All Questions
Tagged with real-analysis inequalities
339 questions
4
votes
3
answers
407
views
On some inequality (upper bound) on a function of two variables
There is a problem (of physical origin) which needs an analytical solution or a hint. Let us consider the following real-valued function of two variables
$y (t,a) = 4 \left(1 + \frac{t}{x(t,a)}\right)...
- 371
6
votes
2
answers
399
views
Maximal (minimal) value of $S=x_1^2x_2+x_2^2x_3+\cdots+x_{n-1}^2x_n+x_n^2x_1$ on condition that $x_1^2+x_2^2+\cdots+x_n^2=1$ [closed]
since $x_1^2+x_2^2+\cdots+x_n^2=1$ is sphere,a compact set,so $S$ has a maximal(minimal) value. But when I try to solve it using the Lagrangian multiplier method, I don't know how to solve these ...
- 81
5
votes
2
answers
924
views
How to show this symmetric function inequality
Question: let $x_{i}>0$ $(i=1,2,\cdots,n)$, such that $x_{i}\neq x_{j},\forall i\neq j$, find all real numbers $p$ that satisfy the following inequality
$$\sum_{i=1}^{n}\dfrac{x^p_{i}}{\prod_{j\neq ...
- 4,280
1
vote
0
answers
157
views
Prove that $\frac{(1-x)^{y(1-x)}}{(1- xy)^{1-xy}} < 1$, when $0 < x < 1$, $y > 1$ and $xy < 1$ [closed]
If $0 < x < 1$, $y > 1$ and $xy < 1$,
$$
\frac{\binom{n}{nx,nx,\cdots,n(1 - yx)}}{\binom{n}{nx,n(1-x)}} \sim \left(\frac{(1-x)^{y(1-x)}}{(1- xy)^{1-xy}}\right)^{\!n} = o(1)
$$
only if
$$
\...
- 271
-2
votes
1
answer
169
views
Question about Lipschitz conditions
Let $f$ be a function on some real interval $[a,b]$. Suppose that $\forall x\in [a,b]$, there exists a positive constant $C$ such that
$$ |f(x)-f(y)| \leq C|x-y| $$
for all $y \in [a,b]$.
Does each $x ...
- 419
23
votes
2
answers
2k
views
Are such functions differentiable?
In my recent researches, I encountered functions $f$ satisfying the following functional inequality:
$$
(*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}.
$$
Since $f$ is convex (because $\...
- 995
7
votes
2
answers
455
views
On a monotonicity property of Fourier coefficients of truncated power functions
Is it true that
$$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$
is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$?
This question is related to this previous one.
Twice integrating by parts, one ...
- 128k
2
votes
1
answer
137
views
Approximating a limit of an integral
How can we prove the following asymptotic lower bound for the regularized Beta function when $n\rightarrow\infty$?
$$\int_0^{1} I_{2 t - t^2}\left(\frac{n - 1}{2}, \frac{1}{2}\right) dt=\Omega\left(\...
- 1,677
0
votes
0
answers
32
views
Minimization of a palindromic-like sequence and asymptotics
Suppose that I have a sequence of $n$ numbers $x_1, \ldots, x_n$ all taken from
the real interval $[0,1]$.
I am interested in minimizing the infinity norm of the vector
$$ v = \left( \frac{x_{1}}{x_2},...
- 313
8
votes
1
answer
662
views
Can this inequality be proved using weighted maximal function estimates?
I am trying to understand the following fact:
Suppose $\{B_i\}_i$ are disjoint balls in $\mathbb R^n$, and $A_i \subset 100 B_i$ is a subset with $|A_i| \geq c |B_i|$. Then for any nonnegative $f$, ...
- 613
1
vote
1
answer
225
views
Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$
Working with Slater's inequality (a companion of Jensen's inequality) I found this statement:
Let $f(x)$ be a continuous, twice differentiable function, convex or concave and non constant on $(0,\...
12
votes
1
answer
525
views
An inequality about unit vector orthogonal to $(1,1,...,1)$
Does there exist a constant $\alpha>0$ such that the following holds?
$$\liminf_{n\to\infty}\inf_{x\in\mathbb{R}^n, \sum_{i=1}^nx_i^2=1, \sum_{i=1}^nx_i=0}\frac{\sum_{i<j, |i-j|\leq\frac{n}{4}}(...
- 1,495
0
votes
1
answer
32
views
Finiteness of a bilinear combination
For $j\in\mathbb{N}$, consider continuous functions $f_j:[0,1]\to\mathbb{\mathbb{R}^+}$ such that
$$\sup_{t\in[0,1]}\sum_jf_j(t)<+\infty,$$
namely $f_j(t)\in L_t^{\infty}((0,1),l_j^1(\mathbb{N}))$. ...
- 943
0
votes
1
answer
159
views
Best bounds on the integral of an increasing function
The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval.
Let $F\colon[0,1]\to[0,1]$ be a ...
- 128k
0
votes
1
answer
60
views
Bounding the ratio of the $\ell_1$-norms of two real-valued $n$-vectors as a linear combination of their $n$ element-wise ratios
Let $a_1, a_2, \ldots a_n$ and $b_1, b_2, \ldots b_n$ be two sequences of $n\gg 1$ real numbers such that, for all $1\le i\le n$, we have $0<a_i \le b_i\le 1$. Let the ratio $R$ defined as follows:
...
- 1,677
5
votes
5
answers
623
views
Elementary inhomogeneous inequality for three non-negative reals
I need the following estimate for something I am working on, but I don't immediately see how to establish it.
For $x, y, z \in \mathbb{R}_{\ge 0}$, show that
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
- 543
7
votes
1
answer
371
views
An elementary inequality for three complex numbers
The following problem arose in asymptotic analysis of difference equations.
Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have
$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
- 128k
8
votes
3
answers
296
views
Shrinking subset and product
Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|...
- 1,209
1
vote
1
answer
117
views
Shrinking subset with disjoint unions
Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be disjoint finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that ...
- 1,209
5
votes
2
answers
243
views
Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
- 804
3
votes
3
answers
128
views
Detecting slow growth in a finite number of queries
The following question was asked at Can you solve this problem using a finite number of queries?
:
Let $g:[0,1]\to[0,1]$ be a continuous monotonically-increasing function. You can access $g$ using ...
- 128k
1
vote
1
answer
181
views
Optimization problem with definite integral inequality constraints
Question: How can we prove that there exists a real constant $c\ge 1$ such that the following inequality holds for all integers $d>1$ and all real numbers $r\in\left[1,\sqrt{d}\right]$?
$$\int_{-1}^...
- 1,677
2
votes
2
answers
214
views
A question about asymptotic affinity and strict convexity with unbounded means
Let $F:[0,\infty) \to [0,\infty)$ be a $C^1$ strictly convex function.
Let $\lambda_n \in [0,1],a_n\le c<b_n \in [0,\infty)$ satisfy
$$ \lambda_n a_n +(1-\lambda_n)b_n=c_n \tag{1}$$ and assume that
...
- 6,741
4
votes
1
answer
161
views
Does strict convexity plus asymptotic affinity imply bounded mean?
I am not sure if this is exactly research-level, but I am struggling to find a proof for the following claim:
Let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function.
Let $\lambda_n \in [...
- 6,741
1
vote
0
answers
151
views
Log-concavity inequality
Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$
$$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{...
- 3,209
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
1
vote
1
answer
484
views
Convexity at a point and Jensen inequality
I am looking for a reference for the following claim:
Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.
Suppose that "$\phi$ is convex at $c$". ...
- 6,741
-5
votes
1
answer
184
views
Two inequalities in $\mathbb{R}$ [closed]
How to prove that for real numbers $a$ and $b$, the following inequalities hold?
$(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq 2^{2-p}|a-b|^p$,if $p\geq 2$
$(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq (p-1)\frac{|a-b|^2}{(|...
0
votes
1
answer
179
views
How to show $\max_{1\leq i\leq n}(X_i+Y_1)\preceq \max_{1\leq i\leq n}(X_i+Y_i)$?
Let two collections of random variables $\{X_i\}$ and $\{Y_i\}$ be independent and let $\{Y_i\}$ be i.i.d. Then
$$\max_{1\leq i\leq n}(X_i+Y_1)\preceq \max_{1\leq i\leq n}(X_i+Y_i).$$
where $\...
- 288
4
votes
1
answer
338
views
Minimization of a discrete valued function
$$
\min_{f} \sum_{i=1}^n \max \left( 0, 2f(i) - f(i-1) -f(i+1)\right),
$$
where the minimum is taken over all the functions $f$ from $\{0,1,2,\ldots,n+1\}$ to $[0,x]$, $x <1$, such that $f$ is non-...
user83947
4
votes
1
answer
256
views
Is there a non-convex function with non-decreasing average rate of change?
$\newcommand{\R}{\mathbb R}$
Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...
- 128k
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 $$\int_{-\infty}^\infty F(x)...
- 319
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
5
votes
2
answers
301
views
Euler–Maclaurin formula in $\mathbb{Z}^d$
I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...
- 446
9
votes
2
answers
2k
views
An inequality involving square roots and sums
I've been trying to prove (maybe even disprove) the following inequality:
$$
\sum_{n=1}^{N} \frac{a_n}{\sqrt{\sum_{i=1}^{n}a_i}} \leq C \sqrt{\sum_{n=1}^{N}a_n}
$$
Where $ a_1,...,a_N\geq 0 $ are ...
- 109
3
votes
1
answer
237
views
Isoperimetric inequality for analytic functions on an annulus
Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
- 434
2
votes
1
answer
200
views
Proof of a discrete isoperimetric inequality
The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions:
$$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{...
- 434
4
votes
0
answers
271
views
An inequality in harmonic analysis with the BMO flavour
I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes).
Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in
$[0,...
- 489
1
vote
1
answer
148
views
Understanding a family of Sobolev-type inequalities
I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following:
Denote the following inequality as $S_{r,s}^{\theta}$: $\...
- 11
2
votes
2
answers
301
views
"Strengthening" the mean value theorem for the sine function
The present discussion arises from this MO question. Below, $e$ stands for Euler's number and let
$$\tau:=\arccos\left(\frac{\sin e-\sin 1}{e-1}\right)\approx 1.82\cdots.$$
An application of the Mean ...
- 43.2k
14
votes
3
answers
2k
views
How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\...
- 1,326
1
vote
2
answers
2k
views
Simple bound on $\log(x)/x$
I would like to pick $x$ as small as possible while guaranteeing that $\log(x)/x \leq \epsilon$ where $\epsilon \ll 1$. Clearly $x$ should (roughly) be of the order $1/\epsilon$; I would like simple ...
- 1,703
1
vote
1
answer
170
views
Convergence of local means implies converge ae?
Let $f,f_n \in L^1(\mathbb{R},\mathbb{R}_+)$ with $\int_{\mathbb{R}} f = \int_{\mathbb{R}} f_n = 1$, $(\sqrt{f_n})'$ bounded in $L^2$, $\nabla \sqrt{f}\in L^2$ and such that $$\int_{ p+[0,1/n]} f_n = \...
- 23
5
votes
1
answer
391
views
Proving a specific case of Robin's Inequality
Edit: It turns out that this is equivalent to the RH which gives the idea that this might a a little difficult to show. As such we could consider an even simpler case in which the number $n$ is ...
- 315
-2
votes
1
answer
286
views
Why this function is monotonic?
Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \...
- 395
1
vote
1
answer
161
views
Inequalities involving Gamma function
I am having some difficulty in proving the following inequality:
\begin{equation*}
\frac{1-e^{-\gamma b}}{b^\eta}-\frac{1-e^{-\gamma s}}{s^\eta}\geq \gamma(1-\eta)\int^b_sy^{-\eta}e^{-\gamma y}dy
\end{...
- 33
1
vote
2
answers
295
views
Monotonicity of maximum of convex combination of two scaled concave functions
Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...
- 13
3
votes
2
answers
127
views
Comparing the tails of two related convergent series
Let $b_1,b_2,\dots$ be positive real numbers such that
$$s_1<\infty\quad\text{and}\quad z_1<\infty,
$$
where
$$s_k:=\sum_{j=k}^\infty b_j\quad\text{and}\quad z_k:=\sum_{j=k}^\infty\frac{b_j}{\...
- 128k
2
votes
3
answers
216
views
Equivalence of operators
let $T$ and $S$ be positive definite (thus self-adjoint) operators on a Hilbert space.
I am wondering whether we have equivalence of operators
$$ c(T+S) \le \sqrt{T^2+S^2} \le C(T+S)$$
for some ...
- 23
0
votes
1
answer
359
views
Dual norm of a max function [closed]
I am attempting to find the dual norm of
$$\|(x,y)\|_K=\max\{|x|,|y|,|x-y|\}.$$
I have obtained $\|(x',y')\|_K^* = |x'|+|y'|$, but don't think that this is correct.
I obtained this as follows :
$$K = ...
- 9