Questions on proving, manipulating and applying inequalities.

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3
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328 views

Trigonometric inequality proof

Can anyone help me in proving that $$\cos\theta > \frac{\left(x^a\cos\theta-(x-1\right)^a\cos\frac{\ln x\theta}{\ln(x-1)})\cos(\theta+\gamma)}{\cos\gamma},$$ where $a<1$, $x\in \mathbb{N}$, and ...
3
votes
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680 views

A special case of Young's inequality for convolutions

The problem: Suppose $f,g\in L^1(\mathbb{R})$. Let $x\in \mathbb{R}$ and $\phi_x(y) = f(y)g(x-y)$. Show that for almost all $x$, $\phi_x$ is integrable. For such $x$ let $\psi(x) = ...
3
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206 views

Inequality of ODE solutions

Says I have two (scalar) ODE: $u' = f(u,t)$ and $v' = g(v,t)$ where Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and ...
2
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28 views

Prove $\sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$ for positive $a,b,c$

For $a,b,c >0$, prove $$\sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$$ My notation $$\sum_{cyc}a^2b= a^2b+b^2c+c^2a$$ What I try: 1. Using C-S ...
2
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55 views

Prove $\sum_{cyc}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$ for positive $x,y,z$

$x,y,z > 0$, prove $$\sum_{\text{cyc}}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$$ While this inequality can be proved by brute force, the elegant ...
2
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46 views

Generalized weighted mean inequality

Let ${p_{1}},{p_{2}},\ldots,{p_{n}}$ and ${a_{1}},{a_{2}},\ldots,{a_{n}}$ be positive real numbers and let $r$ be a real number. Then for $r\ne0$ , we define ...
2
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67 views

$\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$

Let $a$, $b$ and $c$ be positive numbers. Prove that: $$\frac{a^3}{13a^2+5b^2}+\frac{b^3}{13b^2+5c^2}+\frac{c^3}{13c^2+5a^2}\geq\frac{a+b+c}{18}$$ A big problem around $(0.785, 1.25, 1.861)$. In ...
2
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57 views

Inequality involving fourth powers .

I have been into inequalities lately and I am stuck with this. I used a famous inequality at first $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge 3 (\frac{a^4+b^4+c^4}{3})^{\frac{1}{4}}$. From this ...
2
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121 views

When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there exist some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, ...
2
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28 views

Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
2
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34 views

$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}$ inequality is used to prove the theorem

In the book Randomized Algorithms from Motwani and Raghavan, it is stated in page 44 that $$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}.$$ ...
2
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32 views

Equality in Hardy's inequality via Hölder's

I'm working on Exercise 3.14 in Rudin's Real and Complex Analysis. I was able to answer part (a): that for real $p$ satisfying $1<p<\infty$, for every function $f$ in $L^p(0,\infty)$, when $F$ ...
2
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15 views

Plot complex inequalities $|z^2| > Im(z)^2$

I need to draw the set of all complex numbers, which satisfy the following inequality: $|z^2| > Im(z)^2$ This is what I've already done: $|z^2| > Im(z)^2$ $|z|^2 > Im(z)^2$ - use $z = a ...
2
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0answers
28 views

How does this inequality imply this one?

I am having a little trouble understanding this part of a proof. There is an integral $\text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx$ Now, $\text{J}_0 = \frac{\pi ^3}{24} $ The part of ...
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16 views

Requesting for details of Grun's result on odd perfect numbers

This post is an offshoot of this earlier MSE question. According to this Wikipedia page: Grun $1952$ proved that the smallest prime factor $p_1$ (of an odd perfect number $N$) is $< ...
2
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48 views

Is this trigonometric expression always strictly positive?

Let us define a function $f(k,n)$ by \begin{equation} f(k,n)=n \left (\cos\frac{k\pi}{n}\right) \left(1-\cos\frac{k\pi}{n}\right) - \sin \frac{k\pi}{n} \end{equation} Where $\frac{k}{n}$ is ...
2
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0answers
38 views

Cases in which a certain inequality holds true.

I previously asked this question on this forum, and have been demonstrated counterexamples to the claim that $|a| > |b|$ implies $\big|\frac{b+b^{2}}{a+a^{2}}\big| < 1$, which I had previously ...
2
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32 views

Finite sequence created by reducing $n$ with each prime under $n$ ends in $0$?

Given $n$ a fixed integer we constuct the following sequence: $a_0=n$, $a_i=\lfloor \frac{a_{i-1}(p_i-1)}{p_i}\rfloor$. For what values of $n$ do we have $a_{\pi(n)}=0$? Computer calculation shows ...
2
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60 views

Elementary-Looking Inequality on n Complex Numbers

Let $z_1,z_2,\ldots,z_n$ be complex numbers. Is it true that $\displaystyle\sum_{1\le i,j\le n} |z_i+z_j| \ge \displaystyle\sum_{1\le i,j\le n} |z_i-z_j|$? I know the inequality holds for reals and ...
2
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28 views

Choosing three integers to satisfy an equation under a specific condition

Find three integers $(a,b,c)$ such that: $x*a + y*b + z*c = a + b$ only when $x = 1, y = 1, z = 0$ where $x, y$ and $z$ can be chosen as any non-negative integers. For example, choosing $a = 1$; $b = ...
2
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0answers
58 views

I am trying to show an inequality involving the product of three inner product terms

Define the inner product $\langle\cdot,\cdot\rangle$ for continuous functions defined on $[0,1]$ as: $$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$ where $\rho$ is a real number. I ...
2
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0answers
35 views

Prove that $(n-1)!S_m\geq (n-m)!m!P_m.$

If $a_1, a_2,\cdots a_n$ be all positive rationals such that $S_n=a_1^m+a_2^m+\cdots +a_n^m$, $P_m=\sum a_1a_2\cdots a_m$ (the sum of products m taken m at a time). Prove that $$(n-1)!S_m\geq ...
2
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36 views

Proving $n\cdot\bigg(\sum_{i=1}^{n}|x_{i}|^{2}\bigg) \leqslant (n\cdot|x_{n}|)^{2}$

May this be proved by induction? Let $x \in \mathcal{R}^{n}$, where $|x_{n}| \geqslant |x_{i}|$, $i \neq n$. In other words, $|x_{n}|$ is the maximal element of $x$. Then, $$ ...
2
votes
0answers
51 views

Inequality with three variables

Let $a,b,c\ge 0$,show that $$\sqrt{a^3+2}+\sqrt{b^3+2}+\sqrt{c^3+2}\ge \sqrt{\dfrac{9+3\sqrt{3}}{2}(a^2+b^2+c^2)}$$
2
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0answers
34 views

Doob's $L^p$ inequality - Case $p = 1$

I have found in the wikipedia page following generalisation of Doob's so-called $L^p$ inequality, for general nonnegative submartinagles $X_s$: $$E[\sup_{0 \le s \le T} X_s] \le \frac{e(1 + ...
2
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0answers
46 views

Solving Higher Order Polynomial Inequalities in Two Variables

Could someone please point me to a solver (software) and also some techniques to solve equations of the kind shown below? $$ x^{3}y+ay^{3}+by^{2}+cy+dx+exy+fxy^{2}\geq0 $$ Here, $x,y$ are the ...
2
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30 views

Inequality of expansion of $\| \ \ \|$

The following is from a proof of a paper Distributed subgradient Methods for Multi-Agent Optimization, Nedic & Ozdaglar - Lemma 5 Note: ...
2
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0answers
47 views

Is Stirling's Approximation used here, to prove the asymptotic inequalities?

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, ...
2
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0answers
14 views

Prove this inequality $|t_{1}-t_{2}|<2mp\left[1-(mp)^{-\frac{p}{p+1}}\right]$

Let $$f(x)=x+\dfrac{m}{x^p},m,p>0$$ if $t_{1},t_{2}$ such $$f(t_{1})=f(t_{2})=mp,t_{1}\neq t_{2}$$ show that $$|t_{1}-t_{2}|<2mp\left[1-(mp)^{-\frac{p}{p+1}}\right]$$
2
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33 views

Problem related matrix and counting

I have a following matrix related problem: Let $F$ be a $n \times n$ discrete Fourier matrix defined as $F_{j,k} = \frac{1}{\sqrt{n}}exp{(\frac{i 2\pi jk}{n})}$, for $0 \leq j,k \leq n$, where $i^2 ...
2
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0answers
42 views

Khintchine inequality (question about Holder inequality)

I'm reading a proof of a Khintchine inequality : Let $(r_{1}, \dots , r_{n})$ be iid random variables with $P(r_{i} = \pm1) = \frac{1}{2}$. Let $f = \sum\limits_{j=1}^{n}a_{j}r_{j}$, where ...
2
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0answers
394 views

Integral inequality - lower bound on $L^1$ norm.

I was wondering if one can make an estimate of form: Assume $f\in C^\infty(\overline{\Omega})$ where $\Omega$ is a bounded domain in $\mathbb{}R^d$. Is there a constant $C>0$ independent of $f$ ...
2
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0answers
27 views

Dospinescu's Inequality

Consider $a_1,\dots, a_n$ be positive real numbers such that $a_1 a_2 \dots a_n = 1$. Prove that $$n^n \prod_{i=1}^n \bigg ( 1 + a_i^n \bigg ) \geq \bigg( \sum_{i=1}^n a_i + \sum_{i=1}^n \frac 1 a_i ...
2
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0answers
115 views

(algorithms) Show that in any base b >= 2, the sum of any three single-digit numbers is at most two digits long

So, I'd like someone to review my 'proof' and pick on it for incompleteness, and state how it could be improved. The question (reviewing algorithms) asks, "show that in any base b>=2, the sum of any ...
2
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0answers
32 views

A inequality of area in Complex analysis

Let $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}a_{n}z^n$ is a univalent holomorphic function in$B(0,1)\setminus\{0\}$,How to prove that$$\sum_{n=1}^{\infty}n|a_{n}|^2\leq1.$$
2
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0answers
29 views

Doubt about inequalities

My original question is if $f\in \mathcal{S}_{\alpha_1}^{\beta_1},\: g \in \mathcal{S}_{\alpha_2}^{\beta_2}$, where does $(f\cdot g) (x)=f(x)g(x)$ belong ? where $\mathcal{S}_\alpha^\beta$ is defined ...
2
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0answers
38 views

Around an inequality

I have a very general question, hopefully not too general. Assume that we have real numbers $a_{ij}, b_{ij}$ $(1 \leq i, \: j \leq n)$ such that $-1 \leq a_{ij}, b_{ij} \leq 1$ for all $i,j,$ for ...
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0answers
57 views

Is the following inequality involving matrix exponential true?

Let $X$ and $L$ be real positive definite matrices. $$\operatorname{Trace}(X^{-1}(X - e^{\log(X) - L})^2) \leq \operatorname{Trace}(XL^2)$$ where the exponential and the log are matrix exponential ...
2
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0answers
44 views

Question about the assumption of a version of Grönwall's inequality.

According to Wikipedia, A version of Grönwall's inequality for the integral of continuous functions is the following: Let $I$ denote an interval of the real line of the form $[a,\infty)$ or $[a,b]$ ...
2
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0answers
108 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
2
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0answers
63 views

Is my logic on general Proof-Solving techniques correct?

I've just recently started working through proofs for what's really the first time in my life. Throughout high school, and thus far in college I've never really had to prove things too often, and if I ...
2
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0answers
38 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, I try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
2
votes
0answers
24 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
2
votes
0answers
63 views

Upperbound for $\sum_{i=1}^n\frac{1}{x_i^2}$?

Suppose that $x_i>0$, $i=1,\ldots,n$. I'm looking for an upperbound (doesn't have to be particularly tight) of $\sum_{i=1}^n\frac{1}{x_i^2}$ in terms of some symmetric function of ...
2
votes
0answers
58 views

One-sided Bound on Sum of Fourth Moments

I'm interested in methods for proving one-sided bounds of the form $$ \mathbb{P}[\frac{1}{n}\sum_{i=1}^n X^4_i \geq 3+t]\leq Ce^{-nt} $$ where $X_i$ are standard normal random variables. I've run a ...
2
votes
0answers
49 views

Is this system of inequalities (and equality) tractable?

I have some real parameters here. The $\mu_i$ - for $i=1,2,3,4,5$ - are 'convex coefficents' in that $\mu_i\geq 0$ and $\sum_{i}\mu_i=1$. The $x$ and $z$ are such that $x^2+z^2\leq 1$. The ...
2
votes
0answers
21 views

An inequality involving an exponential rate of sum

I'm having trouble understanding the conclusion in the proof of Cramér's Theorem in $\mathbb{R}^d$ in the book by Dembo/Zeitouni: We have the following: $\delta>0$ is fixed, $B_{y,\delta}$ is the ...
2
votes
0answers
28 views

Optimization by Symmetry?

Let $$f(x,y,a,b) := \frac{xa+yb}{\sqrt{xa^2+yb^2}},$$ where $x,y,a,b$ are all positive. Define $$g(a,b) = \min_{x+y=1,\,x,y\ge 0}f(x,y,a,b).$$ How would one solve for $g(a,b)$? I have solved this by ...
2
votes
0answers
37 views

Schwarz Inequality of function from upper half plane to disc

So I've been working on this problem and I have everything nailed down (I think) except for the very end. In particular I get a bound, but I can't seem to reduce it down to the one the question is ...
2
votes
0answers
82 views

Is the Schwarz inequality a special case of the Cauchy-Schwarz inequality?

Given two vectors $\mathbf{x},\mathbf{y}$ in $\mathbb{R}^n$, we all know that:$$\left | \mathbf{x}\cdot\mathbf{y} \right | \le \left \| \mathbf{x} \right \| \cdot\left \| \mathbf{y} \right \|$$ ...