Questions on proving and manipulating inequalities.

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2
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84 views

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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82 views

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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51 views

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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52 views

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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34 views

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 ...
2
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61 views

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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63 views

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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70 views

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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211 views

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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98 views

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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20 views

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.
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103 views

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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183 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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154 views

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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77 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 ...
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52 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) ...
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51 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 ...
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71 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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83 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 ...
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82 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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51 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, ...
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94 views

Geometrical Inequality

Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals $AC$ and $BD$ intersects at $E$. If the shortest height of the triangle $ACD$ equals the radius of the incircle of the triangle ...
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66 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 $n > 1$, $x \ge 1$: ...
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30 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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73 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 ...
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263 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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83 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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117 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 ...
2
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130 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 ...
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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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63 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) ...
2
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266 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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137 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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116 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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0answers
52 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 ...
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45 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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111 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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128 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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97 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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112 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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146 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 ...
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136 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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167 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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0answers
173 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 ...
2
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0answers
259 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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0answers
86 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 ...
2
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0answers
213 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 ...
2
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0answers
272 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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11 views

Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...