Questions tagged [convexity-inequality]
This is useful method for an estimation convex or concave functions on a closed segment.
154 questions
-3 votes
2 answers
137 views
Show that $a^2+b^2+c^2-ab-bc-ca \leq 4$ if $|a|,|b|,|c|\leq 1$.
The problem Let $a,b,c\in \mathbb R$ such that $|a|,|b|,|c|\leq 1$. Show that $a^2+b^2+c^2-ab-bc-ca \leq 4$ My idea For $a\geq b\geq c$ we obtain: $$(a-b)(b-c)\geq0,$$ which gives $$ab+bc-b^2\geq ac$$ ...
- 1,199
0 votes
1 answer
67 views
The epigraph of continuous function on $[-1,1]$ is closed, but why is its convex hull also closed? [duplicate]
I am reading a paper where the author defines a function, takes its epigraph, then takes the convex hull closure of the epigraph to make it equal to the epigraph of the biconjugate of that function. ...
- 9
0 votes
1 answer
66 views
Is the outer product of a vector with itself a convex function?
Let $v$ and $w$ be any two vectors in $\mathbb{R}^N$. Given any $t\in[0,1]$, is it true that $$ [ (1-t)v + tw ] \otimes [(1-t)v + tw ] \le (1-t) v\otimes v + t w \otimes w? $$ The inequality above is ...
- 1,204
1 vote
1 answer
101 views
convexity of $x^{1\over\sqrt{x}}=y^{1\over\sqrt{y}}$
Suppose we have a function $y=f(x) (x>1)$. For all $x\ne e^2$ greater than $1$, $x^\frac{1}{\sqrt{x}}=f(x)^\frac{1}{\sqrt{f(x)}}$ and $f(x)\ne x$. Also $f(e^2)=e^2$. I found two asymptotes, $y=1$ ...
- 23
0 votes
0 answers
38 views
For what $t$ does $t$-convexity imply $\frac{1}{2}$-convexity?
Let's call a real-real function $f$ $t$-convex if $f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$ for all $x,y\in\mathbb{R}$. My question is for what values of $t$ does $t$-convexity imply $\frac{1}{2}$-convexity. ...
- 189
1 vote
3 answers
168 views
How to prove $|a+b|^p\leq|a|^p+|b|^p$ for $0\leq p\leq 1$ from concavity of $x\mapsto x^p$ for $x\geq 0$
In this answer, it is shown by elementary means that $|a+b|^p\leq|a|^p+|b|^p$ for $0\leq p\leq 1$. Can we show the same elementary fact by using concavity of $\phi(x)=x^p$ for $0\leq p\leq 1$, $x\geq ...
- 1,283
0 votes
1 answer
57 views
Using Convexity to Compare Functions
I've been given the question: Show $\frac{e^a + e^b}{2}$ > $e^\frac{a+b}{2}$. The solution involves checking the convexity of both sides of this inequality, but I don't understand the intuition ...
- 243
1 vote
1 answer
122 views
How to prove this simple inequality based on convexity of $e^{x}$?
Suppose $\theta > 0$ and $x>0$. I would like to show that $$ e^{\theta(x+1)} - e^{\theta x} - \frac{ e^{\theta x}-1}{x} \geq \frac{e^{\theta x}-1}{x} - (1-e^{-\theta}) $$ Another way to put it: ...
- 121
0 votes
1 answer
60 views
A question on strict convexity vs strong convexity
Is it possible to build a function $f: [0, +\infty)$ such that $f, f', f'' > 0$ and $x \mapsto x/ f(x)$ is strictly increasing? Of course, if $f, f'>0$ and $f''> \epsilon > 0$, this is not ...
- 419
1 vote
0 answers
111 views
Prove $f(x)=\log\left(\frac{1-x^a}{1-x}\right)$ is convex
Prove that $f(x)=\log\left(\frac{1-x^a}{1-x}\right)$, $x\in (0,1)$ is convex for $a\geq 5$. I've tried with the classical characterization of convex functions: $$f(\theta x + (1-\theta)y) \leq \theta ...
0 votes
1 answer
119 views
Choice of a constant in Lieb and Loss' text on Analysis
I am currently reading Lieb-Loss' book on Analysis. In the proof of Theorem 1.9 (Brézis-Lieb Lemma), whose statement is not relevant here, they use the following statement: Let $p \in (0,\infty)$. For ...
- 142
0 votes
1 answer
77 views
For a fixed positive definite $A$ anb vector $x$, is $B\mapsto x^T B^{1/2} A B^{1/2} x$ always concave?
Let $x\in R^d$ and $A\in R^{d\times d}$ positive definite. Is the map $$ B \mapsto x^T B^{1/2} A B^{1/2} x $$ always concave? One known result that gives a little hope is the Lieb inequality (cf. ...
- 2,257
2 votes
1 answer
115 views
Proving the Increase of $(a+\delta)_m - (a)_m$ Given Decrease of $\frac{(a+\delta)_m}{(a)_m}$.
The question I'm struggling with is as follows: Let $a>0$ and $\delta > 0$ (fixed). Suppose $ a \mapsto \frac{(a+\delta)_{m}}{(a)_{m}}$ is decreasing then prove that $ a \mapsto (a+\delta)_{m} ...
user745447
0 votes
1 answer
74 views
A new type of convexity related to the exponential function
I am new to the study of (undergraduate) convexity and I have recently come across a new definition which generalizes to the classic concept of convex function and is the following. Let be $I\subset \...
0 votes
1 answer
80 views
Proving the Preposition: Log-Convexity of a Function and the Monotonicity of Ratios
While reading a research paper on log convexity, I encountered a preposition (which is my question). I tried to prove it. I'm not getting any idea how to proceed. The statement is as follows: Suppose ...
user745447