Tagged Questions

Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

4
votes
5answers
68 views

Prove $(b-a)\cdot f(\frac{a+b}{2})\le \int_{a}^{b}f(x)dx$

Let $f$ be continuously differentiable on $[a,b]$. If $f$ is concave up, prove that $$(b-a)\cdot f\left(\frac{a+b}{2}\right)\le \int_{a}^{b}f(x)dx.$$ I know that (and have proved) $$(b-a)\cdot ...
3
votes
1answer
25 views

$\phi(v)/\Phi(v)$ is decreasing for $\phi$ and $\Phi$ being the PDF and CDF of $N(0,1)$

Let $\phi(v)$ and $\Phi(v)$ denote, respectively, the PDF and CDF of the standard normal distribution. How would one show that $$ \frac{\phi(v)}{\Phi(v)} $$ is decreasing? I tried the quotient rule ...
0
votes
2answers
24 views

Bounds on algebraic equation

I have to show that: $$ \frac{-\sqrt{(-\beta -\kappa \sigma -1)^2-4 \beta}+\beta +\kappa \sigma +1}{2 \beta } < 1 $$ I am not sure it is possible. The constraints on the coefficients are: ...
0
votes
0answers
14 views

Problem similar to Kolmogorov's inequality using martingale.

Suppose that $X_k$ is a sequence of independent random variables with mean zero and variance $1$. Let $S_k=X_1+\cdots+X_k$ and let $$ h(\lambda)=\limsup_{n \rightarrow \infty}P\left(\max_{1\leq k\leq ...
0
votes
0answers
8 views

An inequality in Zygmund space

How to derive (3) from (2)? Thanks for help.
0
votes
0answers
17 views

inequalities of probabilities: why are these two systems equivalent?

Hope you can help me with this. Let A,B,C be three events such that the three following inequalities are verified: $P(A|BC)>P(A|\bar{B}C)$, $P(A|B\bar{C})>P(A|\bar{B}\bar{C})$, ...
2
votes
3answers
192 views

How to find a solution for this inequation?

what's the best way to find a solution for the following inequation: $$ \sqrt{x^2-1}>x $$ The result is as Wolfram says: $$ x \leq-1 $$
9
votes
4answers
590 views

Beautiful inequality on positive numbers

Showing that $ \frac{2a_1^2}{a_1+a_2}+\frac{2a_2^2}{a_2+a_3}+...+\frac{2a_n^2}{a_n+a_1}\geq a_1+a_2+...+a_n$ holds for positive $a_i$s. I've tried adding $a_1+a_2, a_2+a_3,...,a_n+a_1$ respectively ...
-2
votes
4answers
41 views

Show that for real $a,b,c$, $a^2+b^2+c^2>ab+bc+ca$

Show that for real $a,b,c$, $a^2+b^2+c^2\ge ab+bc+ca$ We can do this in two obviously trivial ways, that is AM-GM inequality, and its equivalent system, the whole square method. Another way could be ...
0
votes
3answers
51 views

Inequality (absolute value)

$$|x-4|^2 -5|x-4| +6 > 0$$ How can I get rid of the absolute value? Does it work the same way equations with absolute value work?
1
vote
1answer
41 views

For every $n$ there exists $k_n \in \mathbb{N}$ such that $a+k_n/2^n$ is an upper bound while $a+(k_n-1)/2^n$ is not

Let $ \mathcal{P} \subset \mathbb{R}$,\ $\mathcal{P}\neq \emptyset $ and let $b$ be an upper bound of $\mathcal{P}$. Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
0
votes
1answer
36 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
2
votes
2answers
57 views

Find max of $P=\frac{2}{\sqrt{a^2+1}}+\frac{1}{\sqrt{b^2+1}}+\frac{1}{\sqrt{c^2+1}}$

Give $a,b,c>0$ and $ab+bc+ca=abc$ Find maximum of $$P=\frac{2}{\sqrt{a^2+1}}+\frac{1}{\sqrt{b^2+1}}+\frac{1}{\sqrt{c^2+1}}$$ Could someone help me solve this?
1
vote
2answers
48 views

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number?

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number? I have tried it both with cartesian and polar coordinates but did not get a solution. I got that far: $z=x+yi$ and then I got: $$\pm x ...
3
votes
2answers
81 views

Given $a,b,c$ are the sides of a triangle. Prove that $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}<2$

Given $a,b,c$ are the sides of a triangle. Prove that $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}<2$. My attempt: I could solve it by using the semiperimeter concept. I tried to transform this ...
0
votes
1answer
35 views

Inequalities textbook request [duplicate]

At university I have got a problem set with lots of inequalities. Unfortunately there are no explanations given how to do them. In Highschool we only did very easy inequalities. Therefore I am looking ...
2
votes
1answer
29 views

How to solve this Complex inequality system

$1 \leq z \overline {z} \leq 4$ and $|\Im(z)|<\Re (z)$ How can I solve this system of inequalities? ($\Im$ is the imaginary part and $\Re$ is the real part of a complex number). I have tried so ...
0
votes
0answers
13 views

Chebyshev's Inequality/Markov's Ineq [on hold]

Please help explain the answer to this question.
0
votes
1answer
25 views

Let $a_n \rightarrow a$. Show that $\liminf(a_n-b_n)=a-\limsup(b_n)$.

Assignment: Let $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ be two sequences of real numbers with $a_n \rightarrow a \in \mathbb{R}$. Show that: $$\liminf(a_n-b_n)=a-\limsup(b_n)$$ ...
0
votes
0answers
31 views

A question on logic and some functional inequalities

Suppose that I have a (generic) function $g$ and arguments $a, b \in \mathbb{N}$. I know that $g$ satisfies the inequalities $$1 < \frac{g(b)}{b} < \frac{g(a)}{a} < 2.$$ I also know that ...
0
votes
0answers
17 views

Question about Cauchy-inequality

Let $f:\mathbb{C} \to \mathbb{C}$ a holomorphic function and assume that there exist $M > 0$ and $r>0$ such that $$ |f(z)| \leq M |z|\ln |z| $$ $\forall z \in \mathbb{C}$ with $z \geq r$. I ...
0
votes
0answers
27 views
+50

How to proof the following Gronwall type inequality?

Suppose that $g,k: [0,a] \to \mathbb R$ are continuous, $a >0 $, $\,k(t) \ge 0$,$\ c(t) \in C^1([0,a])$, $\, \dot c(t) \ge 0 $ (i.e. $c(t)$ is non decreasing) and $g(t)$ satisfies $$g(t) \le ...
0
votes
2answers
16 views

inequation with complex solutions

Could somebody please help me solve this inequality: $|x-2| < x|x|$ I tried to solve it by using three different values of $x$: 1. $x < 0$ Solution : $1/2 - \sqrt{7}i/2 < x < 1/2 + ...
0
votes
2answers
25 views

Proof of inequality of sums

I have the following to prove, with induction and any help would be appreciated! :) $n\in \mathbb{N}, \quad \left(\, x^{1},\ldots,x^{n}\,\right)\in\mathbb{R}^{n}$ $$ \left(\,\sum^{n}_{i\ =\ ...
2
votes
3answers
69 views

Inequality proof for $1+x^3\geq x+x^2$

I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. I tried it based on previous one, but without success... The prev. ...
0
votes
2answers
20 views

Proof of an inequality in $\mathbb{R}$

I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. $x,y\in\mathbb{R},\quad \epsilon\in\mathbb{R}:\epsilon\not=0$ $$ ...
0
votes
0answers
29 views

Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
0
votes
1answer
24 views

Show $1+2^2+…+n^n$ ≤ $(1+1/n-1)*n^n$

$1+2^2+...+n^n$ ≤ $(1+1/(n-1))*n^n$ Well what I come up with is it's left to prove $1+2^2+...+n^{n-1}$≤ $n^n/(n-1)$ I think I need to somehow come up with a summation of the former then compare it ...
0
votes
0answers
34 views

Inequality involving the inverse of a covariance matrix

Consider the following covariance function: $$ k(s_i, s_j) = e^{-|s_i - s_j|/2}. $$ Take $s_i \leq 0$ for $i = 1, \dots, n$ and construct the following matrix and vector: $$ A = (k(s_i, s_j))_{i,j ...
1
vote
1answer
26 views

differential inequality involving the square of the function

It is written in a book, (Bertozzi- Majda, vorticity and incompressible flow page 106) that given a differential inequality of the following type: $ \frac{d}{dt}\|u^{\epsilon}(t)\| \leq ...
2
votes
4answers
21 views

Inequality: showing that $p(1-p)\leq \frac{1}{4}$ if $0<p<1$.

I was wondering if there was another way to derive the inequality shown in the title of this question apart from calculus. Doing it with calculus is pretty straightforward: Let $f:(0,1)\to\mathbb ...
0
votes
3answers
44 views

Is a nonzero number infinitely greater than zero? [on hold]

So many years ago, I posted this question on Yahoo! Answers, and was not really happy with the response. I ran across it again recently and decided to try and breathe new life into this in the hopes ...
2
votes
1answer
24 views

Induction proof with inequalities

Consider the following claim: $$5^n > 4^n + 3^n + 2^n$$ (a) For what natural numbers is this claim true? (b) Prove that your answer to (a) is correct using induction on n.
0
votes
0answers
47 views

Roots of sides of triangle “An inequality”

$a, b, c $ are sides of triangle . Prove that : $$ ...
6
votes
2answers
59 views

Minimum of $ay+az+bz+bx+cx+cy$ with $ab+bc+ca=xy+yz+zx=1$

Let $a,b,c,x,y,z\in\mathbb{R}^+$, and $ab+bc+ca=xy+yz+zx=1$. What is the minimum value of $ay+az+bz+bx+cx+cy$? When $a=b=c=x=y=z=\dfrac{1}{\sqrt{3}}$, the desired value is $2$. When ...
0
votes
1answer
21 views

Algebra operation with real numbers

This is in my GMAT prep book ... is it wrong? Given: [y] denotes the greatest integer less than or equal to y. Is d<1? (1) d= y-[y]. Is this sufficient to answer the question? YES (I agree ...
5
votes
5answers
156 views

Show: $(x+y)^4 \leq 8(x^4 + y^4)$ Using Cauchy-Schwarz Inequality

I wish to show the following statement: $ \forall x,y \in \mathbb{R} $ $$ (x+y)^4 \leq 8(x^4 + y^4) $$ What is the scope for genralisaion? Edit: Apparently the above inequality can be shown ...
0
votes
2answers
19 views

Inequality Question find the value of x+y

If $x^{\frac{y}{2}}=64$ and it is given that $y>3$, $x<64$ and $x>y$, what is the value of $x+y$. I reduced the given equation to $x^y=4096$ and formed an a new inequality $x+y>6$ but ...
1
vote
0answers
40 views

How prove exists $z_{0}$ such $|z_{0}|=1$,and $|f(z_{0})|\ge\frac{1}{2^{n-1}}\prod_{j=1}^{n}(1+|a_{j}|)$

let $a_{1},a_{2},\cdots,a_{n}\neq 0$ be given complex numbers and $$f(z)=\prod_{j=1}^{n}(z-a_{j})$$ I need to show that there exists a complex number $z_{0}$ such that $|z_{0}|=1$ and ...
0
votes
1answer
20 views

How to prove Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$?

As the subject states, how can Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$ be proven? Is the proof distribution-dependent or there is a general way to prove it?
1
vote
0answers
17 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
2
votes
1answer
64 views

How prove this $|z|>1$ with $1+z+\frac{z^2}{2!}+\cdots+\frac{z^n}{n!}=0$

For give the postive integer $n$,and $z\in C$ such this $$1+z+\dfrac{z^2}{2!}+\cdots+\dfrac{z^n}{n!}=0$$ show that $$|z|> 1$$ maybe we Assmue that exst $z$ such $$|z|\le 1$$ then we ...
0
votes
1answer
35 views

Derivative of the fundamental solution of the heat equation

Let $\Gamma$ be the fundamental solution of the heat equation in $(0,\infty)\times\mathbb{R}^n$, that is \begin{equation} \Gamma(t,x)=\frac{1}{(4\pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}} \mbox{ per ...
4
votes
0answers
47 views

Are inequalities harder to prove than equalities?

Browsing through the inequalities tag, I see a lot of straightforward-looking arithmetic statements that I nevertheless have no idea how to prove (and apparently I'm not alone). With equalities it's ...
1
vote
3answers
36 views

Inequality with a mixture of logs and trig

Please help me solve this (rather difficult) inequality (or give a hint as to how to do it) -- thanks! $$ \frac{\log _{3-x}(4 x-5) \cdot \log _{4 x}\left(\log _2(7)-x\right)}{\cos x}\leq 0 $$
7
votes
1answer
85 views

How prove this inequality$4(a^2+b^2+c^2)+9a^2b^2c^2\ge 21$

let $a,b,c\in R$,and such $$ab+bc+ac=3$$ show that $$4(a^2+b^2+c^2)+9a^2b^2c^2\ge 21$$ if this equlity condition is $a,b,c$ be postive numbers,and I can use $pqr$ to prove it. But this is $a,b,c$ ...
0
votes
0answers
29 views

Proving the inequality $3+8\sum_{cyc} {\frac{b^2c^2(3b^2+3c^2+7a^2)}{7a^2+6bc}}\ge 9(a^2+b^2+c^2)$

Prove the following inequality: $$3+8\sum_{cyc} {\frac{b^2c^2(3b^2+3c^2+7a^2)}{7a^2+6bc}}\ge 9(a^2+b^2+c^2)$$ $ ab+bc+ca=3$ and $a,b,c\ge0$ The inequality is really hard so I have not even a ...
7
votes
3answers
96 views

Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$

So, a,b,c are non-negative real numbers for which holds that $a+b+c=3$. Prove the following inequality: $$4\ge a^2b+b^2c+c^2a+abc$$ For now I have only tried to write the inequality as ...
2
votes
1answer
52 views

Proving the inequality $a^4+b^4+c^4+2abc(a+b+c)\ge \frac{(a+b+c)^4}9$

If $a,b,c$ are non-negative real numbers prove the following inequality $a^4+b^4+c^4+2abc(a+b+c)\ge \frac{(a+b+c)^4}9$.
0
votes
0answers
5 views

Integer linear problem (ILP) example, with 3 constraints and an additional restriction?

Say I have the following Integer Linear problem: $x_1 \geq$ 0 No. of Item A $x_2 \geq$ 0 No. of Item B $x_3 \geq$ 0 No. of Item C $x_4 \geq$ 0 No. of Item D Maximize z = ...