The functional-inequalities tag has no wiki summary.
13
votes
11answers
972 views
If $f(x)\leq f(f(x))$ for all $x$, is $x\leq f(x)$?
If I have $f(x)\leq f(f(x))$ for all real $x$, can I deduce $x\leq f(x)$?
Thank you.
9
votes
1answer
314 views
Prove that $\int_0^1|f''(x)|dx\ge4.$
Let $f$ be a $C^2$ function on $[0,1]$. $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that
$\int_0^1|f''(x)|dx\ge4.$
Also determine all possible $f$ when equality occurs.
5
votes
1answer
56 views
An estimate of $C^2$
Let $u$ be a function on a domain $\Omega\subset R^n$, and $D^2u=\left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right)_{n\times n}$ be the Hessian of $u$. If $D^2u$ is positively definite and ...
3
votes
0answers
46 views
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 there are $x_1, x_2, \ldots, x_n \in R^d$ such that
$$\sum_{i=1}^n (x_i - x') = x - x'.$$
...
2
votes
2answers
200 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
$$|x|^\alpha\leqslant ...
2
votes
2answers
55 views
Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$
Good day to everyone. I am interested in the geometric intuition for the following statement:
Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and $c,d \in ...
2
votes
1answer
128 views
Inequality for Gamma functions
Let $k, n ,m \in N$ and such that $0\leq k \leq n \leq m$. When the following ineuality is true?
$$
...
2
votes
0answers
146 views
An application of Poincare inequality [solved]
I am woking on Evans PDE problem 5.10. #15: Fix $\alpha>0$ and let $U=B^0(0,1)\subset \mathbb{R}^n$. Show there exists a constant $C$ depending only on $n$ and $\alpha$ such that
$$
\int_U u^2 dx ...
1
vote
1answer
108 views
Show that $\int_0^1f(x)^3dx\le\left(\int_0^1f(x)dx\right)^2$ [duplicate]
Possible Duplicate:
Prove that $\int_0^x f^3 \le \left(\int_0^x f\right)^2$
Let $f$ be a differentiable function on $[0,1]$. $f(0)=0$ and $1\ge f'(x)\ge0$. Show that
...
1
vote
0answers
39 views
Increasing rearrangement and Hardy-Littlewood inequality
Don't know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it.
As you can see in Leoni (or in Lieb & Loss), the decreasing and the ...
1
vote
0answers
50 views
inequality for series
Let $j \in Z_+$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Let $X(i)=|a^{(2i)}_j|j!$. Verify that $X(i)\leq X(1)$ for ...
1
vote
0answers
41 views
Are there (known) bounds to the following arithmetic / number-theoretic expression?
I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know):
Are there (known) ...
1
vote
0answers
46 views
Inequality, with quotient substitution
I do not know how to prove this inequality: Suppose that $x_i>0$ and $x_1\cdot ...\cdot x_n=1$, show that $$\frac{1}{1+x_1+x_1x_2}+...+\frac{1}{1+x_n+x_nx_1}>1$$
The hint is to use quotient ...
1
vote
0answers
75 views
Harnack Inequality…
Consider the eigenfunction $\varphi_R>0$
$$L\varphi_R=\lambda_R\varphi_R, \ \ \ in \ \ B_R,$$
and
$$\varphi_R=0, \ \ \ in \ \ \partial B_R,$$
where $L$ is a elliptic operator and $\lambda_R$ is the ...
1
vote
0answers
42 views
Uniform Poincaré-Wirtinger constant for diffeomorphic domains?
Consider a bounded and connected open set (with regular boundary) $\Omega\subset\mathbb{R}^d$ and a family of $\mathscr{C}^1$-diffeomorphisms $(\varphi_t)_{t\in[0,\alpha]}$ all in ...
1
vote
0answers
165 views
The geometry of functions $\mathbb{R}^2\rightarrow \mathbb{R}$ that satisfy the norm axioms
What are constraints on the "looks" of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, if $f$ satisfies $$f(x_1+x_2)\leq f(x_1) + f(x_2), \ \quad x_1,x_2\in \mathbb{R}^2 \quad \quad (1)$$i.e. the ...
0
votes
2answers
80 views
Counting $2010^2$-tuples
Here's a problem i invented myself, but i'm not sure about my solution. I'll show it later, so people can enjoy trying to find one:
Consider the function $f:\mathbb{R}^2\to\{1,2,...,2012\}$ that ...
0
votes
1answer
36 views
How to use the Mean Value Theorem to find the “Contraction Constant”
Show that the contraction $T(x)= (1+x)^{1/3} $ on the interval $I=[1,2]$ satisfies the definition of a contraction.
It's not just this problem-- on this site and others explanations will say "the ...
0
votes
0answers
66 views
Can we find a function satisfiying the following functional inequation?
Question:
I would be interested know if there exist (and construct)
a function $f:\mathbb N\times \mathbb N \to (0,\infty )$ such that for all integers $p,q$, $f(p,q)=f(q,p)$ and
a constant ...
0
votes
0answers
23 views
Inequality proof in functional analysis of norms [duplicate]
Possible Duplicate:
Proof of an inequality of $L^p$ norms
For any general measure space let $0 < a < b < c < \infty$. Prove that:
$$
\|f\|_b \leq \max \{ \|f\|_a,\|f\|_c \}.
$$
...
0
votes
0answers
243 views
Find a Lipschitz constant
Please help me to find a Lipschitz constant.
Let $S_n$ be a group of permutations of the set $\{1, \ldots, n\}$. Let $a=(a_1, \ldots, a_{2M})$ be a real valued vector with $n$ non-zeroes entries, $M ...
-2
votes
1answer
69 views
Inequalities between an exponential and log function
For the inequality $(3^n)\gt (n^4)$, what would be n be equal to such that it is the smallest integer possible?
