Tagged Questions
1
vote
0answers
41 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
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 ...
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 ...
1
vote
1answer
110 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
...
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.
