Questions on proving, manipulating and applying inequalities.

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88 views

Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
2
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57 views

Search for a candidate function with specific properties?

Given the following expression: $$ \mathcal{F(p,c,r,s)} = \frac{c^2 p^2 \left(s f'(s)-2 f(s)\right)^2}{4 f(s) \left(c^2 f(s) \left(c^2 p^2 f(s)+s^2 \left(r^2-p^2\right)\right)+\left(-r^2-1\right) s^4\...
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54 views

Least upper bound for a positive real sequence satisfying $|x_u−x_v|\cdot|u−v|>1$

The starting point of this question is the: IMO 1991, problem 6: Prove that for any $\alpha>1$, there exist a bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ such that, if $u,v$ are distinct ...
2
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105 views

Convex function Inequality (3 - point version)

I was reading this article on inequalities (which some of you may find useful) here. On page 7, I came across this question by Titu Andreescu, which I shall reproduce here: Question: Let f be a ...
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106 views

How to obtain the infimum of this inequalities?

Let $A$ be the family of functions $f(z)=z+a_2z^2+\cdots$ that are analytic in unit disk $D:\{z:|z|<1\}$ and $S$ is the subfamily of functions that are univalent in $D$. $R(a)$ is the subfamily of $...
2
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212 views

How prove the following inequality

Let $x,y,z\ge 0$,$x+y+z=3$,prove: $$3{{x}^{2}}(1+2y){{(1+z)}^{3}}+3{{y}^{2}}(1+2z){{(1+x)}^{3}}+3{{z}^{2}}(1+2x){{(1+y)}^{3}}\le {{\left( 3+xy+yz+zx \right)}^{3}}$$ my idea:let $p=x+y+z=3,q=xy+yz+xz,...
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90 views

Norm inequalities in a reflexive space

I am reading an article about reflexive spaces, with a specific example. The article mentions inequalities that I haven't been able to get around to. Here's the setup. The space $X = (\prod_n \...
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56 views

Inequality with $\|\cdot\|_p$ norm

Let $x_1, \ldots, x_{2m}$ be $\{0,1\}$ Bernoulli random variables, i.e. variables which takes values $0$ and $1$ with equal probability. Let $S_m$ be group of all permutations $\pi$ on $\{1, \ldots, m\...
2
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75 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: $$\frac{1}{n}\...
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32 views

An application of Minkowski iequality.

I am Reading this work on page 21. In the said page, the author reach the inequality: $$\int|\zeta\nabla u^{\beta/p}|^p\leq\Big(\frac{\beta}{\beta-p-1}\Big)^p\int|u^{\beta/p}\nabla\zeta|^p$$ By using ...
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77 views

Probability Theory: How to prove inequality $\mathbb{E}|X - m|^3 \leq \mathbb{E}|X|^3 (1 + \frac{m}{\sigma})^3$

Let's define $X$ - random variable with $F(x)$ distribution function. Also, denote $m = \mathbb{E}X$ and $\sigma^2 = \mathbb{D}X$. Suppose, that $m>0$. How to prove this inequality in these ...
2
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94 views

Azuma's inequality with high probabilistic bounds

Let $(X_n)_{n \geq 0}$ be a super-martingale, that is $\mathbb{E}[X_{n+1} \mid X_1, \dots, X_n] \leq X_n$. Let's further assume that $\Pr[|X_n - X_{n-1}| < c_n] \geq 1-\delta$. Does there exist any ...
2
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135 views

Inequalities involving regularized incomplete Gamma functions

I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
2
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157 views

Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B ...
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223 views

Proving that $|x|^p,p \geq 1$ is convex

I want to show that $|x|^p,p \geq 1$ is convex, for this i have to prove the inequality $|(1-\lambda )x+\lambda y)|^p \leq (1-\lambda)|x|^p+\lambda |y|^p $ for $\lambda \in (0,1)$ Can anyone prove ...
2
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37 views

For which minimal $k$ true is that ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$, ${a}_{i}\in {1,2,3,4,5,6}$?

I've got the following inequality, which bounds Minkowski distance. ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$ and values of ${a}_{i}\in {1,2,3,4,5,6}$ We know all ...
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66 views

Lower bound on diophantine system of inequalities with all but one non-linear constraint

I have a system of $n+1$ diophantine inequalities, in the following form: $$f_{1}(x_1, x_2, \dots, x_n) \geq 0$$ $$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$ $$\vdots$$ $$f_{m}(x_1, x_2, \dots, x_n) \...
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347 views

Find a tight upper bound of the following expectation.

I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
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226 views

How to solve systems of polynomial inequalities?

I am currently working on a project that deals with systems of inequalities and so far I have found algorithms for the basic case of a system of inequalities as well as the non-strict linear ...
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128 views

Inequality with binomial coefficient

Let $n$ be a natural number, $m\in [-n, n]$. Let $p=0,\ldots, \frac{n+m}{2}$. Show, that for all $p$, $$ {n \choose \left[{\frac{n+m}{2}}\right]}\geq \frac{2^{n+1/2}}{\sqrt{n-p/2}}. $$ Thank you for ...
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55 views

Symmetric matrices $M$ such that $x,y\geq 0$ implies $(x^t M y)^2\geq (x^t M x)(y^t M y)$

Is there a name for $n$-by-$n$ symmetric matrices $M$ such that for all $n$-dimensional non-negative-valued vectors $x,y$ we have $$(x^t M y)^2 \geq (x^t M x)(y^t M y)?$$ In particular I am ...
2
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46 views

Randomized Solution to a System of Inequalities

Given a set of $\mathbf v_i \in \{0,1\}^k$ for $i=1,\dots,n$ and a vector $\mathbf x \in [0,1]^k$, we want to decide if the following inequality holds or not: $$ \mathbf x \le \sum_{i=1}^n \alpha_i \...
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117 views

Inequality with even powers of trigonometric functions

For $m>0$, $0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that $$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...
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39 views

The sign of a sum of integrals over a probability measure

Let $\mu$ be a probability measure, $f$ a function taking values in $[0,1]$. I am trying to determine the sign of the expression $$3\left( \int f^2 d\mu \right)\left( \int f d\mu \right) - 2 \...
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138 views

Inequality involving norms.

Suppose $p,q,r \in[1, \infty)$ and ${1\over r} = {1\over p} + {1\over q}$ . How can I use Minkowski's Inequality for prove below? $$||fg||_r \le ||f||_p||g||_q$$
2
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102 views

Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
2
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121 views

Generalizing an approach to proving AMGM

This problem is Exercise 5.5.30 of "The Art and Craft of Problem Solving" by Paul Zeitz. The problem asks to use the identity $$ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) $$ to prove the AMGM ...
2
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148 views

Can anyone point me to an elegantly simple proof of Muirhead's Inequality?

I am currently finishing a project for a module I have in Mathematical Investigations. I have been looking at inequalities and ways to produce true inequalities in homogeneous symmetric form. I have ...
2
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162 views

another inequality involving complex numbers.

Let $\{z_i\}$, $i=1,2,\ldots,n$ be a set of complex numbers. Then I know that there is a set $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{\pi} \sum_{k=1}^n |z_k|. $$ However, how do I ...
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192 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: $$\int_0^11-(1-x^{c^n})(1-x)^{c-1}-[F(F(x))]^{c^{n-1}}+(c^n-c^{n-1})(1-F(x))\beta_{[F(F(x))]^{\frac{1}{c}}}\left(c^n,\frac{c-1}{c}\right)...
2
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195 views

A trigonometric inequality involving sine

Let $0<a<\pi/2,0<b<\pi/2$, $0<\lambda<1, \mu=1-\lambda$. Does anyone see a good proof of the inequality: $$\sin(\lambda a)\sin(\lambda b)+\sin(\lambda a)\sin(\mu b)\cos(b)+\sin(\...
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303 views

Two vague steps in the proof of Harnack inequality

I am reading the book Elliptic and Parabolic Equations and the proof is excerpted from page 133-136. In Theorem 5.1.3: it claims that $$C((1-\theta)R)^{-n/2}\left(\sup_{B_R}u\right)^{1-p/2}\left(\...
2
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89 views

Single solutions to an inequality

Suppose we had an inequality $ax < by < cx$ where $a,b,c,x,y \in \mathbb Z$. If we fix $a,b,c$ and let $x,y$ vary how can we find the values of $x$ for which only a single $y$ satisfies the ...
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0answers
13 views

A false identity involving $2^{\frac{1}{\zeta(s)}}$ for $\Re s>1$, from these particular values of the Riemann Zeta function and its alternating

Yesterday when I was exploring symbolic calculations $\dagger$ about specializations in $z=\frac{1}{n}$ with $n>1$ an integer, of $$\zeta(z)=(1-2^{1-z})^{-1}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^z}:=...
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33 views

Hardys Inequality on Integrals

My PDE script lists a form of Hardy´s inequality which is not the common one: Let $u\in C^\infty([0,\infty))$ and $\delta < -\frac{1}{2}$. Then: $(\int_0^\infty |r^\delta u(r)|^2 dr)^{\frac{1}{...
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41 views

Prove Inequality

I'm trying to prove the inequality $$\left(\frac{\sum_j (x_j+y_j)^p}{\sum_j (x_j+y_j)^r}\right)^{\frac{1}{p-r}} \le \left(\frac{\sum_j x_j^p}{\sum_j x_j^r}\right)^{\frac{1}{p-r}} + \left(\frac{\...
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30 views

Integration in an inequality

Does integrating on both the sides of inequality with the same upper and lower limits with respect to same variable somehow affect the inequality. I saw an example lets say, Sin x < x ,x>0 ...
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33 views

Upper bounding a sum of products

Let $a_k$ be an integer valued sequence, $a_k \in \mathbb{N}^+$ and let $b_k = \#\{i: a_i=1,\; i \leq k\}$ and assume that $b_k=o(k)$ (little o notation). How to prove that there exists a constant $...
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0answers
29 views

Inequalities in weak $L^1$ norm

I have the following lemma: Suppose that for $j=1,2,\dots,$ $g_j(x)$ is a nonnegative function on a measure space for which $\left|\left\{x: g_j (x) >s\right\}\right|<1/s$. Let $\{c_j\}$ be a ...
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22 views

prove Fisher's inequality not using matrix method .

Fisher's inequality, is a necessary condition for the existence of a balanced incomplete block design which satisfies certain prescribed conditions in combinatorial mathematics. Let: v : be the ...
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28 views

Doob's maximal inequality with stopping time

I have been searching for a version of Doob's maximal inequality with stopping time insides the time index, i.e. given $\Lambda_n$ is a positive sub-martingale and N is a stopping time is there any ...
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30 views

Probability problem related to Markov inequality

Problem Let $p$ be the probability of a person chosen at random to support Bernie Sanders. A sample is taken of $50$ persons chosen at random, each of them is asked if he or she would vote for ...
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0answers
22 views

Constant of Holder-type Inequality for Polynomial Function

Is anybody aware of an inequality in the following form $$ \Vert f \Vert_{L_p(\Omega)} \leq C(p) \Vert f \Vert_{L_q(\Omega)} $$ where $f$ is a polynomial function of degree $p$ on $\Omega \subset \...
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32 views

Prove trigonometric inequality with sin

Let $ n\in \mathbb{N}^{*},x\in \mathbb{R} $. Prove that $ sin^{2}(x)\cdot sin^{2}(2x)\cdot ...\cdot sin^{2}(2^{n}x)\leq \left ( \frac{3}{4} \right )^{n},\forall x\in \mathbb{R} $. The only result I ...
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0answers
43 views

Minimum variance, fixed mean , discrete random variable

Consider the ordered set $\mathcal{S}$ $=$ $\{0,a_i,a_2,\ldots,a_n\}$, where $a_i$ are all stricly positive real numbers and $a_i< a_{i+1}$ forall indices i. What is the random variable $X$ which ...
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0answers
17 views

Trying to find the asymptotic behaviour of an inequality involving integers

Let $m,q,v$ be integers with $m\geq 2$, and $v|q-1$. A certain result that I have which is not important for this question, holds when $$q^{\frac{m}{2}-2}(q-mv)\geq v^{m-1}. \quad (1)$$ I would like ...
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0answers
57 views

Prove that $(xy+xz+yz)^2 \leq (2x^2 + y^2)(2z^2 + y^2)$

What is the easiest way way to prove that $$(xy+xz+yz)^2 <= (2x^2 + y^2)(2z^2 + y^2)$$ holds for real $x$, $y$, $z$? I've solved it using some "advanced" mathematics, but I would like to know ...
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0answers
33 views

Does this inequality $e^{H_n}\ln[H_n(H_n+0.5)]-e^{H_n-1}(H_n+1)\le 2n$ Hold?

Valid for all values of $n\ge 1$ Does this inequality hold? $$e^{H_n}\ln[H_n(H_n+0.5)]-e^{H_n-1}(H_n+1)\le 2n$$ Where $H_n=\sum_{i=1}^{n}\frac{1}{i}$ Let $A=H_n$ I simplified to $$\ln(A(A+0.5))...
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0answers
16 views

Terminology for asserting truth of equality/inequality based on symbolic equalities/inequalities

This may seem silly, but I am curious about algorithms used to computationally assert the truthiness (true, false, or unknown) of symbolic statements subject to a set of inequality constraints, for ...
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0answers
54 views

Is it true for $a,b,c$ positive reals that $\frac{a}{\sqrt{a^2+2b^2}}+\frac{b}{\sqrt{b^2+2c^2}}+\frac{c}{\sqrt{c^2+2a^2}} \ge \sqrt3$

Is it true for $a,b,c$ positive reals that $$\frac{a}{\sqrt{a^2+2b^2}}+\frac{b}{\sqrt{b^2+2c^2}}+\frac{c}{\sqrt{c^2+2a^2}} \ge \sqrt3$$ My thoughts: The LHS is equal to $$\frac{a^2}{\sqrt{a^2(a^2+...