For questions about proving and manipulating functional inequalities.

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0
votes
1answer
40 views

Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
3
votes
1answer
186 views

Use of cut-off functions and partitions of unity

This is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold, $$\int e^w \sqrt{g}\, dx \leq C \exp ...
0
votes
1answer
36 views

How to prove that $A B A^* \leq \|B\| A A^*$ for operators A,B?

Let $A$, $B$ bounded operators on a Hilbert space $H$. Further let $B$ be self-adjoint. Then we have that $A B A^* \leq \|B\| A A^*$. I wanted to ask how to prove this inequality or where I can find ...
10
votes
0answers
218 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
1
vote
1answer
140 views

sum of ceiling function inequality

I need to show the following inequality: $$\sum_{j=1}^n \left(\frac{D_j}{Q_j} \left\lceil\frac{Q_j}{T_c}\right\rceil \right) \ge \frac{\sum_{j=1}^n D_j}{\sum_{j=1}^n Q_j} ...
2
votes
3answers
202 views

Inequality for Expected Value of Product

Let $(\Omega, \mathbb{P}, \mathcal{F})$ be a probability space, and let $\mathbb{E}$ denote the expected value operator. Consider the random variables $f: \Omega \rightarrow \{0,1,2\}$ and $g: \Omega ...
2
votes
1answer
84 views

Minkowski's Inequality for $0<p<1$.

I need to prove: for non-negative functions $f,g\in L^p[0,1]$, $||f+g||_p\geq||f||_p+||g||_p$ for $0<p<1$. For $1\leq p<\infty$, the inequality is reversed and the proof is like: The cases ...
2
votes
0answers
40 views

I want to solve the inequality $z′(t)+1≥0$ for all $t≥0$

Let $z(t)$ be a differentiable function for all $t≥0$. I want to solve the inequality: $$z′(t)+1≥0$$ for all $t≥0$. where $z′(t)$ is the derivative of $z(t)$.
0
votes
0answers
50 views

Continuation principle

In the following $g$ is a function related with some norms of solution of a certain differential eqution: $g$ be a nonnegative continous (if necessary, it is monotone increasing) function satisfying ...
1
vote
1answer
141 views

Bessel's inequality implying convergence

Define Bessel's inequality $$ \sum_{n=1}^N|a_n|^2\leq \|x\|^2 $$ where $a_n=\langle x,e_n \rangle$. Lemma Let $\{e_n\}_{n\geq 0}\subset H$ be a complete orthonormal set. If $x\in H$ and ...
2
votes
1answer
76 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 ...
2
votes
1answer
159 views

fraction power vector-norm inequality

If X is a Banach Space and $x,y\in X$. Then by the definition of a Banach algebra we know $$\|x.y\|\leq\|x\|\|y\|$$ and thats how we can have relation for any positive power. i.e. $n\in N$, ...
4
votes
2answers
71 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
0
votes
1answer
33 views

Convert an inequality into limiting equality.

Given $f(x)/g(x) \lt 1.5/h(x)$ where all three functions are increasing and positive in nature. My question is, if I can deduce $\lim_{x \to \infty} [f(x)/g(x)]=1$ (then how if yes) or not .?
4
votes
1answer
154 views

Inequality $\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y-1)$

I need to prove $$\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y),$$ for $x>\alpha$ and $y>\beta$ with $0<\alpha,\beta \leq \frac{1}{2}$ constants, and $C(\alpha, \beta)$ is a ...
0
votes
1answer
40 views

Finding range of transformation of function from range of original

I'm asked to find the range of $y = f(x-2)+4$, if the range of $y=f(x)$ is {$y| -2 \geq y \geq 5, y \in R$}. How do I go about finding this? I have no idea where to even start. I'm doing the course ...
0
votes
1answer
94 views

An inequality involving norms.

I have to know how we can show the following inequality: $\|u\|_{2}\leq\|u_{0}\|_{2}+\int_{0}^{t}\|u_{t}(t,x)\|_{2}dt$ where $\|u\|_{2}=\Big(\int_{\Omega}u^{2}dx\Big)^{1/2}$, $u_{0}=u(x,0)$, ...
0
votes
1answer
52 views

Probability bounds

My question is the following: if my random variable $X$ has finite or bounded second moment $\mathbb{E}[X^2]\leq B$ can anyone develop any bounds on pdf of $X$. For example something like this ...
1
vote
1answer
74 views

What proportion of the natural numbers satisfy the following inequalities?

Let $\sigma_1(n)$ be the sum of the divisors of $n \in \mathbb{N}$, and let $$I(n) = \frac{\sigma_1(n)}{n}$$ be the abundancy index of $n$. What proportion of the natural numbers satisfy the ...
0
votes
2answers
208 views

Strong Induction on Inequalities

I'm asked to indicate which natural numbers $n$ each of the below inequality is true, and then I am required to prove this via induction, but I'm wondering what that means... Strong induction? ...
2
votes
2answers
166 views

Convexity of log sum function

Is $f\left( x \right)=\log \left( \sum_i \beta_i e^{-\alpha_ix} \right)$ a convex function where $\beta_i,\alpha_i\in \mathbb{R}$?
1
vote
2answers
65 views

Prove that $\frac1{x}-x$ is multiplicatively subadditive in $(0, 1)$

If $f(x) = \dfrac1{x}-x$, prove that $f(x)$ is multiplicatively subadditive in $(0, 1)$, that is, if $a, b \in (0, 1)$, then $f(a)+f(b) < f(ab)$.
1
vote
2answers
180 views

Does the following inequality hold if and only if $N$ is an odd deficient number?

Let $N \in \mathbb{N}$. (That is, let $N$ be a positive integer.) This is in reference to two of my earlier questions here at MSE: Does the following inequality hold true, in general? Does this ...
4
votes
0answers
193 views

A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and ...
0
votes
2answers
465 views

How to prove equality from poincare inequality?

Let $$D = \{y \in C^1(0,1) : y(0) = y(1) = 0\}$$ Suppose there exists a $C_0$ such that $$\int_{0}^{1} y^2 \ dx \leq C_0 \int_{0}^{1} (y')^2 \ dx$$ for all $y \in D$, and for all $C < C_0$ ...
4
votes
1answer
39 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
172 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
520 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
63 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
171 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
125 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
67 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
113 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
66 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
103 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
93 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
159 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
228 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
58 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
183 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
1answer
72 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
51 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
170 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
80 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 ...
2
votes
2answers
273 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
139 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 ...
2
votes
0answers
261 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
153 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 ...