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4
votes
1answer
31 views

Finding appropriate function

In order to obtain an estimate I'm wondering if there is a positive function $f:\mathbb{R}\to \mathbb{R}_+$ such that $f(x) < |x|$ for $|x|$ large enough (say $|x|\ge C_0>0$) so that the ...
1
vote
2answers
155 views

Does this inequality hold true, in general?

Let $$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$ be the prime factorization of the positive integer $N$. Does the following inequality hold true in general? ...
5
votes
2answers
424 views

Weighted Poincare Inequality

I'm trying to prove a result I found in a paper, and I think I'm being a bit silly. The paper claims the following: By the Poincare inequality on the unit square $\Omega \subset \mathbb{R}^2$ we have ...
1
vote
3answers
59 views

What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?

What is the maximum value of $$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$ if $x \in \mathbb{R}$ and $x > 1$? A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here. Lastly, note that ...
1
vote
1answer
158 views

What is the minimum value of $\sqrt{\frac{2(x - 1)}{x}} + \frac{x + 1}{x}$, if $x > 1$?

What is the minimum value of $$f(x) = \sqrt{\frac{2(x - 1)}{x}} + \frac{x + 1}{x},$$ if $x \in \mathbb{R}$ and $x > 1$? Note that $f$ has a global minimum value of $$f(1) = 2$$ if we allow $x ...
1
vote
1answer
118 views

maximize the expected value of the logarithm of the weighted average of random variables

I'm trying to do the following. $$\max_{m\in\mathbb{R}} \mathbb{E}\left[\log (wA + (1-w)B_m)\right],$$ where $0<w<1$ and $A, B_m > 0$ are correlated random variables. $A$ does not depend ...
-2
votes
3answers
66 views

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum for $0 < x \in \mathbb{R}$?

This question is related to this one. My question here is: Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2}$$ have a global minimum for $0 < x \in \mathbb{R}$? Thank you!
2
votes
4answers
111 views

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?

Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2},$$ where $x \in {\mathbb{R} \setminus \{0\}}$, have a global minimum? I tried asking WolframAlpha, but it appears to give an inconsistent ...
2
votes
1answer
65 views

Does this inequality have any solutions for composite $n \in \mathbb{N}$?

Does this inequality have any solutions for composite $n \in \mathbb{N}$? $$\sqrt{2} < \frac{\sigma_1(n^2)}{n^2} < \frac{4n^2}{(n + 1)^2}$$ Note that $\sigma_1$ is the sum-of-divisors ...
3
votes
1answer
99 views

Does this inequality have any solutions in $\mathbb{N}$?

Does this (number-theoretic) inequality have any solutions $x \in \mathbb{N}$? $$\frac{\sigma_1(x)}{x} < \frac{2x}{x + 1}$$ Notice that we necessarily have $x > 1$.
1
vote
1answer
72 views

Does this function have a (global) minimum?

A good day to everyone. Does the following function have a (global) minimum: $$1 + \frac{1}{x} + {\left(1 + \frac{1}{x}\right)}^\theta,~~x\in\mathbb R$$ where $$\theta = {\displaystyle\frac{3\log ...
8
votes
2answers
90 views

A functional equation with inequality

Find all (at least one) functions $f\colon \mathbb{R}\to \mathbb{R}$ (or show there is none), such that $$ f(x^3+x)≤x≤f(x^3)+f(x), \quad \text{for all $x\in \mathbb{R}$}. $$ This is a problem asked ...
0
votes
1answer
119 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 ...
1
vote
0answers
174 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 ...
-2
votes
1answer
177 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?
1
vote
0answers
59 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) ...
3
votes
0answers
50 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
1answer
161 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? $$ ...
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 ...
2
votes
2answers
262 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 ...
1
vote
0answers
129 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 ...
0
votes
0answers
26 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: $$ \|f\|_b \leq \max \{ \|f\|_a,\|f\|_c \}. $$ ...
2
votes
0answers
251 views

An application of Poincare inequality [solved] [duplicate]

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 ...
14
votes
11answers
1k 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.
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
357 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.
1
vote
0answers
55 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 ...
0
votes
0answers
310 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 ...
0
votes
2answers
85 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 ...
5
votes
1answer
64 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 ...
2
votes
2answers
73 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 ...
0
votes
0answers
182 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 ...