Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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252 views

Does this inequality hold true, in general?

Let $$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$ be the prime factorization of the positive integer $N$. Does the following inequality hold true in general? $$\prod_{i=1}^{\omega(N)}{\frac{...
2
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2answers
34 views

How does $\| \nabla u \|_{L^2}^2 \leq C \|\nabla u \|_{L^2}$ imply $\| u\|_{W^{1,2}}^2 \leq \| u\|_{W^{1,2}}$?

So I'm working on some notes and I found this inequality that I really can't make sense of it. This is what is going on: Let $u \in W_0^{1,2}$ and let $f \in L^2$ both on some $\Omega \subseteq \...
2
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2answers
50 views

For an increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$, do we always have $f(n)^{f(n)} \leq f(n^{n})$?

I have asked the question "Which is bigger, $n!^{n!}$ or $(n^{n})!$ ?", which there is an elementary proof. So now my question is: For an increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ ...
2
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3answers
64 views

If $a, b, c$ are positive and $a+b+c=1$, prove that $8abc\le\ (1-a)(1-b)(1-c)\le\frac{8}{27}$

If $a, b, c$ are positive and $a+b+c=1$, prove that $$8abc\le\ (1-a)(1-b)(1-c)\le\dfrac{8}{27}$$ I have solved $8abc\le\ (1-a)(1-b)(1-c)$ (by expanding $(1-a)(1-b)(1-c)$) but do not get how to show ...
2
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2answers
68 views

Weird norm of the functional

I found wierd (for me) example about computing norm of the functional. We have $$\varphi : \mathcal{C}([0,1])\ni f\mapsto f(1)\in\mathbb{R}$$ with the norm in $\mathcal{C}([0,1])$ given by: $$\Vert f\...
2
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1answer
40 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 ...
2
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2answers
50 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 ...
2
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1answer
34 views

How is this a bound: $|(\iota(Tx))(\varphi)| = |\varphi(Tx)| \leq \Vert \varphi \circ T\Vert \cdot \Vert x \Vert \leq \Vert \varphi \circ T \Vert$

I'm wondering how this normed composition---from this answer---works as a bound. Please read question too. $$|(\iota(Tx))(\varphi)| = |\varphi(Tx)| \leq \Vert \varphi \circ T\Vert \cdot \Vert x \...
2
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2answers
73 views

Inequality for $\sin(20°)$

Prove that $$\frac{1}{3} < \sin{20°} < \frac{7}{20}$$ Attempt $$\sin60°=3\sin20°-4\sin^{3}(20°)$$ Taking $\sin20°$=x I got the the equation as $$8x^3-6x+\sqrt{3} =0$$ But from here I am not ...
2
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1answer
65 views

RMO practice problem inequality

Let $a_n$ & $b_n$ be two sequences such that $a_0$ , $b_0$ > 0 and $a_{n+1}$ = $a_n$ + $\frac{1}{2b_n}$ & $b_{n+1}$ = $b_n$ + $\frac{1}{2a_n}$ $\forall$ n $\geq$ 0. Then prove that $$max(a_{...
2
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1answer
128 views

Find min of $P = \dfrac{1}{(a-b)^2} + \dfrac{1}{(b-c)^2} + \dfrac{1}{(c-a)^2}$

Let $a, b, c \in \mathbb{R}^+$ such that $a^2 + b^2 + c^2 = 3$. Find the minimum value of $P = \dfrac{1}{(a-b)^2} + \dfrac{1}{(b-c)^2} + \dfrac{1}{(c-a)^2}$? In fact, we have that $P \geq \dfrac{4}{...
2
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1answer
115 views

An inequality: Bound of first derivative by the sup norms

Let $f \in \cal{C}^2[-\epsilon, \epsilon]$. I want to show that $$|f'(0)|^2 \leq \frac{4}{\epsilon^2}\|f\|^2_{\infty} + 4\|f\|_{\infty}\|f''\|_{\infty}$$ Using Taylor's theorem, it is easy to deduce $...
2
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1answer
66 views

Find function such that $\sum y_i\le\sum x_i\Rightarrow\sum f(y_i)\le\sum f(x_i)$

What kind of a function $f$ must be to satisfy the following? If $\sum_{i=1}^{n} y_i \leq \sum_{j=1}^{n} x_j$, where $x_j, y_i \in [0,1],\forall i,j$ then $$ \sum_{i=1}^{n} f(y_i) \leq \sum_{j=1}^{n} ...
2
votes
1answer
117 views

Integral inequality involving derivative in integrand

I'm trying to show that \begin{align} \left|\int_{-\pi}^\pi [f(t)\cos t - f'(t) \sin t]\, dt\right| \le \sqrt{2\pi}\left(\int_{-\pi}^\pi [|f(t)|^2 + |f'(t)|^2]\, dt\right)^{1/2}. \end{align} However ...
2
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3answers
84 views

Range of $\tan^2\frac A2+\tan^2\frac B2+\tan^2 \frac C2$ with $A,B,C$ angles in a triangle

What is the range of $\tan^2\frac A2+\tan^2\frac B2+\tan^2 \frac C2$ if $A,B,C$ are angles in a triangle? For $\tan^2\frac A2+\tan^2\frac B2+\tan^2 \frac C2$, I know we have to apply the AM–GM ...
2
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2answers
152 views

Counterexample or proof for the following matrix inequality?

I would like to know whether or not (e.g., via a counter-example) the following inequality is preserved when $P$ is a non-negative (symmetric) positive definite matrix for all $i,j$: $$ \| P_i – k \ ...
2
votes
1answer
143 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 ...
2
votes
1answer
66 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
1answer
874 views

Minkowski's Inequality for $0<p<1$.

I need to prove: for non-negative functions $f,g\in L^p[0,1]$, $||f+g||_p\geq||f||_p+||g||_p$ for $0<p<1$. For $1\leq p<\infty$, the inequality is reversed and the proof is like: The cases $...
2
votes
2answers
483 views

An inequality about the gradient of a harmonic function

Let $G$ a open and connected set. Consider a function $z=2R^{-\alpha}v-v^2$ with $R$ that will be chosen suitably small, where $v$ is a harmonic function in $G$, and satisfies $$|x|^\alpha\leqslant v(...
2
votes
2answers
94 views

Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$

Good day to everyone. I am interested in the geometric intuition for the following statement: Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and $c,d \in \...
2
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1answer
61 views

Poincare type inequalities

I want to prove if following inequality holds: $$\int_0^1(f')^2\ dx\geq f^2(1)-f^2(0)$$ where $f$ is a function in $H^1([0,1])$ satisfying $\int_0^1f \ dx=0$. It is actually a one dimensional case of ...
2
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2answers
176 views

Russian MO 2004 $\sqrt{a} + \sqrt{b} + \sqrt{c} \geq ab + bc + ca$

I have a doubt on a proof included in "Secrets in Inequalities" by Pham Kim Hung. The exercise is to prove $$\sqrt{a} + \sqrt{b} + \sqrt{c} \geq ab + bc + ca$$ for a, b, c whose sum is 3. His ...
2
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1answer
46 views

Function of functions maps closed ball to itself

Let $$P: C[0,1]\to C[0,1]; P(f)(x)= 1+kxf(x)\int_0^1 \frac{f(s)ds}{x+s},$$ where $k$ is a constant, with $|k|<\frac{1}{4\ln 2}$. Let $f_1(x)\equiv 1$ on $[0,1]$. I want to prove that the set $\...
2
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1answer
401 views

Can Big-O be used to prove inequalities?

In Number Theory, the Big-O notation is defined $f=O(g)$ if there exist $C>0$ and $x_0$ such that $|f(x)|<Cg(x)$ for all $x>x_0$. So I'm just wondering if this can be used as a method to ...
2
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1answer
52 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} \...
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2answers
530 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 ...
2
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1answer
460 views

Sketch $ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $

Sketch the following $$ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $$ I have considered this geometrically and ended up thinking that the complex numbers $z$ must satisfy $$0 < \frac{Im(...
2
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1answer
444 views

Improving the inequality $x\sigma_1(x) \leq \sigma_1(x^2)$ for $x \in \mathbb{N}$

Let $\mathbb{N}$ be the set of positive integers. For $x \in \mathbb{N}$, $\sigma_1(x)$ gives the sum of the divisors of $x$. (For example, $\sigma_1(3) = 1 + 3 = 4$.) We call the ratio $I(x) = \...
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votes
2answers
70 views

Trigonometric inequality $|\sin{a_1}|+|\sin{a_2}|+…+|\sin{a_n}|+|\cos{(a_1+a_2+…+a_n)}| \ge1$ for all real $a_i$

Prove that for all real numbers $a_1,a_2,...,a_n$ the following inequality holds: $$ |\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge 1 $$
2
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1answer
122 views

Range of inner product of a sequence and its permutation

$a^n :=(a_i)_1^n$ is a finite sequence of real numbers of length $n$, where $\sum\limits_{i=1}^n a_i=0$ and $\sum\limits_{i=1}^n a_i^2=1$. Consider $s_n(a^n,\sigma):=\sum\limits_{i=1}^n a_ia_{\sigma(i)...
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2answers
234 views

Does the following inequality hold if and only if $N$ is an odd deficient number?

Let $N \in \mathbb{N}$. (That is, let $N$ be a positive integer.) This is in reference to two of my earlier questions here at MSE: Does the following inequality hold true, in general? Does this ...
2
votes
1answer
72 views

Does this inequality have any solutions for composite $n \in \mathbb{N}$?

Does this inequality have any solutions for composite $n \in \mathbb{N}$? $$\sqrt{2} < \frac{\sigma_1(n^2)}{n^2} < \frac{4n^2}{(n + 1)^2}$$ Note that $\sigma_1$ is the sum-of-divisors function....
2
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1answer
31 views

How to Calculate the norm of a matrix transform from $(R^n,\|\cdot\|_{\infty})$ to $(R^n,\|\cdot\|_{1})$

suppose we have a matrix operator $A=(a_{ij})_{n\times n}:(R^n,\|\cdot\|_{\infty})\to(R^n,\|\cdot\|_{1})$, how to calculate $\|A\|$? or if there really is an exactly closed form for $\|A\|$? if $x\in ...
2
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1answer
42 views

Functional Space Inequality for Sobolev Space and Lp Space

Let $X = C_{0}(\Omega) := \{ u \in C(\overline{\Omega})\,|\,u|_{\partial\Omega}=0\}$ and define $F : X \to X$ as Lipschitz continuous function and $F(0) = 0$. Let $\Omega\subset \mathbb{R}^{N}$ be a ...
2
votes
2answers
32 views

Deriving a particular Poincare equality by a lemma

What I want to prove is the following inequality $\Vert u \Vert_{L^2[0,1]} \leq \Vert u' \Vert_{L^2[0,1]}, \quad u \in W^{1,2}_0 [0,1], $ which I am trying to derive from $$ \int_\Omega \vert u(x+...
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1answer
68 views

An inequality for $2$-convex and concave functions

Let $I$ be an interval of the real line and $f:I\rightarrow \mathbb{R}$ a function. Define the divided difference of $f$ at the points $x_0<x_1<\cdots<x_{n+1}$ in $I$ by $$ [x_0;f]=f(x_0)\; , ...
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2answers
2k views

Jensen's inequality proof explanation

I was reading a proof of Jensen's inequality on convex functions, and I need some help understanding it. The proof is as follows: $$f(t_1x_1+...+t_nx_n) = f((1-t_n)(\frac{t_1}{1-t_n}x_1+...+\frac{t_{...
2
votes
1answer
191 views

Which matrix satisfies the following condition?

$P$ is a real (symmetric) positive definite matrix. Let $P_i$ and $P_j$ represent the $i$'th and $j$'th columns of $P$, respectively. Further, let $P_{ki}$ represent the element situated at the $k$'th ...
2
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1answer
329 views

Show that: $nf(\frac{x_1+\cdots+x_n}{n})+n\left(\frac{f(M)+f(m)}{2}-f(\frac{M+m}{2})\right) \ge f(x_1)+\cdots+f(x_n)$

I am looking for a proof of the problem as following: Let $f(x)$ is a real continuous function that is strictly convex ($f''>0$) on $[m, M]$, let $m \le x_i \le M$, for $i=1,2,\ldots,n$ then show ...
2
votes
1answer
53 views

Existence of a functional inequality

Does there exist $f=f(x)$ satisfying $f(x)\ge0$ for $x\in\mathbb{R}$, $f(x)=f(-x)$ for $x\in\mathbb{R}$ (i.e. $f$ is even), $\int_{\mathbb{R}} f(x)\,dx<\infty$, and $\int_{\mathbb{R}} x^2\,f(x)\,dx&...
2
votes
1answer
117 views

Functional equation $(n-1)^2 < f(n) f(f(n)) < n^2 +n$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$(n-1)^2 < f(n) f(f(n)) < n^2 +n$$ for all $n \in \mathbb{N}$.
2
votes
1answer
111 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: ...
2
votes
1answer
65 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 ...
2
votes
1answer
59 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 ...
2
votes
1answer
352 views

$t\in (0,1)$ and $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$. Show that if strict inequality holds for even one $t$, then it holds for all $t$.

This is a part of a solution to a problem in showing that if $f$ is continuous and satisfies the condition $f([x+y]/2)\lt [f(x)+f(y)]/2$, then $f$ is convex. Let $t\in (0,1)$. We have the weak ...
2
votes
1answer
319 views

Proof of a Caccioppoli inequality for non-symmetric operators

I am reading the book Elliptic Partial Differential Equation by Fanghua Lin and I got stuck at the lemma 1.36 ( the Caccioppoli inequality). The conditions of this lemma are: $u\in C^1(B_1)$ ($B_1$ is ...
2
votes
1answer
79 views

Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=1+|a_{1}|+\cdots +|a_{n-1}|+|a_{n}...
2
votes
1answer
139 views

A question on (odd) perfect numbers

(Note: This has been cross-posted to MO.) Let $\sigma(x)$ be the (classical) sum of the divisors of $x$. A number $N \in \mathbb{N}$ is called perfect if $\sigma(N)=2N$. An even perfect number $U$ ...
2
votes
2answers
52 views

$ \int_{\mathbb R^d}\int_{\mathbb R^d}b(x,y)\,f(x)\,f(y)dx\,dy \leq ||b_+||_{L^2(\mathbb R^d\times \mathbb R^d)}||f||_{L^2(\mathbb R^d)}^2 $

We are given the following, $$ b:\mathbb R^d \times \mathbb R^d \rightarrow \mathbb R,\;\; f:\mathbb R^d\rightarrow \mathbb R $$ and $$ f\in L^2(\mathbb R^d)\; ,\;b\in L^2(\mathbb R^d\times \mathbb R^...