Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

1
vote
0answers
58 views

Is this an increasing function?

Let $d,N_1,N_2\in\mathbb{Z}$ with $N_2\leq N_1(N_1-1)$, $d\in\{1,\ldots,N_2\}$ and $d\mid N_2$. Let $p_1(d)=N_1^2-d(N_1+3)-d^2$, and $p_2=27N_2+2N_1^3+(-18N_1-3N_1^2)d+(9-3N_1)d^2+2d^3$. Let ...
1
vote
0answers
185 views

Proof of Karamata Inequality/Hardy-Littlewood Inequality.

Can anyone provide the proof for the Karamata Inequality?
1
vote
0answers
195 views

Bound on Bessel function of the first order

Let $I_1(z)$ be the Bessel function of the first order with purely imaginary argument. Can we explicitly bound $I_1$ on $[0,x]$, where $x>0$ is a real number in terms of $x$?
1
vote
0answers
213 views

General method to bound a power series

Let $x$ be a small real number, say $1/10 <x< 3/10$. Let $(a_n)$ be a sequence of integers with $\vert a_n\vert \leq 3^n$. How can I find a positive lower bound for $$\left\vert\sum_{n=1}^\infty ...
1
vote
0answers
46 views

Lower bounds for holomorphic functions on annuli with explicit bounds on their power series

Let $f$ be a holomorphic function on $\mathbf{C}$ and consider its restriction to the annulus $X=B(0,1) - B(0,3/4)$ in the complex plane. (Here $B(0,r)$ is the open disc with radius $r$ around $0$.) ...
1
vote
0answers
85 views

Solving a non-linear inequality related to geometric Brownian motion

Consider the non linear inequality $$\sum_{i=1}^{n}a_{i}u^{\sum\limits_{j=1}^{i}y_j} > c$$ $$y_j \in \{0,1\}, j=1,2,\dots,n$$ $$a_i \in \mathbb{R}, i=1,2,\dots,n$$ $$n \in \mathbb{N}, u>0, c ...
1
vote
0answers
125 views

Inequality for Trigonometric Polynomials

Problem statement: Define $p(t) = \sum\limits_{j=-N}^{N}c_{j}e^{ijt}$ be a real-valued trigonometric polynomial. Suppose there exists an $x_{0}\in\mathbb{R}$ such that $p(x_{0}) = \|p\|_{\infty}$. ...
1
vote
0answers
139 views

Inequality involving KL divergence

Following is a part of an answer which was not resolved when I tried to answer a question in mathoverflow. I thought it would be nice to discuss that here. Let $P$ and $Q$ be two distinct ...
1
vote
0answers
83 views

Conjecture:$ \forall x , \exists m,n$, $ x<m<n $ and make $\pi(p_{m}+m) - m > \pi(p_{n}+n) - n$

$p_i$ is the $i^{\rm th}$ prime. $\pi(x)$ is prime counting function. Firstly, I think that Prime gap inequality holds true for any $i>0$: $p_{i+1} - p_{i} \leq i$. Very often, $\pi(p_{m}+m) - m ...
1
vote
0answers
83 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 ...
0
votes
0answers
17 views

prove $\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$

If $a,b,c$ are positive real numbers,prove:prove $$\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$$ Additional info:We can use AM-GM and Cauchy ...
0
votes
0answers
22 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
0
votes
0answers
19 views

Deriving inequalities featuring bounded variables

I have a model which fits certain thermodynamic data, of the form $$y = \frac{x}{ 1 + (a - 1)x} + bx(1 - x) \quad a,b \in \mathbb{R} \quad 0 \leq x \leq 1$$ Thermodynamics dictate that ...
0
votes
0answers
19 views

Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
0
votes
0answers
33 views

A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
0
votes
0answers
39 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
0
votes
0answers
25 views

How to solve the inequality: $\prod_{k=1}^N\left(x^k-k^2\right)\gt0$

Given the inequality: $$\displaystyle\prod_{k=1}^N\left(x^k-k^2\right)\gt0$$ how can I solve it? I suppose there is a difference if $N=2n$ or $N=2n+1$ with $n\in\mathbb{N}$, but I'm unable to find a ...
0
votes
0answers
65 views

Bound for this integral

Using the fact that $$\sqrt{(1+y^2)} - \sqrt{(1+x^2)} \geq \frac{x}{\sqrt{1+x^2}}(y-x)$$ for each $x,y\in \mathbb{R}$. We need to show that $$L(k)- L(h) \geq \int_a^b \frac{h'}{\sqrt{1+{h'}^2}} ...
0
votes
0answers
61 views

Prove the inequality $\sum_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $

A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} ...
0
votes
0answers
46 views

Little o notation inequalities involving $n^{\log n}$

Apologies as this is a minor re-post, but I didn't think the other would get answers as it diverged into a discussion and got pushed down... I'm struggling with asymptotic notation a little bit... ...
0
votes
0answers
20 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
0
votes
0answers
18 views

exponential inequality for sum of dependent random variables

I have proved an inequality for the expectation in the context of dependent random variables. Can you please confirm it and give me some feedbacks? If $X_1,X_2,X_3,\ldots,X_m$ are $m$ dependent mean ...
0
votes
0answers
29 views

Can we find some expressions for $p$ and $q$?

Let $f\colon\mathbb R\to\mathbb R$ be a real analytic function. Assume also that $f$ has a zero at $s=1$ of order $m$. Assume that there exists an integer $r$ such that ...
0
votes
0answers
48 views

An extremal problem using AM-GM inequality

Let $x, y, z$ be nonnegative real numbers and such that $$ x^2+y^2+z^2=2. $$ Find the maximum value of $$ P=\frac{x^2}{x^2+yz+x+1}+\frac{y+z}{x+y+z+1}-\frac{1+yz}{9}. $$ My attempt. I guess that $P$ ...
0
votes
0answers
30 views

AM-GM-HM on expression with parameters?

This question, which I believe is easier to answer, is related to my previous question: Finding a value that makes an expression negative I am persistent - and need some ideas to help me prove ...
0
votes
0answers
15 views

Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
0
votes
0answers
44 views

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
0
votes
0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
0
votes
0answers
16 views

Vysochanskij Petunin vs. Cantelli inequality for random variables

The well known Cantelli inequality states: $$Pr(|X-\mu|\ge\alpha)\le\frac{2\sigma^2}{\sigma^2+\alpha^2}$$ where $X$ is a real valued random variable, $\mu$ the mean value and $\sigma^2$ the variance ...
0
votes
0answers
19 views

Application of mean value theorem to function $x \to (x-y)^{a-1-d/2}$

How can I show the following inequality by mean value theorem, for a constant $C>0$ $2|(x+h-y)^{a-1-d/2} - (x-y)^{a-1-d/2}|^p \leq C (x-y)^{(a-2-d/2)p}h^p$ Proof: Let $f(b) =(x+h-y)^{a-1-d/2}$, ...
0
votes
0answers
27 views

How about integral version of Holder's inequality?

In light of the fact that Minkowski's inequality have integral version, I thought there might be one for Holder's as well. I cannot find any through searching (there is an infinite product version in ...
0
votes
0answers
43 views

Help with an inequality of probability distribution functions

There are six random variables $X_{1}$, $X_{2}$, $X_{3}$, $Y_{1}$, $Y_{2}$, and $Y_{3}$ on $[0,c]$. Their cumulative distribution functions are $% F_{1}(t)$, $F_{2}(t)$, $F_{3}(t)$, $G_{1}(t)$, ...
0
votes
0answers
24 views

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 ...
0
votes
0answers
21 views

ML Inequality - don't understand this step?

I'm going through an example question but I don't understand one of the steps they've made. Let $C_R$ be the circle with radius R, centret at 0 and positively oriented. Show $$lim_{R \to ...
0
votes
0answers
30 views

Taylor expansions, inequalities and more

(Part A) I have to find the Taylor expansion of order 2 around (0,0) of $$f: \mathbb{R}^{2}\rightarrow \mathbb{R}$$ $$(x,y)x \mapsto f(x,y) = x\log (1+y)+sin(x+y) $$ Furthermore I have to prove if ...
0
votes
0answers
36 views

Want to find a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies these inequalities

I want to find a function $f \in C^\infty([0,T])$ such that $$0 < L \leq f \leq M$$ $$f' \geq C \geq K_1 + K_2M$$ where $K_1$ and $K_2$ are fixed positive constants and are given. Is it possible ...
0
votes
0answers
21 views

Inequality testing using numerical substitution

I am seeing that CASs might fail in inequality testing, so a vague approach can be to test them for very large number of values sampled at some interval and it might give at least some approximation. ...
0
votes
0answers
19 views

Finding matrix index from triangular array offset

I have a mapping from a lower triangular matrix, A, to a vector,v: A(i,j) -> v( $\lfloor i(i+1)/2 \rfloor + j$ ) $i,j\in[0,N]$, $j\leq i$, $N\in\cal{N}$, $N\geq 0$ (so, my first row is row 0, and ...
0
votes
0answers
23 views

Apollonius’ Identity inner product space

$||z-x||^2+||z-y||^2=\frac{1}{2}||x-y||^2+2||z-\frac{x+y}{2}||^2$ I proved it by expanding both sides and i found both sides are equal. Are there any easy way to prove it?
0
votes
0answers
188 views

Please help compare two integrals

There are two cdf (cumulative distribution function) $F_{1}(t)$ and $F_{2}(t)$ with a support on $[0,c]$. Suppose $F_{1}(t)<F_{2}(t)$ for any $t\in(0,c)$ (i.e., first order stochastic dominance). ...
0
votes
0answers
47 views

Expectation of maximum of product of normal random variables

Let $X_i Y_i \sim N(0,\sigma^2) N(\mu,\sigma^2b)$, $\mu \neq 0 ,b >0$. then is there any inequality for the maximum of these products. What I mean is $E(\max{X_iY_i, 1 \leq i \leq m})$. I have ...
0
votes
0answers
24 views

Suggestions on proving this inequality

I've been having a lot of trouble proving this inequality (I am really bad at proving even easy inequalities, but I'm trying to change that), but I really want to do it. So if you can give me some ...
0
votes
0answers
19 views

complete logic for proving inequalities

Last semester I took a course on algorithm analysis a big part of which was proving that the running time function of a program was in the set $O(f(x))$ for some $f$. To prove $f\in O(g(x))$ one ...
0
votes
0answers
27 views

How to Prove An Integral Inequality---period of a Conservative system?

$a_0=\frac{(n-1)^{n-1}}{n^n}$,$0 < a<a_0$,Proof $$2\int_{x_1}^{x_2}\frac{x^{n-3/2}dx}{\sqrt{x^{n-1}-x^n-a}}>\sqrt{2a_0}\pi$$ $x_1,x_2$is two root of $x^{n-1}-x^n-a=0$,$n>2,n\in \mathbb{Z}$ ...
0
votes
0answers
7 views

Feasability of infinite number of linear inequalities

Consider a continuous function $f:A\to\mathbb{R}^N$ for a closed interval $A\subset \mathbb{R}$. Are there suffieint or necessary conditions for the existence of a solution $w\in\mathbb{R}^N$ such ...
0
votes
0answers
20 views

$\sum (a_i + b_i)^p $ and $\left(\sum a_i\right)^p +\left(\sum b_i\right)^p $ for $a_i, b_i \geq 0,p\geq 1$

Is there a relation between $\sum (a_i + b_i)^p $ and $\left(\sum a_i\right)^p +\left(\sum b_i\right)^p $ for $a_i, b_i \geq 0,p\geq 1$? At first sight, when $a_i = b_i$, we have $$2^p\sum a_i^p ...
0
votes
0answers
36 views

Prove solution does not exist for inequalities system

I have an inequalities sytem like the following: Example > x+y+z <= A > x+y <= B > x+z > C > y+z > D > x >= E Let A,B,C,D,E be any ...
0
votes
0answers
50 views

Help inequality with $O(\cdot)$ and $\Omega(\cdot)$

Suppose,$$f(T)\le O\left(\sqrt{\dfrac{\log( T/\delta)}{T}}\right).$$ If we let $\delta=\dfrac{1}{n^2}$ and $T\ge\Omega\left(n^2\log n\right)$, then: $$f(T)\le \dfrac{1}{n}.$$ Can anyone ...
0
votes
0answers
19 views

Prove that for holomorphic function is inequality $M|a_1| \le M^2 - |a_0|^2$

Let $$f(z) = \sum_{k=0}^{\infty}a_kz^k$$ be holomorphic function in unit disc and $f(z) < M$ for $|z|<1$. Show that $$M|a_1| \le M^2 - |a_0|^2$$ I have any ideah how can I prove this ...
0
votes
0answers
7 views

if $f_{1}(x)<f_{2}(x)$, is it true that $ \underset{B}{\min } \underset{Bx=0} {\max }f_{1}(x)<\underset{B}{\min }\underset{Bx=0}{\max }f_{2}(x)$?

One "obvious" question but I hope I can get some explanations... if $f_{1}(x)<f_{2}(x)$, is it true that $ \underset{B}{\min } \underset{Bx=0} {\max }f_{1}(x)<\underset{B}{\min ...