For questions about proving and manipulating functional inequalities.

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3
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1answer
64 views

Functional inequality $\sum_{1\le i<j\le n}f(x_i+x_j)\ge \frac{n(n-1)}{2}f(a_1x_1+a_2x_2+…+a_nx_n)$

Let $n\in\mathbb N, n\ge 2$. Does there exist a set of non-zero real numbers $a_1, a_2,..,a_n$ with this condition: If function $f: \mathbb R \rightarrow \mathbb R$ satisfies the inequality $$...
0
votes
2answers
52 views

Proof of $\Gamma(x)\Gamma\left(\dfrac{1}{x}\right)\gt 1$

Is it possible to prove the following inequality: $$\forall x\gt0,f(x)=\Gamma(x)\Gamma\left(\dfrac{1}{x}\right)\gt 1?$$ $f(x)$ has a minimum where $f(x)'=0$ which means: $$\Psi(x)\Gamma(x)\Gamma(1/x)=\...
4
votes
2answers
57 views

A sufficient condition on $C^1$ positive functions for $f(x+y)<f(x)+f(y)$

I am trying to show that if $f:(0,+\infty)\rightarrow\mathbb R$ is a $C^1$ function such that $$f'(x)<\frac{f(x)}{x}\quad \forall x\in (0,+\infty) \tag{$\star$}$$ then $$f(x+y)<f(x)+f(y)$$ ...
1
vote
0answers
24 views

Kind of Gronwall Inequality

Does somebody knows if it is possible to obtain an inequality (like for Gronwall inequality) on $f$ if $f$ verify $$ f(t) \leq A+\int_0^{2t} g(s)f(s) ds $$. Where $f$ and $g$ are as smooth as ...
1
vote
1answer
30 views

For what value of $x$: $ n^ {(x+1)} + n^ {2x} < n^2$ ? Where, $0\leq x <1$ and $n$ is constant integer value & $n>1$.

How to find the optimal value of $x$ and what is the relation between $x$ and $n$ i.e. How to get dependency between $x$ and $n$? As per my understanding, solution should be in term of $n$ like like ...
1
vote
2answers
42 views

How to Prove that this Function is Constant?

Let be $h: \mathbb{R} \rightarrow \mathbb{R}$ and c a positive constant, if $\forall a, b \in \mathbb{R}$ we have that: $$\frac{|h(a)-h(b)|}{a-b}\leq |a-b|^c$$ Prove that $h$ is constant. ...
0
votes
1answer
39 views

$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ then Holder's inequality [duplicate]

If $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ and $ f\in L_p $ $g\in L_q $ and $h\in L_r $ so how can I prove $$ ||fgh ||_1\le||f||_p\ ||g||_q\ ||h||_r $$
1
vote
0answers
29 views

Lower bound on the difference between max. and min. values of a polynomial over $[-1, 1]$

Problem: $P(x)$ be a monic, n-degree polynomial with real coefficients. Prove that it is not possible that for all $t \in [-1, 1]$, $$\frac{-1}{2^n} < P(x) < \frac{1}{2^n}$$. I tried it to put ...
4
votes
1answer
754 views

Limit of a function satisfying an inequality

If $f(x)+f(y)\leq f(x+y)$ and $f:\mathbb{R}\to\mathbb{R}$, then can we find $\lim_{x\to 0} \frac {f(x)}{x}$? I am not sure whether the question is correct.Thank you.(I tried this idea: $f(x)=f(x+y-y)\...
0
votes
0answers
16 views

A functional equation with an inequality

I have an increasing function on $[0,1]$, $p \mapsto \Pi(p)$, that has the following properties. $$\Pi(0) = 1 - \Pi(1) = 0$$ $$\Pi(p) + \Pi(1-p) < 1 \quad \forall{p} \in (0,1)$$ $$\Pi(p) > p \...
1
vote
1answer
42 views

Cauchy functional inequality

Given a function on a closed interval $f\colon I\subset \mathbb{R}\to \mathbb{R}$ with $$f(x+y) \leq f(x) + f(y).$$ Moreover, I know that $f$ is monotonic increasing continuous on all points except ...
4
votes
0answers
42 views

Find how many such complex numbers exist

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\text{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers ...
2
votes
3answers
40 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 $f(x)...
2
votes
1answer
32 views

Showing a certain map is a norm.

Define $$\|x\|=\sqrt[3]{(|x_1|^2+|x_2|^2)^{3/2}+|x_3|^3}$$ for any $x=(x_1,x_2,x_3)\in \Bbb R^3$. Show that $\|\cdot\|$ is a norm on $\Bbb R^3$. I stuck on showing triangle inequality. I don't know ...
0
votes
1answer
18 views

Is decimal part of inequality whole number?

If you have an inequality, x is "greater than -7" but "less than -5." Is -6 the only number that will satisfy this inequality, or will there be multiple solutions (e.g. -5.5, -6.5, etc)? In other ...
1
vote
1answer
37 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
1
vote
0answers
34 views

On the roots of certain functional inequality

I have a question to share, if someone can help me. Let, for each prime $n>2$ $L_{n}:[0,+\infty)\longrightarrow \mathbb{R}$ the function given by $ L_{n}(x):=\sum_{k=1}^{n-1}(-1)^{\Omega(k)}k^{x},$...
0
votes
0answers
31 views

I have a proposal on recursive functions, but I need a second voice to be sure it makes sense.

My main field is computer science. Although I am mostly familiar with computer science theory, I lack in math theory. So, to an average computer scientist, the term ‘recursive function’ will invoke ...
0
votes
0answers
14 views

Maximum Piecewise-Linear Lower-Bound

I've run into a problem, which I've formalized here: Given: $n \geq 1, m \geq 1$ fixed integers. $x_0, \cdots, x_n$ evenly spaced list of reals, with $x_0=0, x_n=1$. $y_0, \cdots, y_n$ ...
9
votes
0answers
833 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. ...
0
votes
2answers
90 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 \...
2
votes
1answer
24 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
votes
1answer
25 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: ...
1
vote
0answers
42 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 ...
9
votes
1answer
107 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
vote
1answer
44 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}} $ ...
0
votes
1answer
27 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 ...
2
votes
2answers
61 views

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I've recently been going over the mean value and intermediate value theorems, however I'm not sure where to start on this.
5
votes
1answer
60 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
vote
1answer
129 views

Numerical methods to solve nonlinear system of inequalities?

I know some methods to solve nonlinear system of equaltites: Relaxation Method, Newton method, nonlinear Jacobi method, nonlinear Seidel method. Is it exist some analogous method to solve nonlinear ...
1
vote
1answer
83 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 ...
3
votes
1answer
230 views

Why are so many inequalities and estimate in PDEs?

I want to study PDE, but when I read some PDE books, I notice that there are so many inequalities there and always lots of estimates. I really want to know why before I continue my study. I mean, ...
1
vote
0answers
21 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] \|_{\mathcal{...
2
votes
1answer
53 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
votes
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 ...
0
votes
1answer
49 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
votes
1answer
28 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
34 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
votes
0answers
17 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 $m\leq n$...
1
vote
1answer
41 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
votes
1answer
122 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 $F(n)$.
0
votes
2answers
19 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 ...
1
vote
0answers
22 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
votes
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 ...
6
votes
2answers
188 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 ...
0
votes
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 c\...
0
votes
0answers
36 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 $
0
votes
0answers
25 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
vote
2answers
52 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 $k^...
1
vote
0answers
29 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 Brunn-...