5
votes
1answer
110 views

Is this graph of reciprocal power means always convex?

Let $$ p = (p_1, \ldots, p_n) $$ be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$. Is the function ...
2
votes
0answers
67 views

proving quasi convexity of multivariable function

Given an arbitrary $(N \times N)$ square matrix ${\bf X}$ a positive definite $(M\times M)$ matrix ${\bf T}$ a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is ...
3
votes
1answer
195 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) + ...
1
vote
0answers
206 views

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, $ ...
1
vote
2answers
198 views

Construction of a convex function nondifferentiable on a countable set [closed]

Let $H$ be a countable subset of $[0,1]$. Construct a convex function $f:[0,1]\rightarrow\mathbb{R}$ such that $f$ is nondifferentiable on $H$ and differentiable in the rest.
3
votes
1answer
391 views

Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$. (Notation: in the following, the $a_k$ ...
3
votes
1answer
268 views

Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
0
votes
1answer
140 views

necessary and sufficient conditions for a function to be DC

Hi, Does anyone know the necessary and sufficient conditions for a function to be a DC-function? Definition: A function is a DC-function if and only if it can be written as a differnece of 2 convex ...
2
votes
0answers
129 views

A subclass of log-concave functions satifying a sum inequality

Suppose $f:[0,\infty)\to[0,\infty)$ is continuous and for all positive integers $m$ and real $x\geq0$, $\alpha,\beta>0$: $$ ...
6
votes
0answers
194 views

Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$. Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are ...
4
votes
2answers
624 views

gradient of convex functions

Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail? Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n ...
2
votes
0answers
173 views

Radon transform and Log-concavity

This question is related to (but different from) that of Darsh Ranjan. Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat ...
17
votes
1answer
728 views

If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a diffeomorphism, what can one conclude about the behavior of $f(x)$ at infinity?

This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, ...
2
votes
1answer
245 views

A differential inclusion relating to the slope of a convex function

This question is concerned with a possible lemma which would be very useful in one of my current research projects, but which I am currently unable to prove. The project as a whole relates to the ...
7
votes
9answers
3k views

Ways to prove an inequality

It seems that there are three basic ways to prove an inequality eg $x>0$. Show that x is a sum of squares. Use an entropy argument. (Entropy always increases) Convexity. Are there other means? ...