Tagged Questions

For questions about proving and manipulating functional inequalities.

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0
votes
1answer
37 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 $x > 0$ $$ ...
1
vote
1answer
27 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 $$ ...
0
votes
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
votes
1answer
19 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 ...
0
votes
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 ...
1
vote
0answers
17 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
vote
1answer
32 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 ...
1
vote
0answers
41 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
vote
1answer
27 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 ...
5
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0answers
82 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 ...
2
votes
2answers
58 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
votes
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 $ ...
0
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0answers
34 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
votes
1answer
39 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
votes
1answer
50 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 ...
0
votes
0answers
32 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
votes
1answer
335 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
votes
1answer
57 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
votes
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
votes
0answers
62 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
votes
1answer
62 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
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
50 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 ...
1
vote
0answers
42 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 ...
1
vote
0answers
35 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 ...
3
votes
0answers
46 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 ...
1
vote
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
votes
1answer
64 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 ...
2
votes
1answer
107 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 ...
1
vote
4answers
325 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
votes
1answer
54 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
votes
1answer
64 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
190 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 ...
0
votes
1answer
37 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
176 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
35 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
186 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
115 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
172 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
65 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
37 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
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 ...
1
vote
1answer
115 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
73 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
150 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
49 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
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 .?
4
votes
1answer
142 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
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 ...
0
votes
1answer
90 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 $\|u\|_{2}=\Big(\int_{\Omega}u^{2}dx\Big)^{1/2}$, $u_{0}=u(x,0)$, ...