Questions on proving and manipulating inequalities.

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8
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2answers
264 views

Proving Integral Inequality

I am working on proving the below inequality, but I am stuck. Let $g$ be a differentiable function such that $g(0)=0$ and $0<g'(x)\leq 1$ for all $x$. For all $x\geq 0$, prove that ...
1
vote
2answers
201 views

How to find the minimum value of $px+qy$ when $xy=r^2$?

The question says: "Find the minimum value of $px+qy$ when $xy=r^2$." No information is given on $p,q,x,\text{and }y.$ However assuming the obvious I tried using this, but I am not able reduce it to ...
2
votes
1answer
869 views

A lower bound on the chromatic number of regular graphs

Given a $k$-regular simple graph $G$ of $n$ vertices, show that $\chi(G) \geq n/(n-d)$, where $\chi(G)$ is the chromatic number of $G$. I'm rather unsure how to start this one, however my initial ...
1
vote
1answer
158 views

Markov's inequality for general random variables

Suppose $E(2^{X})=4$. Prove that $P(X\geq3)\leq1/2$. Using Jensen's inequality we know $E(X)\leq2$ but since $X$ is a general random variable we can't use Markov's inequality to get a bound for ...
2
votes
1answer
219 views

Justifying a pair of inequalities involving the exponential function

I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page ...
0
votes
3answers
118 views

Proving an inequality with Taylor polynomials

This is a homework question I was asked to do Of a twice differentiable function $ f : \mathbb{R} \to \mathbb{R} $ it is given that $f(2) = 3, f'(2) = 1$ and $f''(x) = ...
7
votes
1answer
375 views

Is there a discrete-time analogue of Doléans-Dade exponential?

For a continuous martingale $X$, we have the Doléans-Dade exponential: $$\epsilon(X)_t=\exp\left(X_t-\frac{1}{2}[X]_t\right)$$ What is the "correct" analogue, if one exists, for some discrete-time ...
4
votes
3answers
495 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
7
votes
1answer
2k views

A simpler proof of Jensen's inequality

Jensen's inequality states that if $(X,\mu)$ is a measure space with $\mu(X) = 1$, $\phi$ is convex, and $f:X \rightarrow \mathbb R$ is integrable, then $$\phi\left(\int fd\mu\right) \leq \int \phi ...
20
votes
2answers
878 views

Geometric proof for inequality

While on AOPS, I saw this interesting problem. I was wondering how many different approaches could be used to tackle the problem. In other words I am looking for interesting and unique ways to solve ...
2
votes
4answers
747 views

Prove integral inequality

Assume that a function $f$ is integrable on $[0, x]$ for every $x > 0$. Prove that for any $x > 0$, $\displaystyle\left (\int_{0}^{x}fdx \right )^2\leq x\int_{0}^{x}f^2dx$. I have no idea ...
2
votes
0answers
163 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: ...
3
votes
1answer
2k views

Why Mathematica (Reduce) can't find clear solution for almost trivial inequalities?

Suppose we want to solve the inequality $ax^2+bx+c<0$. For simplicity, presume that $a>0$ and $b^2-4ac>0$. In this form, this is almost a trivial problem. Despite that, if we want to solve it ...
4
votes
2answers
149 views

Problem based on Range

Find $a$ and $b$ such that the inequality $a \le 3 \cos{x} + 5\cos\left(x - \frac{\pi}{6}\right) \le b$ holds good for all x.
8
votes
3answers
1k views

Hardy's Inequality for Integrals

I am trying to prove Hardy's Inequality for integrals: Let $f:(0,\infty) \rightarrow \mathbb R$ be in $L^p$. Let $F(x) = \frac{1}{x} \int_0 ^x f(t)dt$. Then $F \in L^p$ and $\|F\|_p \leq ...
0
votes
2answers
85 views

Proving a multivariable inequality

I would like to know if it is possible to select R1, R2, R3, C1, and C2 such that the quadratic equation yields complex roots. $s_{1,2}=-\frac{b\pm \sqrt{b^2-4ac}}{2a}$ where $a = ...
3
votes
1answer
1k views

Reverse Markov Inequality for non-negative unbounded random variables

I need to lower bound the tail probability of a non-negative random variable. I have a lower bound on its expected value. I am aware of a reverse markov's inequality that does the job when the random ...
4
votes
1answer
343 views

Proving that a polynomial is positive

A Finnish mathematics competition asked to prove that for all $x$ we have $x^8-x^7+2x^6-2x^5+3x^4-3x^3+4x^2-4x+\frac{5}{2}\geq 0$ for all real $x$. I heard that it follows from Hilbert's problem that ...
3
votes
1answer
282 views

Applications of Young's convolution inequality

Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided ...
-4
votes
1answer
95 views

Inequality question/proof?

I'm a little confusing in proving this inequality $$\frac{a+b}{|c-b|}<1$$ where $a,b,c$ are positive real numbers, and $a<c$. any help!
2
votes
2answers
225 views

Hölder Inequality Application

I am trying to follow a step in a proof which uses the Hölder Inequality. The setting is: $f\in L^{p}(\mathbb{R}^{d})$, $g\in L^{1}(\mathbb{R}^{d})$, with $1\leq p\leq \infty$. The claim is that: ...
1
vote
1answer
123 views

Berry-Esseen inequality for the event $a<S_n<b$

Suppose that $X_i$ are independent identically distributed with finite variance and $S_n=X_1+\cdots+X_n$. One can use the Central Limit Theorem to estimate (a) $P(S_n \leq b)$ and (b) $P(a<S_n \leq ...
1
vote
3answers
201 views

Trouble with an Inequality

In showing that if $f,g\in L^p$, then $f+g\in L^p$, one can use the fact that $$|f+g|^p\leq 2^p\left(|f|^p + |g|^p\right).$$ The result I'd like help in proving is this: Given that $1\lt p ...
3
votes
2answers
82 views

a plausible maximum or minimum

Is the following statement true? Let $a_1\ge a_2\ge \cdots \ge a_n>0$, $b_1\ge b_2\ge \cdots \ge b_n>0$, then $$\max\limits_{\sigma\in S_n}\;\;\prod\limits_{i=1}^n(a_i+b_{\sigma ...
0
votes
1answer
274 views

Question related to Chebyshev Sum Inequality

The continuous version of the Chebyshev Sum Inequality say that if $f,g:[0,1] \to \mathbb{R}$ are either both decreasing or both increasing, then $$\int_0^1 fg \;dx \geq \int_0^1 f \;dx \int_0^1 g ...
3
votes
1answer
151 views

Fourier transform inequalities on a probability distribution

I am reading a paper and the following came up: Given a probability density function, $\rho(x)$, such that for $\epsilon > 0$ $$ \int_{-\infty}^{\infty} |\rho(x)|^{1+\epsilon}dx < \infty ...
1
vote
2answers
215 views

Sequence inequalities

A bit rusty on this stuff. The whole problem is proving this is true: $$ 2 \leq 1 + \sum_{m=1}^{n} \frac{1}{m!} \leq 1 + \sum_{m=1}^{n} \frac{1}{2^{m-1}} < 3. $$ I have figured out the first two ...
9
votes
3answers
1k views

An application of Jensen's Inequality

Given that $\{\phi_n\}$ is a sequence of non-negative numbers whose sum is $1$ and $\{\psi_n\}$ is a sequence of positive numbers, how can I show that $$ ...
1
vote
4answers
128 views

Proving inequality $x^{10}-x^6+x^2-x+1>0$

How can the inequality $x^{10}-x^6+x^2-x+1>0$ be proved a) using elementary mathematical methods? b) using higher mathematical methods?
4
votes
2answers
115 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
2
votes
1answer
154 views

Inequality involving $\log$

Let $g$ be a non-negative measurable function on $[0,1]$. How can I show that $$ \int \log ~(g(u))~\text{d}u \leq \log~\int g(u)~\text{d}u $$ whenever the left hand side is defined. If it helps, I ...
0
votes
2answers
110 views

Equivalent conditions to $0\leq x+\frac{1}{2}x(1-x)a\leq 1$?

How can I find a equivalent conditions for the following inequality $$ 0\leq x+\frac{1}{2}x(1-x)a\leq 1? $$ This is from a numerical analysis book, Finite-Volume Methods For Hyperbolic Problems. I ...
3
votes
4answers
200 views

How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$?

How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$?
1
vote
2answers
144 views

Lower bound of an integral

How might I find a constant $k>0$ s.t. $\int\limits_0^1|f(x)|dx\geq k\max\{|f(x)|:x\in[0,1]\}$ for all continuous $f$ defined on $[0,1]$? Thank you.
5
votes
1answer
1k views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
1
vote
1answer
83 views

Show $\sum\limits_{k=2}^{n}{k \over \ln k} \le {n^2 \over \ln n}$

Why is it true that $$\sum\limits_{k=2}^{n}{k \over \ln k} \le {n^2 \over \ln n}, n \ge 2$$ I try to expand the term of the sum in taylor series but it didn't help me. I try to recognize the sum as a ...
3
votes
3answers
93 views

Manipulating this Inequality

What's wrong with this method of trying to find the set of positive values of $x$ that satisfy the following inequality: $$\dfrac{1}{x}-\dfrac{1}{x-1} > \dfrac{1}{x-2}$$ Find a common ...
4
votes
3answers
238 views

Proving $n! > n$ for $n > 2$ using mathematical induction

I have to prove $n<n!$ for all $n>2$ by mathematical induction. I did it as follows. I proved the base case. Then let it be true for $K>2$: $$ K<K! $$ I have to prove, $$ ...
5
votes
1answer
195 views

Lower bounding a ratio of gamma functions

I am trying to show that the following function has a lower bound of $\ \frac{1}{2}$ for all $c\geq 2$. Or, alternatively, that that function increases with $\ c$: ...
1
vote
3answers
125 views

Prove: $|x+y| < \max(|x|, |y|) \Leftrightarrow xy < 0$

I have to demonstrate that $|x+y| < \max(|x|, |y|) \Leftrightarrow xy < 0$. I'm bit lost as how to proceed on this. I know I have to separate in two cases and that the right side is $x$, when $x ...
0
votes
3answers
191 views

Solve inequations over a finite field

The following is a special case of my earlier question which is still not solved. Suppose both $\mathbf{a_i}$ and $\mathbf{v}$ are $1\times N$ vectors over a finite field $\mathbb{F_q}$, where $i\in ...
4
votes
6answers
484 views

How can I prove the inequality $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \frac{9}{x+y+z}$?

For $x > 0$, $y > 0$, $z > 0$, prove: $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \frac{9}{x+y+z} .$$ I can see that this is true, I also checked it with a few numbers. But I guess that ...
3
votes
3answers
283 views

Finding an inequality for a word problem

A little bit of background - I am currently in high school learning system of equations and inequalities. We've done things like linear inequalities, inequalities with parabolas, and a combination of ...
11
votes
10answers
354 views

Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no ...
0
votes
0answers
169 views

How to solve these inequations?

$C_i$ is a $k_i\times N$ matrix over finite field $\mathbb{F}_q$, where $i\in \{1,2,\ldots,K\}$, $k_i<N$, and $q<K$. My questions are 1) how to determine whether there is a $1\times N$ vector ...
6
votes
1answer
131 views

An inequality for graphs

In the middle of a proof in a graph theory book I am looking at appears the inequality $$\sum_i {d_i \choose r} \ge n { m /n \choose r},$$ and I'm not sure how to justify it. Here $d_i$ is the ...
5
votes
2answers
187 views

Smallest integer $n$ such that $\left(1-\frac{n}{365}\right)^n < \frac{1}{2}$

Find the smallest integer $n$ such that $$\left(1-\frac{n}{365}\right)^n < \frac{1}{2}.$$ I cannot use a calculator, and I do not know where to begin.
3
votes
3answers
430 views

Tough Inequality

I was doing some problems for Olympiad training and encountered this: How would you prove that $(a+b+c+d)-(a+c)(b+d)\geq 1$? We are told that $0<a,b,c,d<1$ and the product ...
3
votes
3answers
596 views

Strong Mathematical Induction Recursion Inequality

I have a question that is for a homework assignment and I just would like to ask if I seem to be on the right track or if I'm just doing it completely wrong. Here is the question: The sequence ...
2
votes
0answers
171 views

A trigonometric inequality involving sine

Let $0<a<\pi/2,0<b<\pi/2$, $0<\lambda<1, \mu=1-\lambda$. Does anyone see a good proof of the inequality: $$\sin(\lambda a)\sin(\lambda b)+\sin(\lambda a)\sin(\mu ...