Questions on proving and manipulating inequalities.

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Miklos Schweitzer 2013, Strong lower bound on sumset $|A+qA|$,

Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: $$|A+qA|\ge (q+1)|A|-C_q.$$ This is a problem from ...
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Proving a quantity negative.

For $j\in{1,2,3}$ let $x_j,y_j \in R$ be nonzero and let $v_j=x_j+y_j$. Suppose that following holds: $$x_1x_2x_3=−y_1y_2y_3 \quad \text{and} \quad x^2_1+x^2_2+x^2_3=y^2_1+y^2_2+y^2_3$$ nd that ...
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Prove the inequality for composite numbers

Is it true that $c_m+c_n$ $>$ $c_{m+n}$ for all $m$, $n$ $\in$ $\mathbb{N}$? Though the result seems true, I can't get a solution. Even the bounds on $c_n$ obtained from Prime Number Theorem ...
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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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How to prove or disprove this statement?

For $n\geq1$, $$\int_{0}^{\pi/2}(\theta \sin\theta)^{n+1}d\theta>\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$$ . It is hard to find $\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$ so I have no ...
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85 views

Minimum value of an algebraic expression

What is the minimum value of ($x,y,z$ are positive): $$\sum_{cyc} \frac{(x+y)^2}{x^2+6 x y+5 y^2+4 y z}$$ With the values I have tried it does seem to be smallest has been $\frac34$. However, I can't ...
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116 views

Proof of integral inequality

How does one prove (without making use of any approximations whatsoever) the following inequality: $$\int_1^2 \left(\ln(x)\right)^{2013}dx\leq\dfrac{1}{2^{2013}}.$$
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Proving AM-GM via $n \cdot (a_1^n + a_2^n + \dots + a_n^n) \ge (a_1^{n-1} + a_2^{n-1} + \dots + a_n^{n-1}) \cdot (a_1 + a_2 + \dots + a_n)$

I want to prove the arithmetic–geometric mean inequality. To prove that, I need the following inequality: Suppose that $n$ is an integer which is greater than or equal to $1$ and $a_1, a_2, \dots, ...
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Normal sequences and Montel's Theorem

I am currently stuck on an exam question involving normal sequences and Montel's theorem: Give two examples of non-constant normal sequences one in the $(a)$ unit disk $\mathbb{D}$ and one in $(b)$ ...
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Inequality similar to Hoeffding

I have a coin with heads probability $p_1$. I toss it $n_1$ times. Let $\hat{p}_1$ be the empirical heads probability. Then we know from Hoeffding that $$P\left( \left|\hat{p}_1-p_1 \right| \geq ...
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Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
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Muirhead's Inequality (software?)

I just started learning about inequalities: Schur's, Karamata's, Muirhead's, etc... They are beautiful but it seems that in the case of more than two variables, some of the computations become a ...
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39 views

Solving an inequality involving sum of floors

Suppose I want to find $t_{critical}(u)$, the least $t\in\mathbb{R}^+$ for a given $u\in\left(0\ldots\dfrac{1}{s}\right]$ satisfying $$f(t)=\lfloor rt\rfloor x+\lfloor s (t-u)\rfloor y + y > h$$ ...
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When this inequality true?

If $a$ and $b$ are non-negative integers and $c$ and $d$ are non-negative real numbers, for what values is the following inequality true? $\log((a+b)!) - \log(a!b!) \ge(a+b) \log(c+d) - (a \log(c) ...
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Showing that a logarithmic inequality holds

Given $0 < x_1 < x_2 < x_3 < x_4 < 1$, how can I show that the following inequality holds: $$ \frac{1}{R(x_1, x_3)}+\frac{1}{R(x_2, x_4)}<\frac{1}{R(x_1, x_2)}+\frac{1}{R(x_3, x_4)} ...
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Help me find a less tedious way to prove this theorem about inequalities

I'm currently working through Spivak's Calculus and am currently in Chapter 1. So far, the book has covered commutativity, associativity, existence of an inverse and existence of an identity for ...
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91 views

Apostol Limit Proof

This is an interesting proof of the product limit law. I can see the squeeze theorem but how do you work out the step when applying the triangle-CS inequality with norms? Will this work for any $n$ ...
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How find this maximum and minimum of this $\sum_{k=1}^{n-1}(x_{k}-a)(x_{k+1}-a)$

let $a$ is give positive numbers,and $n$ is give positive integer number,and such $$x_{1}+x_{2}+\cdots+x_{n}=na,x_{i}\ge 0,i=1,2,\cdots,n$$ Find this function maximum and minimum ...
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Name of theorem?

I am trying to understand a proof which uses the following statement without further explanation, so I am wondering if this is a well known theorem? For the unit ball $B$ with radius $r>0$ and the ...
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84 views

Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
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Integral Inequality Question: Effect of Exponents

Is it true that, if $\alpha q < (1-\alpha)q$ then $\int|k(x,y)|^{\alpha q}dy \leq \int|k(x,y)|^{(1-\alpha)q}dy$, where $k(x,y)$ is a positive measurable function?
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Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
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necessary and sufficient conditions for the following inequality to be hold

Let $p=(p_n)$ is a sequence of nonnegative numbers with $p_0>0$ s.t $P_n= \sum_{k=0}^{n}p_k \rightarrow \infty$ as $n \rightarrow \infty$ and let $t\in(0,1)$. find necessary and sufficient ...
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Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
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A probability inequality

Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume $X_1,X_2,Y_1,Y_2$ are four random variables. Assume $X_1$ and $X_2$ are independent. Is it necessarily true that ...
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Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
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Inequality: double jensen, is this correct?

Imagine I have an exponent $a=bc$ where $b>1$ and $0<c<1$, the product $a=bc\in \mathbb{R}$ might be greater, equal or less than 1. Then for a measurable set with measure 1. Can I say the ...
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Upper bound for tail of binomial expansion

Let $P,R,T$ be integer constants with $PR$ much greater than $T$. Suppose I flip a coin $PR$ times, each time (independent of other times) getting heads with probability $1/P$. The probability that I ...
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Should a “good” equation divide the plane?

At the question Is there any equation for triangle? (MSE) the answer given by Henning Makholm received the most upvotes. Therefore let's define the following triangle function $H$ with (a,b,c) the ...
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What is $iav-\log(v)$? Any series expansion or inequality for it?

I am investigating the integral of this question here where \begin{equation} \frac{\exp(i a v)}v=\frac{\exp(i a v)}{\exp(\log(v))}=\exp(iav-log(v)) \end{equation} where I am interested in the ...
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Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
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Prove $\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 + b^2}\right) \ge 9$

If $a,b,c \in \mathbb{R^+}$,then prove that the following inequality holds: $$\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 ...
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Inequalities on Matrix Minimax?

Suppose I have a matrix $M$. How can I get a good bound on the minimax quantity $$ \min_{i}\max_{j}M_{ij} $$ or variations thereof? Links to literature would be greatly appreciated. EDIT: I ...
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Alternative notation to 'y = max(lower bound, min(x, upper bound))'

I was wondering if there is a more succinct way of writing \begin{equation} y = \left\{ \begin{array}{l l} \text{lower bound} & x < \text{lower bound} \\ \text{x} & ...
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198 views

Weak/Variational Gronwall type inequality

I came across the following weak differential inequality while looking through F.Otto's paper on $L^{1}$ contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - ...
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Gagliardo Nirenberg Sobolev inequality

Assume that $f$ satisfies the equality in the Gagliardo Nirenberg Sobolev inequality for the best constant. What can be said about $f$?
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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 ...
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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) ...
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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 ...
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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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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 ...
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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 ...
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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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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, ...
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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 $n > 1$, $x \ge 1$: ...
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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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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 ...
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317 views

Bounds for the exponential integral

In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is found that $$ \frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
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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 ...
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113 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 ...