...
1
vote
3answers
71 views

Why $-1 \leq\frac{\langle A,B\rangle}{||A||\, ||B||}\leq1$?

I'm reading Apostol's Calculus. It says that due to the Cauchy-Schwarz inequality written as: $$|\langle A,B\rangle|\leq ||A||\, ||B||$$ Then $$-1\leq\frac{\langle A,B\rangle}{||A||\, ||B||}\leq ...
6
votes
1answer
66 views

Find multivariable limit $\frac{x^2y}{x^2+y^3}$

Find multivariable limit of: $$\lim_{ \left( x,y\right) \rightarrow \left(0,0 \right)}\frac{x^2y}{x^2+y^3}$$ How to find that limit? I was trying to do the following, but i am not able to find a ...
3
votes
1answer
48 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
0
votes
2answers
36 views

Inequality error possibly. How are two inequalities equal?

Notation: $\underline{x}\in \Bbb R^n,||\cdot||_p =\left(\sum \limits_{i=1}^n |\cdot|^p\right)^{\frac1p}$ $$||\underline{x}||_p\left( \sum \limits_{i=1}^n |x_i + ...
1
vote
0answers
25 views

Simplification of inequalities

I am trying to simplify the following conditions $$\beta\leq1 $$ $$ 1-\alpha-\beta \leq 0$$ Can I say that the above two conditions are equivalent to saying that $$1- \beta \geq 0$$ $$\alpha \geq ...
1
vote
1answer
23 views

Does the following series of transformations of inequalities holds?

I am to calculate limit of the function $f(x,y)$ i am trying to apply squeeze theorem. Is the following series of transformations of this inequality correct? If not how to do this correctly? i.e. are ...
1
vote
1answer
29 views

Logarithm multivariable limit $\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$

Find multivariable limit $$\lim_{\left( x,y \right) \rightarrow (0,0)}\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$$ I was trying to find and inequality i've found out that: ...
1
vote
2answers
85 views

I can't understand how to prove this inequality

I don't understand how we can prove that inequality, without integration $$\frac{1}{x}\int_x^{2x} \left(2-\frac{1}{y+2}\right)\,dy \geq 2 - \frac{1}{x+2}.$$ P.S: Here is what I try... if can someone ...