For questions about proving and manipulating functional inequalities.

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2answers
204 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? ...
3
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1answer
68 views

Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R $ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
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1answer
34 views

Upper bound for integral over boundary in terms of integral over interior

I've encountered quite some papers in which it is simply assumed that $\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq ...
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1answer
65 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This has been cross-posted to MO.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). ...
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1answer
38 views

Is it correct to approach this with Holder Inequality? What am I doing wrong?

I know that $ \forall n\in N, a_0 + a_1 + ... + a_n = 1$, with $a_0, a_1, ... a_n > 0$ and $f(t) = a_0t^n+a_1t^{n-1}+...+a_n, \forall t\in R$ I have to prove that for every $x > 0$ $$ ...
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1answer
32 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 .?
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1answer
37 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 ...
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1answer
50 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 ...
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1answer
181 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?
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0answers
206 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 ...
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0answers
91 views

Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
4
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0answers
188 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 ...
3
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0answers
48 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
3
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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 $$\sum_{i=1}^n (x_i - x') = x - x'.$$ ...
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0answers
66 views

Help explain why the proof works - Gradient Estimate Using the Maximum Principle

I found the following proposition and proof in a textbook for one of my classes, and the end of the proof seems a little "unfinished" to me. In other words, as the author left it, I'm not entirely ...
2
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0answers
39 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 $z(t)$.
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0answers
15 views

Inequality to bound $\sum_i a_i b_i - \sum_i c_i d_i$ (harmonic eigenfunction/graph) type sum with constraints

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
1
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0answers
21 views

An integral inequality

Suppose $K\in C[0,1]^2$, $G\in C[0,1]$ are arbitrary and given. The question is that does there exists $H\in B[0,1]$ continuous a.e. with possibly finitely many discontinuities such that $$ ...
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0answers
19 views

On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
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0answers
43 views

application of Gronwalls inequality

My question comes from a proof in Daniel Stroock's book 'An introduction to the Analysis of Paths on a Riemannian Manifold' (lemma 3.60, page 86). He proves that a function F satisfies the integral ...
1
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0answers
44 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
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0answers
36 views

Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle ...
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0answers
220 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 ...
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0answers
56 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 ...
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0answers
60 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) ...
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0answers
79 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 ...
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0answers
136 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 ...
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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 ...
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0answers
33 views

hint for investigating positivity/negativity of a function

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function ...
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0answers
29 views

bounding supremum with Sobolev embedding theorem

I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am ...
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0answers
65 views

Functional Inequality

I have no idea about this question. Please give a hand whoever can. Characterize twice-differentiable and bounded functions $f$ mapping the set of positive reals into itself and satisfying $ ...
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0answers
37 views

Prove an inequality involving $Si(x)$ and $Si(2x)$

How Is it possible to prove the following inequality? $$xSi(2x)-2Si(x)*\sin(x)\lt x^2$$ for $x\in\mathbb{R}$ Thanks
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0answers
33 views

Reference/confirmation of a result in analysis

Does anyone know of or have a reference for the following result: Let $X$ be a reflexive Banach space with dual $X^{*}$. If there exists a continuous mapping $f: K \rightarrow X^{*}$ on compact ...
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0answers
48 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 ...
0
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0answers
320 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
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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 ...