For questions about proving and manipulating functional inequalities.

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2answers
63 views

An inequality on an arbitrary function

I'm trying to find the complexity of a program and reduced the question to the following one: Let $g$ be a function from natural numbers (including $0$) to natural numbers. Assume that for every $n ...
1
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1answer
19 views

How to show contradiction in the Hardy inequality when the singularity power is greater that 2.

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
2
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1answer
20 views

A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
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0answers
40 views

Study the variation of $\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)}$ with $q$

We define function $g(q)$ as the following: $$\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)},$$ where $n^\prime \ge n+1$, $n^\prime \le m$. We note that $q$, $m$, $n$ $n^\prime$ are all ...
8
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1answer
90 views

The Functional Inequality $f(x) \ge x+1$, $f(x)f(y)\le f(x+y)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function that satisfies the following conditons. $$f(x)f(y)\le f(x+y)$$ $$f(x)\ge x+1$$ What is $f(x)$? It is not to difficult to find ...
1
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1answer
34 views

For all real a, b, order the averages.

I'm taking a proofs class and the textbook says to do this problem: For all real $ a, b > 0 $, show $ \dfrac{2ab}{a + b} \leq \sqrt{ab} \leq \dfrac{a + b}{2} \leq \sqrt{\dfrac{a^2 + b^2}{2}} $ ...
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1answer
19 views

How to prove that the Statistical Entropy $S_{BG}$ is concave

So I am for the moment studying the properties of the Boltzmann-Gibbs statistical entropy \begin{equation} S_{BG}=-k_B \sum_{i}p_i\ln p_i, \end{equation} where of course $k_B$ is the Boltzmann ...
5
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1answer
58 views

If $f(0)=f(1)=1$ and $|f(a)-f(b)| < |a-b|$ then $|f(a)-f(b)| < \frac{1}{2}$

Problem: $f$ be a function on $[0,1]$ such that $f(0)=f(1)=1$ and $f(a)-f(b) < |a-b|$ for all $a$ not equal to $b$. Prove that $|f(a)-f(b)| < \frac{1}{2}$. My attempt: Things I observed are ...
1
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1answer
55 views

Proof of Khintchine inequality

I'm trying to go through a proof of Khintchine inequality, available here: sadly only a polish version available, but perhaps language will not be a barrier here. I understand everything perfectly ...
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0answers
15 views

What is Hoeffding's inequality in Hilbert space?

Suppose I have random variables $X_1, X_2,...,X_n \in \mathcal{H}$, where $ \mathcal{H}$ is some Hilbert space. How can I bound the following term - $ P(\| \sum_{i = 1}^n X_i - E[X_i] ...
2
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1answer
48 views

Prove that $\log x<\sqrt{x}$ for $x\geq 1$

Prove that $\log x<\sqrt{x}$ for $x\geq 1$ Let $f(x)=\sqrt{x}- \log x$. So, $f(1)=1>0$. $f'(x)=\frac{1}{2\sqrt{x}}-\frac{1}{x}>0$ only when $x>4$. When I draw the graph of $f$ in ...
2
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2answers
37 views

How to solve $ | x - a | + | x + a | < b $ where $ a \neq 0 $?

The options are : A ) no solution if $ b\leq 2|a| $ B ) has a solution set $ { ( -b/2 , b/2 ) } $ if $ b > 2|a| $ C ) has a solution set $ { ( -b/2 , b/2 ) } $ if $ b < 2|a| $ D ) All of ...
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1answer
46 views

For $f(x)$ on $(a,b)$ s.t. $f'(x)+f^2(x) \ge -1$ and $\lim\limits_{x \to a} f(x) = - \lim\limits_{x \to b} f(x) = \infty$, prove that $b-a \ge \pi$

$f(x)$ is continuously differentiable here. Using separation of variables, I think I might have shown that the equality form of the statement is true, but I'm a bit wary of trying separation of ...
0
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1answer
25 views

Real Analysis - Manipulating an inequality

I am trying to get the following into a nice expression in terms of $n$ by using inequalities. $$\sup_{x \in [-1,1]}\left| \sqrt{x^2+\frac1n \cos^2(nx)} - |x| \right| \leq \sup_{x \in [-1,1]}\left| ...
2
votes
1answer
30 views

Extending continuous linear functional of the derivative to continuous linear functional of the function

Suppose given $f,g\in L^2(\mathbb{R})$ and $f', g' \in L^2(\mathbb{R})$, the linear functional defined by $$F(g):= \int_{\mathbb{R}} f'g' dx $$ is continuous with respect to the derivative, that is ...
0
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0answers
16 views

Inequality related to Beta functions

Consider two monotonically decreasing functions $f_0(x)$ and $f_1(x)$ for which the following holds: \begin{equation} 1-t\leq f_0(t)\leq f_1(t)\leq1. \end{equation} Let $n,m\in\mathbb{N}$ and $m\leq ...
1
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1answer
37 views

Solving a series of inequalities

If $x_1=1$ and $x_1,x_2,\ldots,x_{100}$ satisfy the following inequalities: $$(x_1 - 4x_2 + 3x_3 )\geqslant0\\ (x_2 - 4x_3 + 3x_4 )\geqslant0\\ \vdots\\ (x_{100} - 4x_1 + 3x_2 )\geqslant0$$ ...
-1
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1answer
102 views

Finding a lower bound of a function which is an inequality [closed]

A function $F(n)$ satisfies the recurrence $F(n) \le 7F(3n/2) + 3n$ for all $n \in \mathbb{N}$. Give a lower bound for $F(n)$.
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2answers
15 views

Can someone clarify in simple terms what it means to “apply an inequality to a measure”?

I am reading wikipedia's entry on Holder's inequality https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality#Counting_measure There are quite a few variations of Holder's inequality "applied to ...
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0answers
21 views

An inequality involved $L_p $ functions [duplicate]

If $p\ge 2$ & $f,g $ are $L_p $ functions, prove that : $||\frac {f+g}{2}||_p^p +||\frac {f-g}{2}||_p^p \le \frac {1}{2}[||f||^p_p +||g||_p^p ]$ At first glance , I thought it is easy noticing ...
2
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1answer
25 views

Functional inequality $f(y)+xf(x)≤yf(x)+f(f(x))$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function sucht that: $$ f(y)+xf(x)≤yf(x)+f(f(x)) $$ for all $x,y\in\mathbb{R}$. Show that $$ f(x)+yf(x+y)≤0 $$ for all $x,y\in\mathbb{R}$. I tried some ...
9
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0answers
826 views

How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
6
votes
2answers
181 views

Is the function characterized by $f(\alpha x+(1-\alpha) y) \le f^{\alpha}(x/\alpha)f^{1-\alpha}(y)$ convex?

[Edit. The question lacks certain important conditions, as kindly pointed out by NeutralElement. Below is the amended version. I apologize for the omissions and many thanks to NeutralElement and ...
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0answers
47 views

$F(a,c)<1$ if $a=c$ and $F(a,c)\geqslant1$ if $a\neq c$

What is a function $F$ (not constructed from step functions) defined for all real or (if possible) complex numbers pairs such that: $F(a,c)<1$ for all $a=c$ and $F(a,c)\geqslant1$ for all $a\neq ...
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0answers
25 views

Softmax inequality

Given $c \lt d$, and x an n-dimensional vector with entries from 0 to 1 inclusive, Prove that: $\frac{e^{cx_i}}{\sum_{j=0}^n e^{cx_j}} + c \lt \frac{e^{dx_i}}{\sum_{j=0}^n e^{dx_j}} + d $
2
votes
3answers
32 views

Rational Inequality - Find the values of a such that range of $f(x)=\frac {x+1}{a+x^2}$ contains $[0,1]$

Find the values of $\text{“}a\text{''}$ such that range of $f(x)=\frac{x+1}{a+x^2}$ contains $[0,1]$ where am i wrong ??: I took $2$ cases : Case $1$ : $a+x^2 \gt 0$ , I solved the inequality ...
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0answers
24 views

Find the range of of $r_i$ from the inequality.

I have a relationship $$ \begin{cases} 0 \le r \le n \\ 0 \le k_1 \le n_1,0 \le k_0 \le n_0\\ r=r_0+r_1\\ 0 \le r_0 \le n_0\\ 0 \le r_1 \le n_1 \end{cases} $$ where $n=n_1+n_0,k_0,k_1,n_0$ are known ...
1
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2answers
50 views

Prove $n^2 \leq 1.1 ^{n}$ by induction

Prove that for all $n \geq 100$ you have $n^2 \leq 1.1^n$ Base Case: $n = 100$ $(100)^2 \leq 1.1^{100}$ (True) Inductive Case: Suppose $(k-1)^2 \leq 1.1^{k-1}$ for some $k \geq 101$ Prove ...
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0answers
27 views

Brunn-Minkowski Inequality : A Partucular Example of a 2 dimensional set.

I have an inequality - $(R_{C+D})^{2} \geqslant (R_{C})^{2} + (R_{D})^{2}$. Representing it as $((R_{C+D})^{4})^{1/2} \geqslant ((R_{C})^{4})^{1/2} + ((R_{D})^{4})^{1/2}$, it follows from ...
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1answer
37 views

Show that a recurrence relation is bounded

I'm studying the following recursive equation as part of my Economics class: $$k_{t+1}={k_t}^\alpha - c_t$$ For $t=0,1,2,...$, $0\leq \alpha<1$, and $k_0>0$ and $c_t\geq 0$ for every $t$. I ...
2
votes
3answers
163 views

An upper bound of binary entropy

Binary entropy is given by $$H_{\mathrm b}(p) = -p \log_2 p - (1 - p) \log_2 (1 - p), \hspace{6 mm} p \le \frac{1}{2}$$ How can I prove that $$H_{\mathrm b}(p) \le 2 \sqrt{p(1-p)}$$
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4answers
37 views

Typical Inequalities for real numbers

If a,b,c and d are real numbers, then (a) $\bigg(\frac{5a}{12}+\frac{b}{3}+\frac{c}{6}+\frac{d}{12}\bigg)^2 \leq \frac{5a^2}{12}+\frac{b^2}{3}+\frac{c^2}{6}+\frac{d^2}{12}$ (b) $a+b+c=2$ with ...
2
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0answers
23 views

What is the purpose of continuous and differentiable dependence

In learning Gronwall's inequality you also get to learn about continuous an differentiable dependence. I know the theorems but I have no idea about their application. What is the big idea of ...
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1answer
18 views

Fractions in L^\infty

Let $f,g \in L^2(\Omega)$, where $\Omega$ is an open bounded subset of $R^n,\; n\ge 1$. We consider the two following properties: (1) $\frac{f}{g} {\bf 1}_{\{x\in \Omega;\; g(x)\ne 0\}} \in ...
3
votes
2answers
192 views

Finding a function satisfying a certain inequality

This is a continuation of this post where I tried to find a function $f(n)$ that would satisfy the induction step of an inductive argument and it was shown that such function does not exist. Trying ...
0
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1answer
41 views

Finding a function such that $(n-2)/2 + f(n-1) \leq f(n)$

By bounding a certain quantity defined on real numbers by $f(n)$ I derived the following inequality arising from an inductive argument. $ (n-2)/2 + f(n-1) \leq f(n).$ A solution to the above ...
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0answers
40 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in ...
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1answer
87 views

Inequalities help! [closed]

Let $a,b,c > 0$ and $a + b + c = 1$. Prove: $$\sqrt{\frac{ab}{c + ab}} + \sqrt{\frac{bc}{a + bc}} + \sqrt{\frac{ac}{b + ac}}\leq \frac32$$
5
votes
1answer
73 views

When does $f\left [g\left (x \right ) \right ]=g\left [ f\left ( x \right ) \right ]$ have roots.

I met a interesting equation: $$\sin\left [\cos\left (x \right ) \right ]=\cos\left [\sin\left (x \right ) \right ]$$ (And of course, the equation has no roots). So, Let $f\left ( x \right )$ ...
4
votes
1answer
165 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
0
votes
2answers
72 views

Need help in solving inequality of this type: $|ax + b| > -c$

This is the equation: $|3x+6|>-12$. I solved it under two cases ($3x+6>-12$) and ($3x+6<12$). More than the answer, what I need to know is - Have I rightly constructed those $2$ cases, if ...
0
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0answers
58 views

If $\|f\|_X\le c_1\|f\|_Y$ then do we have $\langle f,g\rangle_X\le c_2\langle f,g\rangle_Y$?

Let $X$ and $Y$ be two inner product spaces with inner products $\langle \cdot,\cdot\rangle_X$ and $\langle \cdot,\cdot\rangle_Y$, respectively. Suppose we have $\|f\|_X\le c_1\|f\|_Y$ for any $f\in ...
4
votes
1answer
49 views

Suppose that $g(x) = \frac{f(x+1)-f(1)}{f(x)-f(0)} \geq g(1)$, what do we know about $f$?

Suppose $f:\mathbb{R_+} \to \mathbb{R}$ is a continuous and strictly increasing function. Define $g(x) = \frac{f(x+1)-f(1)}{f(x)-f(0)}$. For which $f$ the function $g$ satisfies $g(x) \geq g(1)$ for ...
4
votes
3answers
114 views

How to show $ \Big\vert \frac{\sin(x)}{x} \Big\vert $ is bounded by $1$?

This may be a silly question, but I cannot figure it out. I want to prove that $ \Big\vert \frac{\sin(x)}{x} \Big\vert \leq 1 $ for $x\in[-1,0)\cup(0,1]$, but I don't even know where to start.
0
votes
0answers
22 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
1
vote
2answers
306 views

Quadratic Absolute Value Inequality

Problem: Find all $x$ such that $|x^2-3x+1|<1$ I can't understand how to get started with this. I've never tried to solve quadratic Inequalities before. At first I thought of working with the ...
5
votes
2answers
59 views

Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
1
vote
2answers
51 views

Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
0
votes
0answers
27 views

Inequalities for Laguerre polynomials

The following inequality holds, $$ \Big( 4\int_0^\infty rdr \big|\mathcal{L}_1(4r^2)\big|e^{-2r^2}\Big)^3 \geq 4\int_0^\infty rdr \big|\mathcal{L}_3(4r^2)\big|e^{-2r^2}, $$ where $\mathcal{L}_n(x)$ ...
0
votes
1answer
32 views

Bounds on functions via its derivatives

Suppose, we have a function $f$ where $f$ is: Contionuos. Non-Negative Has a derivative given by $f'$. Can we have a bound on $f$ in terms of its derivative $f'$? That is have an inequality that ...