Questions on proving and manipulating inequalities.

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38 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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122 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? $$||fg||_r \le ||f||_p||g||_q$$
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95 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 ...
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106 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 ...
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137 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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128 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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161 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: ...
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169 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 ...
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242 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 ...
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187 views

Bounding function from below

I would like to obtain a bound from below for the function $$f(p) = \frac{pe - (1-p)^{d+1}v}{p^{v-1}}$$ subject to $0 < p \leq 1$ and $e,v,d > 0.$ The usual method is to check the boundary ...
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260 views

Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq ...
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82 views

Is my proof wrong?

Previously on an answer this, I received many downvotes, So I assume there's something wrong with it and no one did explain the reason, So I am asking a Question to it,I want if someone can tell me if ...
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42 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
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17 views

Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
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76 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
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53 views

Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
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91 views

How to prove this inequality $b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$

let $a,b,c>0$,and $$abc=1$$ show that $$b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$$ since $$b(a-1)(c-1)=b(ac-a-c+1)=abc-ab-bc+b=1-ab-bc+b$$ so we only prove $$3-2(ab+bc+ac)+a+b+c\le 0 $$ oh,this ...
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40 views

Inequality among trigonometric sums of normal random variables

This is an inequality used in a proof which I do not know how to prove. $$\left(\sum_{k = 2^j +1}^{2^{j+1}} \frac{\sin(k\pi t)}{k}G_k\right)^2 \leq \left|\sum_{k = 2^j +1}^{2^{j+1}} \frac{e^{ik\pi ...
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57 views

Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
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52 views

Arithmetic Mean and Geometric Mean Question, Guidance Needed

I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff ...
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83 views

Practicing the arithmetic-geometric means inequality

I am struggling with learning the AM-GM Inequality that is considered vital to know for math olympiads, contests, etc. I just don't really know when to use it, when it is necessary to use, the purpose ...
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57 views

Almost Jensen's Inequality

Let $a,b$ and $c$ three positive reals numbers such that $abc=1$. Define the function $f$ by $f(x)=\frac{^1}{1+(n-1)x^n}$ where $n$ is a positive integer. Prove that ...
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24 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
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35 views

Decomposition of polynomials and inequality

This was asked in comment here by @23rd : If $f$ is a polynomial with $\deg f=n\ge2$, then there exist polynomials $g$ and $h$, such that $$f(x)=2xg(x)−h(x)$$ $$\deg g\le n−1, \quad \deg h \le ...
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95 views

Inequality with size of sets

Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1}+p_2+\cdots+p_k= 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ ...
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20 views

Do these inequalities make sense?

I have two sets of inequalities and i just want to know if they are correct. The parameters $\mu, K, d_1, \sigma_1,\sigma_2$ and dependent variables $H,F$are positive. Also $\sigma_2>\sigma_1$. ...
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25 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
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30 views

Can we prove this inequality in another way?

As explained here, I've managed to prove the following inequality: $\sigma(n)\geq\sqrt n(d(n)-2)+n+1$. This can be proved easily in two cases (one for $n$ being a perfect square and one for otherwise) ...
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26 views

Radical Inequality related to three variables

Prove the following. $$\sum_\text{cyclic}\sqrt[4]{\dfrac{(a^{2}+b^{2})(a^{2}-ab+b^{2})}{2}}\leq\dfrac{2}{3}\left(\sum_\text{cyclic}\dfrac{1}{a+b}\right)\left(\sum_\text{cyclic}a^{2}\right)$$
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52 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
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27 views

max value of difference of square roots of sums

Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the ...
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136 views

An number theoretic inequality

Prove this inequality : $$\displaystyle \prod_{i\le \left\lfloor{\frac{n-1}{2}}\right\rfloor}\left\lfloor{\frac{\left\lfloor{\frac{n-1}{2}}\right\rfloor}{i}}\right\rfloor\le ...
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32 views

How find this maximum $\sum_{i=1}^{2000}\left(\frac{x^{2000}_{i}}{\sum_{j=1}^{2000}x^{3999}_{j}- i\cdot x^{3999}_{i}+2000}\right)$

Question: let $x_{1},x_{2},\cdots,x_{2000}$ be real numbers,and such $x_{i}\in [0,1],i=1,2,\cdots,2000$.and define $$F_{i}=\dfrac{x^{2000}_{i}}{\displaystyle\sum_{j=1}^{2000}x^{3999}_{j}- ...
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18 views

I want to solve these inequalities with respect to $f$

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
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46 views

How prove this inequality $\sum_{cyc}\frac{1}{(ka+(k+1)b+(k+2)c)^2}\le \frac{1}{(k+1)^2(ab+bc+ac)}$

Inequality: Let $a,b,c\ge 0$. and $k\ge\dfrac{\sqrt{21}-3}{3}$ show that $$\sum_{cyc}\dfrac{1}{(ka+(k+1)b+(k+2)c)^2}\le \dfrac{1}{(k+1)^2(ab+bc+ac)}$$ This problem is from: ...
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52 views

Is there an inequality such that $\sum_{k=1}^{n}a_{k}b_{k}\leq \frac{1}{n}(\sum_{k=1}^{n}a_{k})(\sum_{k=1}^{n}b_{k})$?

Is there an inequality such that $$\sum_{k=1}^{n}a_{k}b_{k}\leq \frac{1}{n}(\sum_{k=1}^{n}a_{k})(\sum_{k=1}^{n}b_{k})$$ with some restrictions on $a_{k}$ and $b_{k}$?
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42 views

Prove an inequality with two sigma notations

I am stuck at this inequality question that I cannot solve. The question writes: Given a list of numbers k0, k1, k2, ...,kn that k1+k2+...+kn=0, and k0=0 Prove: I find the (j-i) part extremely ...
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41 views

hard inequalities

I have to find real $x$ that satisfy the equation: $\dfrac{x^7}{7} = 1+10^{1/7}x(x^2-10^{1/7})^2$ I saw that the way is to look for solution of the form: $x = a^{1/7}+b^{1/7}$. my question is: how ...
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39 views

What are the similarities and differences in solving equations and inequalities?

What are the similarities and differences in solving equations and inequalities?
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22 views

What conditions must satisfiy a positively skewed density function to ensure that median is greater than mean

We are collecting environmental Air Quality data. When we validate data, we always plot ECDF and compute basic statistics and percentiles. Our experimental distributions are far away from normality. ...
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20 views

Finding a derivative through the definition

Let $f$ be a function of domain $\mathcal D$ and codomain $\mathcal C$, both subsets of $\mathbb R$, and $\mathcal D_{\mathrm{cl}}$ the set of cluster points of $\mathcal D$. I want to find $f'$ by ...
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36 views

Eigenvalue Inequality Involving Hermitian Positive semi-definite matrices

I am trying to determine if an inequality holds for Hermitian matrices $A$,$B$, and $C$ of the same dimension. Let $A$ be positive definite, $B$ and $C$ be positive semi-definite, and $B-C$ positive ...
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37 views

Prove this statement about inequalities

Can someone help to prove this. For $n$ and $\{a_{11},\dots,a_{nn}\}$, if we know that $a_{ij}$ is either $0$ or $1$ or $-1$, and further assume that the following inequality system on $\{b_n|b_n\in ...
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22 views

What does it mean by First Order Inequality

As said on the title, sorry for my newbie-ness in the terms.
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17 views

Show for any $s \in \mathbb{R}$ that $C_1(1+x^2)^s \le (1+x)^{2s} \le C_2 (1+x^2)^s$.

Let $x \in \mathbb{R}$, then for any $s \in \mathbb{R}$ we must show that there exist $C_1,C_2 \in \mathbb{R}$ such that $$C_1(1+x^2)^s \le (1+x)^{2s} \le C_2 (1+x^2)^s.$$ It seems pretty obvious ...
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30 views

Want to prove an inequality of two norms in a Hilbert space

So here is my problem, Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$. Moreover we have the ...
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39 views

One-Sided Bivariate Chebyshev Inequality

Let $X$ and $Y$ be random variables with finite means $\mu_X$ and $\mu_Y,$ finite variances $\sigma_X^2$ and $\sigma_Y^2,$ and correlation $\rho.$ Let $A$ be the event that $X \leq \mu_X + k\sigma_X$ ...
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38 views

Relation between determinant and L1 norm

Recently, I have coped with a problem about the relation between determinant of positive definite matrices and their L1 norm. More specifically, assume that $\Sigma_{1}$ and $\Sigma_{2}$ are two ...
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31 views

Can proof by contradiction and counterexample by used at the same proof?

Here is a part of a theorem: If $\alpha>1$ and $x\ge-1$ then $(1+x)^\alpha \ge 1 + \alpha x$ I was wondering if I could use proof by contradiction and counterexample at the same time. Assume ...
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39 views

Solving nonlinear matrix inequality - transformation to LMI

I have a nonlinear matrix inequality problem where $A,B,C$ and $M$ are known and T is unknown and I would like to find $T$ that satisfies $\begin{bmatrix} T^T M T + A & B \\ B^T & ...