Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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Conditional Berry–Esseen Inequality

From Wikipedia, the Berry-Essen Inequality is given by $$ \sup_{x\in \mathbb{R}}|\text{Pr}(X\leq z)-\Phi(z)|\leq \frac{C E(|X|^3)}{E(X^2)\sqrt{n}}. $$ From a statistics paper that I am reading, I am ...
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Minkowski inequality and infinite sum

I really need your help, we have $u_{2k-1}(t)=\phi_{2k-1}\cos(\lambda_{k}t)+\frac{1}{\lambda_k}\psi_{2k-1}\sin(\lambda_{k}t)+\frac{1}{\lambda_k}\int_{0}^{t}F_{2k-1}(\tau;u)\sin\lambda_{k}(t-\tau)d\tau$...
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+100

Find the relationship among adjusted 3-pointer (not parameter, but formula like Pythagenport, Pythagenpat), goals scored and goals allowed in football

I was seeking a formula to relate goals scored and goals allowed to points in association football. In association football, three points are awarded to the team winning a match, with no points ...
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I am trying to find an uncommonly cited paper called: "Morrey inequality for Sobolev functions with mean of zero(?)" (I am not sure about given name.)

I am trying to find an uncommonly cited paper called: "Morrey inequality for Sobolev functions with mean of zero(?)" (Or "mean is zero", not so sure about given name.) I can ...
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3 votes
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Prove the uniqueness of solution of $\mathbb{E}\left [ X\left ( s \right )X\left ( t \right ) \right ]=\lambda\min\left ( s, t \right )+\lambda^{2}st$

Given a Poisson process, to check the stationarity, I tried to get $\mathbb{E}\left [ X\left ( s \right )X\left ( t \right ) \right ]$ then adopt $$\mathbb{C}_{X}\left ( s, t \right )= \mathbb{E}\left ...
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1 answer
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Prove point wise inequality

How to prove: $$|x+y| \cdot \mathbb{I}_{\{|x+y| \ge 2a \}} \le 2\big( |x| \cdot \mathbb{I}_{\{ |x| \ge a\}} + |y| \cdot \mathbb{I}_{\{|y| \ge a\}} \big) $$ My attempt, I know that $ \{ |x + y| \ge ...
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Vector inequality regarding p-Laplace euations

In chapter~12, (p.99) of this note it is written that "In the case $p \geq 2$, the inequality $$|b|^p \geq |a|^p + p<|a|^{p−2}a, b − a> + C(p)|b − a|^p$$ holds with a constant $C(p) > 0$....
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Functional inequality with integral function [closed]

Given a function $f:[0;1]\to[0;1]$ such that $f(x)\leq2\int_0^x f(t)dt$, prove that $f(x)=0$ $ \forall x\in [0;1]$. I've observed that the function has to be concave down in his domain and that $...
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Modified Gronwall lemma for $f' \le a + b f^\alpha$

Let $f\colon [0,\infty)\to [0,\infty)$ be a smooth function such that $f(0)=0$ and \begin{equation} f'(t) \le a(t)+b(t) \bigl( f(t) \bigr)^\alpha \label{eq:1} \tag{1} \end{equation} for all $t\ge 0$, ...
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Under what conditions, I can claim that the given inequalities $(1)$ and $(2)$, always, have an intersection part?

Suppose that I have these two inequalities over the real domain $x>0$ and the parameter $a>0$; here, $f(x)$ and $g(x)$ are real functions $$ -|g(x)|+1+a \; \leq f(x) \leq \; |g(x)|+1+a ...
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Analytic proof: $\frac{2}{t^2}-\frac{\frac{1}{2}\cos(\frac{t}{2})}{\sin^2(\frac{t}{2})} \geq 0$ for $t \in [0,\pi]$?

I have a relatively simple question: How can one show that $\frac{2}{t^2}-\frac{\frac{1}{2}\cos(\frac{t}{2})}{\sin^2(\frac{t}{2})} \geq 0$ $\forall t \in [0,\pi]$ or equivalently why $\frac{1}{\sin(\...
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Optimal Poincare constant (with constraints)

Given any integrable function $f\, \colon \mathbb{R}_+ \to \mathbb R$, it is relative easy to show that the best constant for which the following Poincar'e-type inequality $$ \int_{\mathbb{R}_+} f^2\,\...
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Functional inequality $\sqrt{f(x+y)}\leqslant\sqrt{f(x)}+\sqrt{f(y)}$

The following problem was posted on another forum but I wonder about the validity of its claim: Let $f:\mathbb{R}\to [0, +\infty)$ satisfy the functional equation $$f\left(\frac{x+y}{2}\right) +f\left(...
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Function composition with logsumexp

Given the function $g(\vec{x})=\log \sum_i \exp(x_i)$, I am curious which functions $f$ satisfy $f(g(\vec{x})) \geq g(\vec{f(x)})$? Let's let $x \in \mathbb{R}^N$ to be concrete. Useful properties of $...
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How to prove the following superquadratic property

Let $\Omega \subset \mathbb{R}^N$ a bounded smooth domain and let $f:\Omega \times \mathbb{R} \to \mathbb{R}$ an Carathéodory function such that $(f_1): |f(x,s)| \leq c|s|^{\sigma} + d$ for all $x \in ...
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2 answers
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If $f(x + 19) \leq f(x) + 19$ and $f(x + 94) \geq f(x) + 94$ how to prove that $f(x +1) = f(x) + 1$

We know that $f(x + 19) \leq f(x) + 19$ and $f(x + 94) \geq f(x) + 94$ $\forall x \in \mathbb{R} $ And I have to prove that $f(x + 1) = f(x) + 1$. I know that this question has been asked already,...
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1 vote
0 answers
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Inequality searching for an upper-bound

Suppose that there exists a constant $C$ such that the following relation holds for all $G$: \begin{equation*} \vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert \end{equation*} Suppose that ...
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Sufficient Conditions for $f(x_1)g(y_1)+f(x_2)g(y_2)\le f(x_1+x_2)g(x_1+x_2)$

Let $f$ and $g$ be two strictly increasing functions such that $% f:[0,1]\rightarrow \mathbb{R}_{+}$ (with $f(0)=0$) and $g:[2,\infty )\rightarrow \mathbb{R}_{+}$. Can anyone come up with sufficient ...
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How to show $[-3, 1)$ with an Absolute value inequality?

I know I can write the closed form of this interval using this formula: $|x - \text{center}| ≤ \text{radius}$ if it was $[-3, 1]$ then number $-1$ would be the center and the radius is the distance ...
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2 votes
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Show the sequence uniformly converge to null function. [duplicate]

Problem: If $f_0$ is a continuous function in $[0,a]$, $a>0$, show that the sequence $\{f_k\}$ (on that same interval) defined the recursive relatio n$f_k(x)=\displaystyle\int_0^x f_{k-1}(t)dt$ ...
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0 answers
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System of Differential inequalities: Can this lemma be generalized?

The following lemma is known Lemma. Let $p, q, \varepsilon > 0$ satisfy $pq > 1$, and let $0 < T \leq \infty $. Assume that $0 \leq y, z \in C^1(0, T)$, $(y, z) \neq (0, 0)$, and that $(y, z)$...
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Norm $\int_0^1|f^{(n)}(t)|\text{d}t$ for $f$ with $f(0)=f\left(\frac 1{n-1}\right)=...=f(1)=0$

Let $n\geq 2$ an integer and $E=\{f\in\mathscr{C}^n([0,1],\mathbb{R}),\; f(0)=f\left(\frac 1{n-1}\right)=f\left(\frac 2{n-1}\right)=...=f(1)=0\}$, equipped with the norm $\|f\|=\displaystyle\int_0^1|f^...
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1 answer
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An inequality involving the Laplacian and the norm of the gradient

For any given probability density function $p$ (with finite second moments) on $\mathbb{R}^d$ I want to show that the following integral is bigger than $\textit{(or equal to)}$ zero $$ \int_{\mathbb{R}...
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1 vote
1 answer
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Non-trivial solution of $x = r\sin\pi x$

To compute the fixed points of a sine map, I need to solve $$x = r\sin\pi x$$. The question asks me to find the value of r for which the non-trivial fixed point (a second solution of the above ...
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2 votes
1 answer
24 views

Lower bound on the difference of perturbed functions

Let $f,g:\mathbb{R}^n \to \mathbb{R}$ be two continuous functions where $f(\tilde{x}) < g(\tilde{x})$ for some $\tilde{x} \in \mathbb{R}^n$. According to this post, there exist a neighborhood $B(\...
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Find the smallest value at x=25

A function g: is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers m and n. Let f be an adjective function such that the value of $f(1)+f(2)+f(3)+...+f(30)$ is minimized. Find ...
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6 votes
2 answers
196 views

If $g(x+y)\le2\cdot\max\{g(x),g(y)\}$ and $g(xy)=g(x)g(y)$, show that $g(x+y)\le g(x)+g(y)$.

Let $g$ be a multiplicative function from $\mathbb R^+$ to $\mathbb R^+$. If $g(x+y)\le2\cdot\max\{g(x),g(y)\}$ for all positive reals $x,y$, prove that $g$ is subadditive. For the function $g$, ...
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1 answer
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Deducing Inequality from an Equation

I was interested in Solution of this Non-Homogenous Recurrence Relation $f(n)=f(n-1) + f(n-3) + 1$ The Base conditions are: $f(0)=1$ $f(1)=2$ $f(2)=3$ Since, the equation is non-homogenous, it will ...
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1 vote
0 answers
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2-Normed Space Inequality

If $X$ is a vector space over field $\mathbb{R}$, then a real-valued, non negative function $ \Vert \cdot , \cdot \Vert$: $X \times X \rightarrow \mathbb{R}$ on $X^2$ is said to be $2$-normed on $X$ ...
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Prove Holder's inequality with $0<p<r<q<\infty,\ (1-\theta)/p+\theta/q=1/r$ and $0<\theta<1 \implies \|h\|_r \leq \|h\|_p^{1-\theta} \|h\|_q^{\theta}$

I'm trying to prove Holder's inequality using that in a measure space $(X,\mu)$ for every $h:X\to \mathbb{C}$ measurable, $0<\theta<1$ and $0<p<r<q<\infty$, with $$\frac{1}{p}(1-\...
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2 votes
0 answers
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If $f'(x) − f'(y) ≤ 3|x − y|$ then show that $|f(x)−f(y)-f'(y)(x-y)|≤\frac{3}{2}(x-y)^2$

Let $f : \mathbb R \rightarrow \mathbb R $ be a twice differentiable function. Suppose that for all $x, y \in \mathbb R$ the function $f$ satisfies $f'(x) − f'(y) ≤ 3|x − y|$ then show that for all $x$...
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A problem concerning a divergent function on $[0, 1]$

This problem was posted on another forum and was given at the 1992 Miklós Schweitzer Competition. This competition is known for its very difficult problems and this one seems no exception. I also can'...
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How to prove that $f(x)$ is much less than $g(x)$?

In order to prove that $f(x) \ll g(x)$, can I prove that $\lim_{x \rightarrow \infty}\dfrac{f(x)}{g(x)}=0$? Just to clarify my question, I need to prove that $\dfrac{x}{\log(x)-\log(2^t)} \ll \dfrac{x}...
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0 answers
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Norm of laplacian vs norm of gradient

I'm wondering if there exist a constant $C>0$ such that $\|\Delta c\|_{L^2}\leq C\|\nabla c\|_{L^2}$ where $c:\Omega\subseteq\mathbb{R}^2\to\mathbb{R}$ is a $C^{\infty}$ function with $\Omega$ a ...
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1 vote
1 answer
77 views

If $f'(x)≥0$, $f''(x)≤0$ then prove that $f(x)f^{-1}(x)-x^2≤0$

If $f:[0,1] \rightarrow [0,1]$ such that $f'(x)≥0$, $f''(x)≤0$ then prove that $$f(x)\cdot f^{-1}(x)-x^2≤0$$ I was able to form a graphical solution but it doesn't feel rigorous enough to me, can ...
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Understanding Ch.8 Problem 41 on pg. 179 in Royden (4ed)

I have some problems on understanding Problem 41 in Ch.8 (pg. 179) in Royden, 4ed. The corrected statement (according to this and this) of the problem is as follows: Let $E$ be a measurable set and $...
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1 vote
1 answer
51 views

Is there a simple proof for $\sup_{|x|\le \delta} \frac{|e^{ax} - 1|}{|x|} \le \frac{e^{|a|\delta}}{\delta}$?

Suppose that $a \in \mathbb{R}$. There's an inequality $$\sup_{|x|\le \delta} \frac{|e^{ax} - 1|}{|x|} \le \frac{e^{|a|\delta}}{\delta},$$ see "Testing statistical hypotheses" by E. ...
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1 vote
1 answer
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Why do we consider the lower bound in this Inequality?

Given that $a,b,c,d,e$ are real numbers such that $a+b+c+d+e=8$, $a^2+b^2+c^2+d^2+e^2=16$. Determine the maximum value of $e$. Let $t$ be a real number. Now consider the expression $$ \sum_{cyc} (t-a)...
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1 vote
2 answers
70 views

Let $f(x)$ satisfies $f(x)\ge|x|^{\alpha}, \frac12\lt\alpha\lt1$ and $f(x)f(2x)\le|x|$ then find $\lim_{x\to0}f(x)$

Let $f(x)$ satisfies $f(x)\ge|x|^{\alpha}, \frac12\lt\alpha\lt1$ and $f(x)f(2x)\le|x|$ for all $x$ in the deleted neighbourhood of zero then $\lim_{x\to0}f(x)=1/\alpha/0/$Does not exist? $f(x)\ge|x|^\...
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4 votes
0 answers
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Proper name for a "polytope" when defined by linear functionals

Is there a name for the set of functions $f:[0,1]\to\mathbb{R}$ which satisfy a finite set of equations $\int_0^1 f(x)w_i(x)dx\geq c_i$ for $i=1,\ldots,n$? If $f$ had a discrete domain and the ...
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2 votes
2 answers
157 views

Let $f(x)=ax^3+bx^2+cx+5$. If $|f(x)|\le|e^x-e^2|$ for all $x\ge0$ and if the maximum value of $|12a+4b+c|$ is $m$, then find $[m]$

Let $f(x)=ax^3+bx^2+cx+5$. If $|f(x)|\le|e^x-e^2|$ for all $x\ge0$ and if the maximum value of $|12a+4b+c|$ is $m$, then find $[m]$ (where $[.]$ represents the greatest integer function.) I first ...
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1 vote
2 answers
73 views

If $| f(p + q) – f(q)| \le \dfrac pq$ for all $p$ and $q\in \mathbb Q$ & $q\ne 0$, show that $\sum_{i=1}^k| f(2^k ) – f(2^i ) |\le \dfrac{k(k – 1)}2$

The following question is taken from the practice set of JEE exam. If $| f(p + q) – f(q)| \le \dfrac pq$ for all $p$ and $q \in \mathbb Q$ & $q \ne 0$, show that $\sum_{i=1}^k| f(2^k ) – f(2^i ) ...
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  • 5,785
4 votes
1 answer
89 views

Szegő's inequality in approximation theory

Let $T_n$ be the space of all real-valued trigonometric polynomials on $[0,1)$ of degree at most $n\in \mathbb{N}_0$ and $p\in T_n.$ Then $$ \left|p'(x)\right|\leq 2\pi n\sqrt{\|p\|^2_\infty -|p(x)|^...
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-1 votes
1 answer
100 views

For help an inequality that for large $n$, whether $\frac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$ for some $\beta>0$

Can anyone help me to prove that for large $n$ whether $$\dfrac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$$ for some $\beta>0$ where and $\big[x\big]$ denotes the ...
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4 votes
1 answer
117 views

Intuition behind the entropy functional for curvature bounds

What is the intuition behind defining in metric measure spaces a Ricci curvature bound via convexity or concavity of the entropy functional? I know that this is made because in Riemannian manifolds ...
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1 vote
1 answer
85 views

Solve $n!\gt\binom{2n}{n}$

Let $f(n)=n!$ and $g(n)=\binom{2n}{n}$. Obviously $f(7)\gt g(7)$ and $f(6)\lt g(6)$. The functions are increasing on $n\in (1,\infty)$, $n$ being an integer. But this is not enough to prove $$n!\gt\...
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-1 votes
1 answer
105 views

Ask for help a inequality problem, does $\sum\limits_{i=0}^{k-2}\log_2\left(\frac{n-i}{k-i-1}\right)>c\cdot n$ [closed]

Can anyone help me to give me a detailed proof (or disproof) of the following $\sum\limits_{i=0}^{k-2}\log_2\left(\frac{n-i}{k-i-1}\right)>c\cdot n$ for some constant $c>0$, where $k=\Big[\frac{...
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5 votes
1 answer
200 views

Spectral gap and Poincaré inequality

Consider the PDE $$\partial_t u = L u$$ where $L = \Delta + \nabla V \cdot \nabla $ is a self-adjoint operator. I read that if $L$ has a spectral gap $\lambda > 0$ then "[convergence of the ...
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  • 1,504
0 votes
1 answer
27 views

Solution for a particular inequality

I'm following up on this question on MathOverflow. By taking some particular functions, $a(.),\,b(.)$ and $K(.,.)$. This problem is equivalent to the following inequality: $$4\left \| r \right \|_{\...
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  • 670
0 votes
0 answers
89 views

Domination of derivatives by an involution

Let $h:(0,1) \to (-\infty,0)$ be a $C^1$ function, with $h'>0$. I am looking for sufficient conditions on $h$ that imply the existence of a $C^1$ decreasing* function $g:(0,1) \to (0,1)$ with $g'&...
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