2
votes
1answer
44 views

Bounded almost-homomorphisms on the integers

Let $f : \mathbb{Z} \to \mathbb{Z}$ be an "almost-homomorphism": The set $\{f(n+m)-f(n)-f(m) : n,m \in \mathbb{Z}\}$ is bounded. We may assume that $f$ is odd, i.e. $f(-n)=-f(n)$ for all $n$. Assume ...
2
votes
1answer
39 views

Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$

Let $f$ be a twice derivable function and $M_i =\sup_{x \in \mathbb{R}} |f^{(i)}(x)|$ and $|M_0|, |M_2|<\infty $. Preferably using the Taylor series on the interval $[x,x+h]$ show the following ...
4
votes
1answer
63 views

submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
1
vote
1answer
47 views

Taylor's Theorem and inequalities on some interval of the domain?

From the following form of Taylor's Theorem and assuming that $|f(x)|\le 1$ and $|f''(x)|\le 1$ hold on $[0,2]$, $$f(a+h) = f(a) + hf'(a) + (1/2)h^2f''(a+θh),$$ some application of Taylor's Theorem ...
2
votes
1answer
70 views

Range of inner product of a sequence and its permutation

$a^n :=(a_i)_1^n$ is a finite sequence of real numbers of length $n$, where $\sum\limits_{i=1}^n a_i=0$ and $\sum\limits_{i=1}^n a_i^2=1$. Consider $s_n(a^n,\sigma):=\sum\limits_{i=1}^n ...
1
vote
0answers
156 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
55 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
77 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
74 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
143 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
356 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.