Questions on proving and manipulating inequalities.

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1
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1answer
387 views

A generalized Jensen's inequality

Let $(U , \mu)$ and $(V,\nu)$ be probability spaces. Let $f$ be a convex functional on $L^1(\mu)$, i.e. $$f(tX + (1-t)Y) \leq t f(X) + (1-t)f(Y)$$ for all random variables $X$ and $Y$ in $L^1(\mu)$. ...
6
votes
4answers
866 views

Showing inequality for harmonic series.

I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.
2
votes
4answers
208 views

Solving a quadratic Inequality

My question is: Solve $$9x-14-x^2>0$$ My answer is: $2 < x < 7$ Though I know my answer is right, I want to know in what ways I can solve it and how it can be graphically represented. ...
2
votes
1answer
281 views

looking for a norm inequality

I want an inequality of the form : $\Vert a - b \Vert^2 \leq k.(\Vert a\Vert^2 + \Vert b\Vert^2)$ ? where k is a constant. The norm in consideration is the euclidean norm, and $a$ and $b$ are ...
5
votes
1answer
133 views

A Hölder-like inequality

Consider a probability measure $\mu$ on a set $X$. Let $p,q \in (1, \infty)$, $f \in L^{pq} \cap L^1$ (so also $f\in L^p \cap L^q$) by non-negative. Can we say anything about the relationship between ...
1
vote
2answers
94 views

Theorem about two real numbers 2

My question is: Prove- If $a,b$ are two positive real numbers such that their sum is $a+b=k$. Then the product $ab$ is maximum if and only if $a=b=\displaystyle\frac{k}{2}$. I proved the ...
25
votes
6answers
1k views

$m!n! < (m+n)!$ Proof?

Prove that if $m$ and $n$ are positive integers then $m!n! < (m+n)!$ Given hint: $m!= 1\times 2\times 3\times\cdots\times m$ and $1<m+1, 2<m+2, \ldots , n<m+n$ It looks simple but ...
2
votes
0answers
96 views

Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
0
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1answer
1k views

Holder's inequality

Suppose that $f$ and $g$ are two non negative real valued functions defined on a measure space $(X,\mu)$. Let $0<p<\infty$. Holder's inequality says that $\int fg d\mu\le \|f\|_p \|g\|_q$ where ...
-1
votes
3answers
149 views

An inequality with absolute value and a parameter: $|x-4|>a$

Solve : $|x-4|>a$. Case 1: $a>0$; Case 2: $a<0$ Progress I am getting answers which look similar in both cases: Let $a>0$ so $x>4+a$ or $x<4-a$ , Let $a<0$ so ...
2
votes
1answer
332 views

A definite integral inequality with $\pi$ number

I need to prove that the following inequality holds: $$\int_{0}^{1} \sqrt{x}\space e^{-x^2}dx \leq \frac{\pi}{6}$$ No progress on it, yet. Any suggestion is welcome. Thanks.
3
votes
1answer
145 views

Another inequality with definite integrals

Let be $ f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous, monotone function. Then, if $a>0$ i must prove that the following inequality holds: $$\int_{-a}^{a}xf(f(x)) \geq0$$ I wonder if there ...
3
votes
2answers
168 views

Is this inequality true?

Clearly if $a,b >0$ and $p \in \mathbb{N}$ $$ a^{p} + b^{p} \le (a+b)^{p} $$ Is there a constante $C = C(p)$ such that if $a,b >0$ and $p \in \mathbb{N}$ then \begin{equation} a^{p} - b^{p} \le ...
7
votes
1answer
1k views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...
1
vote
2answers
115 views

A hard proof of two matrix's elements

This is not duplicate of A matrix's element proof, but it is harder than that one. Given an constant $\alpha \in (0,1)$, and an $n \times n$ matrix $X$ whose all entries are between 0 and ...
0
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2answers
76 views

A matrix's element proof

Thanks again for copper.hat and Robert Israel's quick immediate reply. While I am modifying the questions, they've already given the answer. Now in this thread, I've changed it back to the original ...
2
votes
1answer
142 views

Equality in the Isoperimetric Inequality

Stein and Shakarchi, in their book Real Analysis, the third volume of the Princeton Lectures in Analysis series, give a proof of the isoperimetric inequality for closed rectifiable curves in ...
9
votes
3answers
339 views

Convex functions in integral inequality

Let $\mu,\sigma>0$ and define the function $f$ as follows: $$ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\mathrm \exp\left(-\frac{(x-\mu)^2}{2\sigma ^2}\right) $$ How can I show that $$ ...
2
votes
1answer
149 views

Sufficient conditions to hold the following inequality.

If I have an inequality: $\lVert u\rVert_{L^p(R^n)} \le C\lVert\nabla u\rVert_{L^q(R^n)}$ , where $C \in (0,\infty)$ and $u \in C_c^1(R)$, is there a relation between $p, q, n$ such that the ...
4
votes
1answer
187 views

Prove an inequality about $\arctan 1/(nx)$ for any $x$ and $n$

How to prove this inequality for any $x$ and $n$? $$ \left|\arctan\frac 1{nx}\right| \leq \frac 1{nx} ;\, 0<x<+{\infty} $$ Is this bounded? But how that can help me in proving? I mean that I ...
8
votes
1answer
185 views

An inequality with strictly positive real numbers

Let be $x_1, x_2, \ldots , x_n$ strictly positive real numbers. Prove that the following inequality holds: $$\frac1{1+x_1}+\frac1{1+x_1+x_2}+\cdots+\frac1{1+x_1+x_2+\cdots+x_n} < ...
1
vote
1answer
81 views

How to prove an L$^p$ type inequality

Let $a,b\in[0,\infty)$ and let $p\in[1,\infty)$. How can I prove$$a^p+b^p\le(a^2+b^2)^{p/2}.$$
3
votes
1answer
65 views

Inequality between two sums of inverses

Let $x_1, \ldots ,x_n$ be positive numbers satisfying $|x_{k+1}-x_k| \leq 1$ for any $k$, and $x_1+x_2+ \ldots +x_n > \frac{n(n-1)}{2}$. For $k$ between $1$ and $n$ put $$ y_k=\frac{x_1+x_2+ ...
1
vote
1answer
136 views

Prove by Induction: Weak Inquality involving Product of Factorial, Factorial in the Exponent, Exponents.

I want to show the below statement. $n!x^{n!} \leq n\cdot x^{n}$ Where $0 \leq x <1$ for all $n\in \mathbb{N}$ My approach is induction. The Induction start This is easy. Set n=1 and the ...
2
votes
2answers
69 views

natural question of inequality

Is there $C=C(p)$ constant that depend only on $p$ such that if $a,b > 0$ we have $$ (a +b)^{p} \le C(a^{p} + b^{p})? $$ where $p \in \mathbb{N}$ is fixed. For example, if $p=2$ $$ (a+b)^{2} \le ...
0
votes
2answers
55 views

Inequality involving exponentials

I have a small doubt and will much appreciate it if someone will clear it up. For large $k$ and $n=\lfloor 2^{k/2}\rfloor $, why are the following inequalities true: ...
8
votes
1answer
457 views

Geometric interpretation of Young's inequality

Is there a geometric interpretation of Young's inequality, $$ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$$ with $\dfrac{1}{p}+\dfrac{1}{q} = 1$? My attempt is to say that $ab$ could be the surface of ...
8
votes
1answer
404 views

Putnam PigeonHole

This is from page 12 of Putnam and Beyond. Problem: Prove that for every set $X ={x_1,x_2, \ldots ,x_n}$ of $n$ real numbers, there exists a nonempty subset $S$ of $X$ and an integer $m$ such that ...
2
votes
3answers
101 views

An inequality problem

I can't work out that inequality problem. If $x\ge0$, $y\ge0$, how to prove that $$ 1+x+y+xy\leq(x+1)\ln(x+1)+e^y? $$ I tried taylor expansions for $$ln(x+1)$$ and $$e^y,$$ I also tried $$ ...
1
vote
1answer
535 views

Poincaré inequality in unbounded domain

Help me please, how can I to show that Poincaré inequality in unbounded domain doesn't holds? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
16
votes
2answers
504 views

How to show that $\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$

Show that: $$\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$$ All I've got so far is that the minimum of $x^x$ is $e^{-1/e}$. At this point I could compare $\pi/5$ to $e^{-1/e}$ but I'm ...
30
votes
6answers
2k views

What is the larger of the two numbers?

What is the larger of the two numbers? $$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$ I solved this, and I think that is an interesting elementary problem. I want different points ...
6
votes
1answer
562 views

Help understanding proof of Schwarz Inequality

I'm working through Spivak's Calculus over the summer, and I'm currently on problem 19 of Chapter 1, which involves proving the Schwarz inequality. The first two parts of the proof are fairly ...
3
votes
2answers
292 views

Lower Bound of Central Binomial Coefficients

I would like to prove by induction the following inequality: $\frac{4^n}{n+1} < \binom{2n}{n}$, for all natural numbers n > 1. Any hints?
6
votes
2answers
2k views

Min/Max of $f(x,y) = e^{xy}$ where $x^3+y^3=16$

Use Lagrange multipliers to find the maximum and minimum values of the function :$$f(x,y)=e^{xy}$$ constraint $$x^3+y^3=16$$ This is my problem in my workbook. When I solve, I'm just have one ...
0
votes
0answers
54 views

How to prove inequality_3

$\Bigl( \sum\limits_{i=1}^{n-1} a_{i}^{2}\Bigr) \cdot \Bigl( \sum\limits_{j=1}^{n-1} b_{i}^{2}\Bigr)-\Bigl( \sum\limits_{l=1}^{n-1} a_{l}b_{l}\Bigr)^{2}-2\cdot\lceil\frac{n-1}{2} \rceil\cdot \Bigl( ...
0
votes
1answer
117 views

How to prove inequality

How to prove this inequality: $$a^2 d^2+b^2 c^2-1-4ac-4bd-2abcd \leq 0,$$ where: $a, b, c, d \in \{0, 1, 2, \ldots\}$ and $|a-c|\leq 1, |b-d|\leq 1$?
1
vote
2answers
218 views

Relating Gamma and factorial function for non-integer values.

We have $$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$ for integers, so if $\Delta$ is some real value with $$0<\Delta<1,$$ then $$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$ because ...
8
votes
3answers
164 views

Prove the inequality

I need to prove that $$\frac{k(k+1)}{2}\left(\frac{a_1^2}{k} + \frac{a_2^2}{k-1} + \ldots + \frac{a_k^2}{1}\right) \geq (a_1 + a_2 + \ldots + a_k)^2\;,$$ where $a_1, a_2, \dots, a_k$ is some set of ...
0
votes
2answers
119 views

Lower bound on a minimum of maximum of a sequence of standard normal random variables

Let $X = (x_{ij}) \in \mathbb{R}^{n \times p}$ be a matrix with independent $N(0,1)$ entries. We know that $\max_j x_{ij} < \sqrt{2\log(p/\delta)}$ with probability at least $1-\delta$. I would ...
2
votes
0answers
110 views

Generalizing an approach to proving AMGM

This problem is Exercise 5.5.30 of "The Art and Craft of Problem Solving" by Paul Zeitz. The problem asks to use the identity $$ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) $$ to prove the AMGM ...
1
vote
2answers
187 views

An inequality for two complex numbers

I recently saw the following inequality for complex numbers: If $a,b\in\mathbb C$ and $|a + b|$ and $|a-b|$ are each less than or equal to 1, then $$|a| + |b^2|/2 \leq 1.$$ How can one prove this?
2
votes
1answer
235 views

Upper bound for $n^{th}$ power of a sum [duplicate]

Possible Duplicate: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$ We can use Young's inequality to show that $(a+b)^2 \leq 2a^2 + 2b^2$. Does a similar ...
2
votes
2answers
109 views

How to solve this inequation

Given two real numbers $0<a<1$ and $0<\delta<1$, I want to find a positive integer $i$ (it is better to a smaller $i$) such that $$\frac{a^i}{i!} \le \delta.$$
1
vote
3answers
217 views

elementary inequality proof

I am working on a howework question, trying to prove the following: $$5a+b > 4\sqrt{ab},$$ where $a$ and $b$ are positive real numbers. I've tried multiplying expression by $\sqrt{ab}$, squaring ...
2
votes
1answer
101 views

the scaling problem of Sobolev inequality

How to show that by replacing $\psi(x)$ with $\psi(\lambda x)$, an inequality of the form $$ \int|\nabla\psi|^2dx\geq C\left(\int|\psi|^qdx\right)^\frac{2}{q} $$ can only hold for $q=6$ in $N=3$?
1
vote
1answer
354 views

An inequality $|x^{1/n}-y^{1/n}|\leq c|x-y|^{1/n}$

I need to prove that $|x^{1/n}-y^{1/n}|\leq c|x-y|^{1/n}$, where $x,y\in [0,+\infty)$, $n\in\mathbb{N}-\{0,1\}$, and $c=2^{(n-1)/n}$. I tried a lot ways for proving cases $n=2$ and $n=3$, but are ...
4
votes
1answer
206 views

Polynomial inequality

I found the following problem on a website and would be curious to find a solution. Let $a_1\ge a_2\ge\cdots\ge a_n$ be real numbers such that for all integer $k>0$: $$a_1^k+a_2^k+\cdots+a_n^k\ge ...
2
votes
1answer
113 views

Help with question involving limits, bounds, inequalities

Would someone like to help with the following question? Prove that for $n=1,2,\ldots$ (a) $5\leq (4^n+5^n)^{1/n}\leq 10$ and that $(4^n+5^n)^{1/n}$ is bounded, (b) $(4^n+5^n)^{1/n}\geq ...
12
votes
1answer
891 views

Factorial Inequality problem

I met an inequality, I ask, do not mathematical induction to prove that: Prove \[ \left(\frac n2\right)^n > n! > \left(\frac n3\right)^n \] without using induction