All Questions
Tagged with convexity inequalities
41 questions
0
votes
2
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244
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Decreasing magnitude of spherical centroid
Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
CommunityBot
- 1
4
votes
1
answer
320
views
Sub-Gaussian random variables and convex ordering
Suppose that $X$ is a $1$-sub-Gaussian real-valued random variable, i.e. for all $t \in \mathbf{R}$, it holds that $\log \mathbf{E} \exp \left( t X \right) \leqslant \frac{1}{2} t^2 $.
Does there ...
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2
votes
0
answers
164
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Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
- 873
7
votes
3
answers
986
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Mixtures of log-convex functions are log-convex: a reference
A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
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11
votes
1
answer
1k
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A square root inequality for symmetric matrices?
In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
- 12.9k
2
votes
2
answers
189
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Elementary convexity example
I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...
- 128k
5
votes
1
answer
151
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On existence of a concave function
Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...
- 10.6k
1
vote
1
answer
225
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Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$
Working with Slater's inequality (a companion of Jensen's inequality) I found this statement:
Let $f(x)$ be a continuous, twice differentiable function, convex or concave and non constant on $(0,\...
- 6,367
2
votes
1
answer
174
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Gradient of a convex function on $\mathbb{R}^d$, maximum on hypercubes bounded by values in corners?
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex.
For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that
$$
(f'...
- 128k
4
votes
2
answers
261
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A convexity question
Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds
$$ \frac{\partial^2}{\partial x_1^2}u <0 $$
...
- 109k
3
votes
1
answer
241
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$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $
I have noticed experimentally that the following question has a positive answer.
Is it true that for all even and convex functions $f$, $g$:
$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
- 128k
2
votes
2
answers
144
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Convexity of the exponential of the negative Renyi entropy
I would like to try my luck here for the following question after failing to elicit an answer to it on math.stackexchange.com.
For $r\ge -1$, the exponential of the negative Renyi entropy is defined ...
- 2,239
19
votes
4
answers
3k
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Strange result about convexity
$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$.
Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ?
Source: AoPS
- 2,222
1
vote
0
answers
115
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How to prove the inequality $\ln\frac{1+e^{-y}}{1+e^{-x}}+\frac{1}{1+e^x}(y-x)\geq 0$?
I am trying to prove that $\ell(\beta) = \sum_{i=1}^n \left (-y_i \beta^{\top}x_i + \ln \left (1 + e^{\beta^{\top}x_i }\right )\right )$ is a convex function. I follow the following steps:
Let $\...
- 109k
2
votes
1
answer
188
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When is a continuous subadditive function (0,1]-superhomogeneous
Continuous version of this Superhomogeneity of subadditive functions
Let $f$ be a continuous function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(...
- 128k
2
votes
1
answer
109
views
Superhomogeneity of subadditive functions
Let $f$ be a function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(0) = 0$. Suppose $f$ is subadditive, i.e. $f(x_1+y_1, \dots,
x_n+y_n) \leq f(...
- 128k
0
votes
1
answer
81
views
Planar function inequality on parallelograms
Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ which is convex and satisfies $\frac{\partial{f}^2 }{\partial{x}\partial{y}} \leq 0$. The last condition is ...
2
votes
2
answers
214
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A question about asymptotic affinity and strict convexity with unbounded means
Let $F:[0,\infty) \to [0,\infty)$ be a $C^1$ strictly convex function.
Let $\lambda_n \in [0,1],a_n\le c<b_n \in [0,\infty)$ satisfy
$$ \lambda_n a_n +(1-\lambda_n)b_n=c_n \tag{1}$$ and assume that
...
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4
votes
1
answer
161
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Does strict convexity plus asymptotic affinity imply bounded mean?
I am not sure if this is exactly research-level, but I am struggling to find a proof for the following claim:
Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function.
Let $\lambda_n \in [...
- 109k
1
vote
0
answers
151
views
Log-concavity inequality
Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$
$$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{...
- 3,209
1
vote
1
answer
484
views
Convexity at a point and Jensen inequality
I am looking for a reference for the following claim:
Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.
Suppose that "$\phi$ is convex at $c$". ...
- 128k
0
votes
0
answers
81
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Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article
To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question.
We consider the ...
4
votes
1
answer
256
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Is there a non-convex function with non-decreasing average rate of change?
$\newcommand{\R}{\mathbb R}$
Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...
- 54.2k
0
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1
answer
339
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Expectations, double integrals and Jensen's inequality
$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and
$c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and
$v$ be $[x,y]$....
- 128k
4
votes
1
answer
1k
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Generalizing inequality relating Euclidean distance & Frobenius norm to Bregman divergences such as relative entropy & von Neumann divergence
Motivation- A Special Case
Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...
- 13.4k
3
votes
1
answer
444
views
Another diameter-perimeter-area inequality
Recently I learnt that $$\DeclareMathOperator{\diam}{diam}\DeclareMathOperator{\per}{per}\DeclareMathOperator{\area}{area}
\inf\frac{\diam(C)(\per(C)-2\diam(C))}{\area(C)}=0$$ where the infimum is ...
- 4,713
0
votes
1
answer
113
views
Inequality involving product-of-minus vs minus-of-product for positive integers
I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows:
Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...
- 63.9k
4
votes
1
answer
269
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An inequality of T. Carleman
I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $...
- 4,034
7
votes
1
answer
716
views
A question on the Mahler conjecture
In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and
$$
K^* := \{y \in \mathbb{R}^n : \langle y, x \...
- 4,713
5
votes
1
answer
477
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An inequality involving a sum of power terms
I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows.
Consider a positive ...
- 128k
20
votes
3
answers
1k
views
mixing convex and concave for convexity
Let $n\in\mathbb{N}$ and $0<x<1$ be a real number. Is the following a convex function of $x$?
$$G_n(x)=\log\left(\frac{(1+x^{4n+1})(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\...
- 61.9k
44
votes
7
answers
4k
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The missing link: an inequality
I've been working on a project and proved a few relevant results, but got stuck on one tricky problem:
Conjecture. If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then
$$F_n(x)=\...
- 128k
3
votes
1
answer
249
views
Sufficient conditions for weak majorization
Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive integer $p$,
$$
\sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p.
$$
Question: Is ...
CommunityBot
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8
votes
1
answer
926
views
Generalization of Popoviciu's inequality
Popoviciu's inequality states that for convex $f$ and numbers $x_1,x_2,x_3$, we have
$f(x_1)+f(x_2) + f(x_3) + 3\cdot f(\frac{x_1+x_2+x_3}3) \geq 2\cdot f(\frac{x_1+x_2}2)+2\cdot f(\frac{x_1+x_3}2)+2\...
CommunityBot
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10
votes
2
answers
496
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Inequalities for averaging over partially ordered sets
Let's start from a classical inequality:
If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then
$(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$.
It can be written also in ...
CommunityBot
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5
votes
1
answer
921
views
About generalized Minkowski inequality
For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + f^{-1}\left(\sum\...
- 28.6k
1
vote
0
answers
291
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Proof of an inequality for a linear combination of three trigonometric functions
Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$
where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ 0<\alpha&...
- 121
4
votes
2
answers
524
views
a diameter-perimeter-area inequality for convex figures
Is the following inequality known? I believe it's true, but I could find no reference.
For any convex body $C$ in the plane
we have
$$\left(4-\frac{8}{\pi}\right)area(C)\leq
> diam(C)(per(C)...
- 27.5k
0
votes
1
answer
264
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Convex functions: bounding the difference
Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x_1, x_2, \ldots, x_n \in R^d$ such that
$$\sum_{i=1}^n (x_i - x') = x - x'.$$
Is it ...
- 54.2k
0
votes
1
answer
269
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Is there relationship between $f\left({\sum_i(\mathbf{v}_i^\top\mathbf{x})^2\lambda_i},\sum_j{(\mathbf{u}_j^\top\mathbf{x})^2\theta_j}\right)$ and $\sum_i\sum_j{(\mathbf{v}_i^\top\mathbf{u}_j)^2f(\lambda_i,\theta_j)}$ if $f$ is jointly convex?
Hello, everyone!
As we know that by Jensen's inequality, for jointly convex function $f$ and $\sum_ix_i^2=1$, we have
$$f(\sum_i{x_i^2\lambda_i},\sum_i{x_i^2\theta_i)}\leq\sum_i{x_i^2f(\lambda_i,\...
- 607
1
vote
0
answers
285
views
Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?
Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and $\mathbf{p}=[p_1\;\ldots\;p_m]^\top,\mathbf{q}=[q_1\;\ldots\;...
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