For questions about proving and manipulating functional inequalities.

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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 ...
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1answer
58 views

Proving inequality $(x^2+y^2)(y-1)+yx-y^2<0$

I have an inequality which came out of Lyapunov function for system of ODE's: $$(x^2+y^2)(y-1)+yx-y^2<0.$$ To prove stability of my solution, I have to prove that the inequalty is true in area ...
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2answers
27 views

L2 norm and L1 norm inequality

In the vector space, we have the following inequality $$ ||x||_2 \leq ||x||_1 $$ where x is a vector. I am wondering that we have similar inequality for function's norm. L1 norm of function f is ...
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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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2answers
44 views

Difficult but Interesting Inequalities Problems

1.) Consider the identity $$(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.$$ Find the minimum of $\max(A,B,C)$ over $0 \leq p \leq 1$. 2.) Let $n$ be a positive integer. Show that the smallest integer ...
6
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1answer
489 views

Does any one-to-one function exist that satisfies this inequality for all real numbers?

Does there exist a one-to-one function $f: \Bbb R \to \Bbb R $ such that $f(x^2) - (f(x))^2 \geq \frac 1 4\ \ \forall x \in \Bbb R$ ? I've tested this with many one-to-one functions but the ...
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1answer
41 views

A question on logic and some functional inequalities

Suppose that I have a (generic) function $g$ and arguments $a, b \in \mathbb{N}$. I know that $g$ satisfies the inequalities $$1 < \frac{g(b)}{b} < \frac{g(a)}{a} < 2.$$ I also know that ...
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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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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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2answers
39 views

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I've recently been going over the mean value and intermediate value theorems, however I'm not sure where to start on this.
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3answers
93 views

If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex

Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then how do we ...
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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
28 views

$ {\|f\|}_p = \sqrt[p]{\int_{a}^{b} |f(x)|^p {\rm d}x}$ is a norm

Consider the space $C([a,b])$ of all continuous functions $f\colon [a,b]\rightarrow \mathbb{R}.$ Show that the function $\|\cdot\|_p\colon C([a,b]) \rightarrow [0,\infty),p>1$, given by $$ ...
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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 ...
0
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1answer
22 views

Find the size of squares cut from a box.?

This has been taking me days to do and I really want to do it for test practice. I actually have absolutely no idea how to even start this, so if I can get a hint, advice, or something to start me ...
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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
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 := ...
1
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1answer
33 views

Inequality regarding exponential function

For every positive $x$ and for every $n$ show that $(1+\sqrt{\frac{x}{n}}+\sqrt[3]{\frac{x}{n}})e^{n\arctan{\frac{x}{n}}}>e^x$. I plotted it, it seems that the inequality holds. Any ideas how to ...
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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 ...
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1answer
29 views

$\| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le C\left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f g''\|_{L^2}\right)$ holds?

I want to know that the following inequality holds $$ \| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le ^\exists C_{>0} \left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f ...
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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 ...
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2answers
62 views

Questioning a Proof of Khinchine Inequality

[Khinchine Inequality] Let $a_1,\ldots,a_n\in R$, $\varepsilon_1,\ldots,\varepsilon_n$ be i.i.d. Rademacher random variables: $P(\varepsilon_i=1)=P(\varepsilon_i=-1)=0.5$, and $0<p<\infty$. Then ...
0
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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
0
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1answer
45 views

$f(x)=\sec(x)$ inequality inconsistency\trouble

I'm currently attempting to find the range of $f(x)=\sec(x)$ by considering $\cos(x)$ in the intervals of $0<\cos(x)\leqslant 1$ and $-1\leqslant \cos(x)<0$ (as $\sec(x)$ is undefined for ...
2
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1answer
51 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 ...
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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 ...
10
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1answer
364 views

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
2
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1answer
58 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 ...
2
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1answer
30 views

Inequality with Bessel Functions of the first kind

How can be proven the following inequality: $$\int_0^1dx|J_k(x)'J_k(x)|\lt\frac{1}{2}\int_0^1dx|J_k(x)'^2|$$ where obviously: $J_k(x)'=\frac{d}{dx}J(k,x)$? Thanks.
2
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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 ...
5
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1answer
64 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
4
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1answer
64 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 ...
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1answer
52 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 ...
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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
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 ...
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1answer
46 views

Sobolev Inequality with powers

The following is one of the Sobolev Inequalities as stated in Gilbarg & Trudinger's text on elliptic PDEs: $\displaystyle W_{0}^{1,p}(\Omega)\subset L^{\frac{np}{(n-p)}}(\Omega)$ for $p<n$. ...
0
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1answer
71 views

Bounds on functions using inequalities?

I'm studying inequalities as part of a course on Numbers, Proofs and Mathematical Induction. There is one type of question that I don't understand, primarily because there's only one example in the ...
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1answer
113 views

Why are so many inequalities and estimate in PDEs?

I want to study PDEs, but when I read some PDEs books, I notice that there are so many inequalities here and always do estimates. I really want to know why there are so many inequalities here and ...
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4answers
396 views

Find the range of values of $x$ which satisfies the inequality.

Find the range of values of $x$ which satisfies the inequality $(2x+1)(3x-1)<14$. I have done more similar sums and I know how to solve it. I tried this one too but my answer doesn't matches the ...
0
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1answer
55 views

Show that $\ c_X(p,q) \le d_X(p,q)$, for $ p, q \in X$

Update I'm trying to show the Corollary, but I have stuck...That is: For any complex space $X$, we have: $$\begin{align} (1).\ c_X(p,q) &\le d_X(p,q),\ \text{for}\ p, q \in X \\ (2).\ ...
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, ...
3
votes
1answer
193 views

Exponential Inequality

I was working on a problem and reduced it to showing the following inequality: $$2x e^{x^2/6} \ge e^x - e^{-x} \text{ for $x \ge 0$}$$ I tried expanding everything in Taylor series to no avail. I ...
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votes
1answer
39 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
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1answer
184 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
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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
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 ...
1
vote
1answer
134 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
190 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 ...