All Questions
Tagged with real-analysis inequalities
62 questions with no upvoted or accepted answers
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
6
votes
0
answers
129
views
Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
- 349
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
5
votes
0
answers
153
views
Inequality for functions on $[0,\infty)$
Let $0<q<1$ and $\varphi(q;x)=\displaystyle \prod_{j=0}^\infty (1+q^jx),\;x\geqslant 0.$ Consider the following functions:
$$l_k(x;q):=\frac{q^{k(k-1)/2} x^k}{(1-q)(1-q^2)\dots (1-q^k)\varphi(x;...
- 783
5
votes
0
answers
205
views
Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 101
5
votes
0
answers
268
views
An integral trigonometric inequality
Problem 1. Suppose that $\xi>0$ and $\sin(2\xi)<0$.
Let $$b_\nu=(N-v+1)\tfrac{\pi}{\xi}\quad\mbox{for}\quad\nu=1,\dots,N:=\big[\tfrac{\xi}{\pi}\big].$$
Prove that
$$\mathrm{sgn}(\sin \xi)\...
- 4,535
4
votes
0
answers
130
views
A "counterbalancing" trigonometric sum inequality
Is it true that
$$s_{n,k}:=\sum_{j=1}^{n-1} r_{n,k,j}
<0$$
for all natural $n\ge2$ and all natural $k\in\{1,\dots,n-1\}$,
where
$$\text{$r_{n,k,j}:=\frac{x_{n,2j}}{y_{n,k,j}\;y_{n,k+1,j}},\quad$
$...
- 128k
4
votes
0
answers
179
views
Approximation by gaussian mollification in Sobolev spaces
I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
- 230
4
votes
0
answers
169
views
Questions on a cone defined by elementary symmetric functions
Let $1\le k\le n$ be given integers. Define the following cone
$$\Gamma_k=\{\lambda\in\mathbb{R}^n| S_j(\lambda)>0, j=1, ..., k\},$$
where $S_j(\lambda)$ is the $j$th elementary symmetric function ...
- 69
4
votes
0
answers
271
views
An inequality in harmonic analysis with the BMO flavour
I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes).
Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in
$[0,...
- 489
4
votes
0
answers
633
views
Problem with an integral equation taken from a paper
I am reading a paper (the 2015 paper by A. Falkowski and L. Slominski Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation, page 353, proof of the Lemma 3.1)...
- 779
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
4
votes
0
answers
197
views
Dynamics of an inequality
The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...
- 128k
4
votes
0
answers
462
views
System of Equations Upper Bound
I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:
For $i=1,2,\...
- 4,952
3
votes
0
answers
118
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
3
votes
0
answers
86
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
3
votes
0
answers
40
views
Bound of a regular function that cancels at some points
Let $K$ be a bounded convex set of $\mathbb{R}^n$ and $x_1,\ldots,x_k\in K$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function that cancels on points $x_1,\ldots,x_k$ . When $n=1$, ...
- 121
3
votes
0
answers
275
views
Maximizing the discrepancy in Jensen's inequality for a certain function
Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter.
Define
$$
D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}}
-\sum_{...
- 959
2
votes
0
answers
101
views
An inequality related to Problem 10210 AMM 1992 No. 3
Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
2
votes
0
answers
94
views
A surprisingly simple and difficult problem on sums of upper bounds
Let $T$ be a large integer, and $C$ be a positive real constant.
Consider a sequence $\{p_t\}_{T\geq t\geq 1}$ of real numbers in $[0,1]$. The sequence $\{b_t\}_{T\geq t\geq 1}$ can be defined as ...
- 353
2
votes
0
answers
159
views
Upper bound of a product of sines
Consider the function
$$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$
I wonder whether it is possible to compute some nontrivial upper ...
- 826
2
votes
1
answer
288
views
How to estimate an integral by the variation and upper bound of the integrand?
Suppose that $f$ is a continuous function on $\mathbb{R}$. I want to estimate the definite integral
$$ I:= \int_{0}^a [f(x)-f(0)]dx $$
by the upper bound $M = \sup_{x\in[0,a]}|f(x)|$ and the variation ...
- 419
2
votes
0
answers
118
views
Inequality for log-likelihood ratio
Let $ p, q $ be two probability densities on $ [0,1] $, strictly positive over $ (0,1) $. Let $ P $ be the cumulative function of $ p $, i.e., $ P(x) = \int_0^x p(x') \, \mathrm{d}x' $, $ x \in [0,1] $...
- 503
2
votes
0
answers
56
views
Inequality for a weighted bilinear form in Fourier variables
Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...
- 1,101
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
150
views
Closeness of a rational approximation
What is
$$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N}
|2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$
where $\mathbb N:=\{1,2,\dots\}$?
In other words, I would like to ...
- 128k
2
votes
0
answers
108
views
Absolute lower bound on derivative of generalized trigonometric polynomial at zeroes
By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\...
- 739
2
votes
0
answers
250
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 197
2
votes
0
answers
70
views
Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$
By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy:
$$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...
user153000
2
votes
0
answers
73
views
Proof of a technical fact in the book of Schapire and Freund on boosting
Disclaimer: I asked this question on math.stackexchange.com two weeks ago but it has not been answered yet so I figured that I might as well try to also post it here.
I am currently looking at ...
- 121
2
votes
0
answers
60
views
Gronwall type inequality involving iterated integrals
Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have:
$$p(t) \leq a(t) + C \int_0^t du e^{-\kappa(t-u)}p(u) \int_0^u ds e^{-\kappa(u-s)} p(s),$$
where $\kappa, C >...
- 31
2
votes
0
answers
65
views
Minimal condition $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $, $\mid \rho_{ij} \mid \leq 1$, $s_i \in \mathbb R$ and $\Psi_{ij} \in \{0,1\}$
Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below
$$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$
$$G = ...
- 123
1
vote
0
answers
100
views
Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
vote
0
answers
269
views
Monotone likelihood ratio of a kernel based on $\log(\cosh(x))$
Let $f(x) = \log(\cosh(x))$, and define the kernel density:
$$p_r(\phi;\theta) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta)\big) \Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}...
- 391
1
vote
0
answers
50
views
Bound an integral with parameter
Let us define, for $x_0 > 0$ and $x_0 \ll 1$,
$$K(x) = \int_{-x_0}^{x - g(x)^2} \frac{f(y)}{x - y}dy, \quad \text{for } x \in [-x_0/4, x_0],$$
and
$$g(x) = \frac{(x_0 - x)|\log(x_0)|}{|\log(x_0 - x)...
- 452
1
vote
0
answers
214
views
Bound for the laplacian of a strictly convex function from above by the gradient of it
Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$
I am trying to show that
$$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
- 185
1
vote
0
answers
158
views
examples of function difficult to prove to be $\geq0$?
I have often wondered whether there has ever come a point in your research,
when you were confronted with an explicit real function $f(x_1,x_2,\ldots,x_n)$ and an explicitly defined compact set $S\...
- 211
1
vote
0
answers
77
views
Divergence between random variables after transformation
Let $X$ and $Y$ be random variables with laws $\mu_X$, $\mu_Y$ and $d$ be some $f$-divergence (e.g. KL, total variation, Hellinger). Writing $d(X,Y)$ for the divergence between $\mu_X$ and $\mu_Y$, ...
- 19
1
vote
0
answers
70
views
An inequality for a recursively defined sequence of numbers
Consider an arbitrary sequence $(x_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}$ and $r \in \mathbb{R}$ with $r > 2$.
Set $y_0 = 1$ and $z_0 = 0$ and for $n \in \mathbb{N}$ recursively define
$$y_n = ...
- 111
1
vote
0
answers
151
views
Log-concavity inequality
Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$
$$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{...
- 3,209
1
vote
0
answers
60
views
Maximum value of $\int (aF^2(x)g(x)+G^2(x)f(x))dx$ over all $f,g$ densities satisfying $\int F(x)g(x)dx=1/2$
I want to maximise $$I(f,g):=\int_{-\infty}^\infty (aF^2(x)g(x)+G^2(x)f(x))dx$$ where $a>0$ is a given constant, over all possible probability densities $f,g$ satisfying $$\int_{-\infty}^\infty F(x)...
- 319
1
vote
0
answers
87
views
An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
- 4,135
1
vote
0
answers
150
views
Proving the existence of a sequence with recursive growth constraints
Fix $0 < c < 1/3$. Show that there exists a strictly increasing sequence $a_k > 0$, a sequence $b_k \geq 0$, and an infinite set $K \subseteq \mathbb{Z}_{>0}$ such that
\begin{align}\...
- 161
1
vote
0
answers
41
views
Necessary additive and multiplicative properties to characterize a mildly growing function
Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we ...
- 13.9k
1
vote
0
answers
99
views
simultaneous smallness
QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that
$$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\
1-(1-(\frac{\...
- 43.2k
1
vote
0
answers
58
views
A question on Integral inequality
Let $0 < \epsilon < 1$. Consider $\{a_n\}_{n \geq 1} \in l_2$ and $L(t) = 1+\epsilon t$. Let $x$ be fixed such that $0 < x < L(t)$. Does there exist $\tau \geq 0$ such that the following ...
- 11
1
vote
0
answers
85
views
What are good bounds on ratios of subdeterminants?
Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...
- 1,893
1
vote
0
answers
225
views
Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?
By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the $n!...
- 1,893