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for questions involving inequalities.

0
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0answers
101 views

Is the sum of inverses always greater than inverse of the sum [on hold]

I've been working on a heat transfer problem and run into something I can't quite prove. I have three independent variables: a,b,c. I want to show that for all values in the range below, the ...
0
votes
0answers
33 views

Function in two variables inequalities

Let $A$ be the set of all subintervals of $[0,1]$. For any function $f:[0,1]\times [0,1]\rightarrow A$, define $g(a,b)=[0,1]\setminus f(a,b)$. What are all functions $f$ such that for any $a,b\in[0,1]$...
3
votes
2answers
251 views

Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove: $$ \int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1. $$ Numerically it seems to hold true. So I have made some attempts to ...
1
vote
1answer
73 views

A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported

I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...
1
vote
0answers
49 views

Show that the norm's bound is an exponent

Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...
1
vote
0answers
81 views

Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...
1
vote
1answer
161 views
+50

A simple two variable analytic inequality, inspired by probability

I'm trying to prove the following inequality: $$ bf_1g_1 + (x-b)f_1g_0 + (y-b)f_0g_1 + (1-x-y+b)g_0f_0 \le (|f_1|^p x + |f_0|^p (1-x))^{1/p} (|g_1|^p y + |g_0|^p (1-y))^{1/p} $$ where $0\le xy\le b\le ...
4
votes
0answers
192 views

Maximize product of sums

Let $n,k\geq 2$ be positive integers. For each $1\leq i\leq n$, let $I_i$ be a nonempty subset of $\{1,2,\dots,k\}$. Let $P_i=\sum_{j\in I_i}x_j$, and let $P=P_1\cdot P_2\cdot\dots\cdot P_n$. (For ...
1
vote
2answers
83 views

One inequality connected with the linear second order ODE

Is the following statement true? Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation $ \ddot{x}+a \dot{x}+bx=h$ with initial conditions $x(0)=u<0 , \dot{x}(0)...
0
votes
0answers
45 views

Bounding a particular ratio of sums

Let $p_i = \frac{1}{w_i + x}$ for $w_i >0$, $1\leq i \leq N$, and $x \in [\min_i w_i^{2/3},\max_i w_i]$. We wish to bound the ratio of the expected value of $w/(w+x)$ under the probability ...
4
votes
2answers
117 views

A kind of exponential concavity for polynomials?

Is there $C > 0$ such that the inequality $$ \prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right) $$ holds for all finitely supported sequences $(a_n)$ with $a_n\geq ...
3
votes
1answer
180 views

Inequality for exponential sum in Dvoretzky 1972

I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
2
votes
1answer
175 views

Computing minimum / maximum of strange two variable funcion

I want to compute $$\max_{\frac{1}{5 \theta }\leq \alpha \leq \frac{1}{2}} \left(\frac{\alpha\log (\alpha)}{1-\alpha} + \log \left( 1 - \alpha\right) + \frac{1}{1-\alpha} \cdot \left( f\left(\frac{...
-2
votes
1answer
131 views

A generalization of Chebyshev's sum inequality

From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference. Inequality: Let $y=f(x,y)$ is ...
1
vote
1answer
84 views

Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?

Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that $$u(t) \leq \psi(t) ...
0
votes
1answer
111 views

A rearrangement inequality for exponentiation function

Inequality: If $n$ be positive integer $n \ge 2$ and $a_1 \ge a_2 \ge \cdots \ge a_n \ge 0$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 0$ then $${\left(\sum_{i=1}^{n}{a_i^{\...
2
votes
1answer
189 views

Some inequalities on chain of circle packing

By my computation, I pose a conjecture as follows and I am looking for a proof: Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...
2
votes
0answers
101 views

A generalization of Bernoulli's inequality and what does it application for?

Let $a_1 \ge a_2 \ge \cdots \ge a_n \ge 1$ or $0 \le a_1 \le a_2 \le \cdots \le a_n \le 1$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 1$ then $$\left(\sum_{i=1}^{n}{\alpha_i} \right)\...
2
votes
0answers
111 views

Volume of critical points decrease under symmetric decreasing rearrangements?

If, $u: \mathbb{R}^n \to \mathbb{R}$ be a non-negative test function, i.e., $u \in \mathcal{D}(\mathbb{R}^n)$ and $u \ge 0$, then does it follow that, $$\mathcal{H}^{n}(\{s < u \le t \} \cap \{ \...
8
votes
1answer
184 views

Expectation inequality for sampling without replacement

Is the following proposition correct? $X_1, X_2, X_3$ are uniformly at random sampled from a finite set $\mathcal X$ without replacement. $f : \mathcal X^2 \rightarrow \mathbb R_{\ge0}$ is symmetric:...
0
votes
2answers
157 views

A symmetric polynomial inequality

I improve my previous question. Because this conjecture is exactly natural development of A Muirhead Like Inequality and Muirhead's Inequality so I think the conjecture is true. But I can not prove it....
1
vote
1answer
87 views

Proof of an inequality $s_m(n) \le f_m(n)$

For fixed $m = 0, 1, 2, ...$ $$f_m(k) = \prod_{j=1}^{m}(k+j).$$ Some examples of $f_m(k)$ are as following: $$f_0(k) = 1, \quad f_1(k) = (k+1), \quad f_2(k) = (k+1)(k+2).$$ The $s_m(n)$ is defined as ...
3
votes
2answers
188 views

Inductive proof of $s(n)≤n+1$

I was able to conclude, numerically, the following: $$s(n) = \sum_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum_{k=j}^n\frac{(k+1)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2}\le n+1$$ for $x\in[0,1]$. For example ...
0
votes
2answers
135 views

An inequality on length of two curves [closed]

I am looking for a proof, reference, comment of an inequality as follows: If $f(x)$ and $g(x)$ be two continuous derivative funcions in interval $[a, b]$. Such that: $f(a)=g(a)$ and $f(b)=g(b)$ $(...
4
votes
2answers
231 views

Combination power elementary symmetric polynomial inequality

Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities as follows: Inequality 1: Let $n>2$ and $1 \le m \le n$...
2
votes
0answers
154 views

In arbitrary cyclic polygon then $\sum_{i<j} x_{ij}^\alpha \ge \sum_{i<j} y_{ij}^\alpha $

I am looking for a proof of the inequality as follows: Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the circle $(O)...
7
votes
1answer
142 views

Distributing $N$ points on the sphere so that the sum of their mutual distances is maximized?

Generaliation the result in our paper for sum and similarly my previous question for product. I have a question: My question: Distributing $N$ points on the sphere so that the sum of their mutual ...
5
votes
2answers
114 views

An inequality related to area and sidelengths of a polygon $Area(A_1A_2…A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$

I am looking for a proof (or a reference) of an inequality related to a rea and the sidelengths of a polygon as follows: Let $A_1A_2...A_n$ be arbitrary polygon, then: $$Area(A_1A_2....A_n) \...
1
vote
1answer
140 views

An inequality of a cyclic polygon

I am looking for a proof of the inequality as follows: Let $A_1A_2....A_n$ be the regular polygon incribed in a circle $(O)$ with radius $R$. Let $B_1B_2....B_n$ be a polygon incribed the ...
2
votes
1answer
110 views

Non-negativity condition for special quartic

I know that a necessary and sufficient condition for the positivity of a quartic polynomial of many variables is in general difficult. I have a somewhat special case, maybe here more can be said. Let $...
8
votes
1answer
232 views

An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm

Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that $$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\ a_j & b_j & c_j \\ ...
3
votes
0answers
139 views

best-possible inequalities for hypergeometric functions

In what follows, let $n$ be a positive integer and $0<a<1/2$. I am interested in the Gauss hypergeometric functions, $_{2}F_{1}( -n, -n-a; 1-a; z)$. Notice that these are polynomials, if that is ...
0
votes
0answers
89 views

Do three inequalities as follows holds $\sum_{k=1}^n \frac{{a_k(x)}}{b_k^\alpha} >0$ and $\sum_{k=1}^n \left( \frac{{a_k(x)}}{b_k} \right)^k >0$?

This topic are some generalizations of my previous question: Let series $b_1, b_2,\cdots, b_n$ and $a_1(x), a_2(x), \cdots , a_n(x)$ so that $$b_n > b_{n-1} > \cdots > b_1 \ge 1 \; \text{...
2
votes
1answer
116 views

$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0 ?$ with $\alpha \ge 1$ and $n=1, 2,\cdots$

Could You give a poof, comment or reference for the inequality as follows: $$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$$ for all $n=1,2,3,\ldots$ and $0<x<\pi$ and $\alpha \ge 1$...
4
votes
1answer
231 views

$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$

The Fejer-Jackson inequality as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ I conjecture that the inequality as follows holds: $...
11
votes
3answers
365 views

norm inequalities

Let $p>2$. I'd like to know the best possible lower and upper bound for $\|x\|_p$ given that $x\in R^n$ and $\|x\|_1$, $\|x\|_2$, and $\|x\|_\infty$ have fixed values. It is well-known that $$\|x\...
4
votes
1answer
125 views

An inequality involving $k$-generalized Fibonacci numbers

I have worked on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, to solve completely the equation I have one complicated case and I proved ...
4
votes
1answer
100 views

Weak closedness of the extremal set to a linear inequality

Let $\mathcal{H}$ be a Hilbert space, let $p\ge 2$, and consider a bounded linear operator $ T\colon \mathcal{H}\to L^p(\mathbb R^d). $ Is the set $M=\{f\in \mathcal H\ :\ \|Tf\|_{L^p}= \|T\|\|f\|...
13
votes
1answer
399 views

A question on the sine function

The Fejer-Jackson-Gronwall inequality involving the sine function is as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ Here I ask the ...
8
votes
0answers
2k views

On the A+B=C conjecture

Could You give a comment or a reference for the conjecture as follows: Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem ...
11
votes
3answers
521 views

Steiner's inequality reference request

I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have $$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$ where $$\...
7
votes
1answer
310 views

A Muirhead Like Inequality

I am looking for a proof of the inequality as follow: Let $n$ be an integer number $n \ge 2$ and $x_1, \cdots, x_n$ and $y_1,\cdots, y_n$ are nonegative real numbers such that $(x_1,\cdots, x_n)$ ...
2
votes
0answers
96 views

An inequality related to Power sum and elementary symmetric polynomial and majorizes

Power sum and elementary symmetric polynomial Let $x_1,. . . , x_n$ be variables, denote for $k \ge 1$ by $p_k(x_1,\dots,x_n)$ the $k-th$ power sum: $$ p_k(x_1,\dots,x_n)=\sum\nolimits_{i=1}^nx_i^k =...
0
votes
2answers
80 views

On OR condition in Linear Programming with exponentially many constraints [closed]

Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
0
votes
0answers
54 views

Transition probabilities for an approximating tree for a couple of correlated stochastic processes

i apologize if the question is not at research level, in that case i'll remove and post it elsewhere. First the motivation for the question. I have two continuous stochastic processes $x(t),y(t)$ (...
3
votes
1answer
316 views

A Poincaré-type inequality: proof or counterexample

The following is a simplified version of a Poincaré-type inequality that I'm studying; I'd like to prove it (the inequality) or find a counter example. Consider a function $f:[0,1]^2\rightarrow\mathbb{...
2
votes
3answers
156 views

Algebraic inequalities on different means

If $a^2+b^2+c^2+d^2=1$ in which $a,b,c,d>0$, prove or disprove \begin{equation*} \begin{aligned} (a+b+c+d)^8&\geq 2^{12}abcd;\\ a+b+c+d+\frac{1}{2(abcd)^{1/4}}&\geq 3. \end{aligned} \end{...
2
votes
1answer
61 views

A gaussian based inequality

I came across an inequality in the paper of 'Estimation of a function with discontinuities ...' (AoS, 1998, p.1374) and tried to prove it, but could not get to the result. Some simulations also seemed ...
12
votes
0answers
218 views

Symmetric strengthening of the Cauchy-Schwarz inequality

In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle ...
41
votes
2answers
2k views

A strengthening of the Cauchy-Schwarz inequality

Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...