Questions on proving, manipulating and applying inequalities.

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1
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1answer
30 views

Proving an inequality about a set of combinations.

Suppose $A$ is a set of $r$ combinations of an $n$ set, with $\alpha \cap \beta \neq \phi$, whenever $\alpha, \beta \in A$. Show that $$|A| \leq \binom{n-1}{r-1}$$ if $r \leq \frac n2$. What does ...
1
vote
1answer
101 views

Inequality $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ [duplicate]

If $\frac {1}{p}+\frac {1}{q}=1$ and $a,b \ge 0$ , then prove $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ . I can't find a simple and short way to prove this. Any hint would work. Thanks in advance!
0
votes
1answer
21 views

Integral Inequality with Monotonic Function

Problem: For continuous, either both increasing or both decreasing functions $f, g$ on $[a, b]$, suppose that $p(x)$ is continuous and positive. Prove that $$\int_a^bp(x)f(x)dx \int_a^bp(x)g(x)...
0
votes
1answer
20 views

Why is $|\psi_n-f|^p \leq2^p |f|^p$ when $|\psi_n|\leq |f|$?

Why is $|\psi_n-f|^p\leq 2^p |f|^p$ when $|\psi_n|\leq|f|$? Is it true that $|a+b|^p\leq 2^p (|a|^p+|b|^p)$?
1
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2answers
34 views

Inequality involving a convex function

I am stuck, showing the following inequality in an easy way (using only inequalities or something): Let $x\in [-a,a]$ for some $a>0$ and $p\in (1,2)$. I want to show that there then exists a ...
2
votes
3answers
120 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
3
votes
1answer
52 views

Does this inequality involving inverse tangent (arctan) hold?

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\...
8
votes
2answers
127 views

Prove inequality $\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$

For any $n\ge2, n \in \mathbb N$ prove that $$\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$$ My work so far: 1) $$\...
3
votes
3answers
46 views

Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$

Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$ without using a calculator. I rearraged to get $4 \cdot 1976^4-1 > 10^{-19.76} \cdot 16 \cdot 1976^7$ and so we have ...
0
votes
0answers
17 views

Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \begin{equation} \vert \vert T \vert \...
1
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3answers
49 views

Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$

Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$ if $n>0$. I didn't see an easy way of proving this without doing a lot of algebra and rearranging. Is there an easier ...
2
votes
2answers
44 views

Inequality involving ArcTan

How to prove that for $x\in[0, +\infty]$ the following inequality is true: $$\arctan x\geq\frac{3 x}{1+2\sqrt{1+x^2}}?$$ I don't have idea from where to start, so any hint is welcome. Thanks in ...
0
votes
0answers
21 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Edit of progress: Since the SDE is linear, I got a solution in the form of $e^\int...$$\cdot e^\int$. By Jensen's inequality I can change the order of the left factor to have the integral on the ...
6
votes
2answers
119 views

The number of positive integer solutions to the equation $x_1+2x_2+…+nx_n=n^2.$

Let $n \ge 2, n \in \mathbb N$. $A_n$ denotes the number of positive integer solutions to the equation $$x_1+2x_2+...+nx_n=n^2.$$ Prove inequality $$\frac{n^n(n-1)^{n-1}}{2^{n-1}\left(n!\right)^...
1
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4answers
45 views

$\cos2\theta +\cos\theta +k = 0 $ - set of all values of $k$ for which there is a solution

The set of all values of $k$ (real), such that the equation $\cos2\theta +\cos\theta +k = 0 $ admits a solution for $\theta$ is? MY ATTEMPT: I substituted $\cos2\theta$ with $2\cos^2\theta - 1 $. On ...
1
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3answers
44 views

Solutions to the inequality $0>-x^2 +2x+3.$

I am trying to solve an inequality of $$0>-x^2 +2x+3.$$ I am aware of two different ways of factorizing this. $$(-x+3)(x+1)\quad\text{ and }\quad(x-3)(-x-1).$$ When I use $(-x+3)(x+1)$, I get the ...
0
votes
1answer
31 views

Would the following series of implications be logically correct?

Let $a$ and $b$ be positive integers, and let $f$ be a generic function satisfying $f(1) = 1$, and taking on only positive integer values. Suppose that I have the following propositions: $$\bf{A} : ...
1
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1answer
41 views

It's true that $ |\log^2(z)| \leqslant |\log(R)|^2 + |i \arg(z)|^2 $ where $z \in \mathbb{C}$

In some residue integral, when one have to prove that an integral vanish at infinity, I've found in some textbooks the inequality: $$ |\log^2(z)| \leqslant |\log(R)|^2 + |i\ \arg(z)|^2 $$ Where $z= ...
2
votes
0answers
33 views

Finding a maximum with some constraints

I would like to maximize the term $ l_1b_1+l_2b_2+l_3b_3-2 $ such that the following conditions hold: $ 1>l_1>l_2>l_3>0 $, $ l_1,l_2,l_3 \in \mathbb{Q} $, $ b_1,b_2,b_3 \in \mathbb{N} $...
0
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0answers
16 views

Convex hull possesses only integer extreme points

I have the following question. Consider given natural numbers $ 1 \le l_m <\ldots < l_1 < L $. Is it possible to prove that the convex hull of $ \left\lbrace a \in \mathbb{Z}^m_{\ge 0} \, \...
-3
votes
1answer
36 views

Neighbourhood set in Graph theory [closed]

Let $G$ be any connected graph with $\Delta(G)$ be maximum degree. If $D \subseteq V(G)$ then how can we say that $\left | \bigcup \limits_{v \in D} N(v) \right | \leq |D| \Delta(G)$.
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1answer
36 views

Prove that $-4\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq10$

Prove that $$\color\red{-4}\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq\color\red{10}$$ My attempt:- I simplified the equation to $$\begin{align} &\;\;\phantom{=} 5\cos\theta+3\cos(\...
0
votes
1answer
35 views

Variant of Holder's inequality: $\|x\|_p \le n^{\frac1p- \frac1r} \|x\|_r$

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
0
votes
1answer
46 views

Cauchy-Schwarz inequality application

I am having trouble verifying the first step where the author makes use of Cauchy-Schwarz inequality. https://gbas2010.wordpress.com/2011/10/16/inequality-53-vo-quoc-ba-can/ I am unsure how he ...
1
vote
2answers
105 views

prove inequation

a,b,c,d $\in \mathbb{R}$ $a,b,c,d \gt 0$ and $ c^2 +d^2=(a^2 +b^2)^3$ prove that $$ \frac{a^3}{c} + \frac{b^3}{d} \ge 1$$ If I rewrite the inequation like $ \frac{a^3}{c} + \frac{b^3}{d} \ge \frac{...
0
votes
1answer
23 views

Automatic tools for solving a set of inequalities

I am trying to solve problems such as the following: Find real numbers $a_1,a_2,a_3, b_1,b_2,b_3$ such that all following expressions are true: $a_1+a_2+a_3=0$ $b_1+b_2+b_3=0$ $a_1>...
0
votes
1answer
46 views

Matrix Trace Inequality [closed]

If $\operatorname{Tr}(A) < \operatorname{Tr}(B)$, is it fair to say that $\operatorname{Tr}(AC) < \operatorname{Tr}(BC)$? All of $A$, $B$ and $C$ are positive definite matrices.
1
vote
1answer
57 views

How to prove this inequality $x\sin^2{A}+y\sin^2{B}\ge xy\sin^2{C}$

In $\Delta ABC$,if $x,y>0$ and $x+y=1$.show that $$x\sin^2{A}+y\sin^2{B}\ge xy\sin^2{C}$$ I have looked at the simpler methods,? Here is one solution $$\dfrac{\sin^2{A}}{y}+\dfrac{\sin^2{B}}{...
2
votes
1answer
138 views

How to prove that $\frac{x^2+y^2}{4}\leq e^{x+y-2}$

I need to prove that given any $x,y\in \mathbb{R}$, such that $x,y\geq 0$, then is true that $$\frac{x^2+y^2}{4}\leq e^{x+y-2}$$ My try was use logarithms and state that without losing generality, ...
3
votes
1answer
79 views

Functional inequality $\sum_{1\le i<j\le n}f(x_i+x_j)\ge \frac{n(n-1)}{2}f(a_1x_1+a_2x_2+…+a_nx_n)$

Let $n\in\mathbb N, n\ge 2$. Does there exist a set of non-zero real numbers $a_1, a_2,..,a_n$ with this condition: If function $f: \mathbb R \rightarrow \mathbb R$ satisfies the inequality $$...
1
vote
2answers
71 views

How can I solve this inequality? [closed]

Have a nice day, how can I solve this inequality? $$a<b<-1$$ $$ |ax - b| \le |bx-a|$$ what is the solution set for this inequality
5
votes
5answers
112 views

Bounding a series: $\frac{\pi}{2} < \sum_{n=0}^\infty \frac{1}{n^2 + 1} < \frac{3\pi}{2} $

I have the following statement - $$\frac{\pi}{2} < \sum_{n=0}^\infty \dfrac{1}{n^2 + 1} < \frac{3\pi}{2} $$ So I tried to prove this statement using the integral test and successfully proved ...
0
votes
1answer
52 views

Special Case of A.M-G.M Inequality. [closed]

How can we prove the special case of A.M-G.M Inequality, that is: The Geometric Mean of $n$ positive real numbers is equal to 1.Prove that their Arithmetic Mean is greater than or equal to 1. I ...
2
votes
5answers
150 views

Prove: if $|x-1|<\frac{1}{10}$ so $\frac{|x^2-1|}{|x+3|}<\frac{1}{13}$

Prove: $$|x-1|<\frac{1}{10} \rightarrow \frac{|x^2-1|}{|x+3|}<\frac{1}{13}$$ $$|x-1|<\frac{1}{10}$$ $$ -\frac{1}{10}<x-1<\frac{1}{10}$$ $$ \frac{19}{10}<x+1<\frac{21}{10}$$ $...
4
votes
5answers
459 views

What is the fastest method to find which of $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ and $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $ is bigger manually?

What is the fastest method to find which number is bigger manually? $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $
1
vote
0answers
59 views

Prove this inequality $xy+yz+xz\ge xyz$ [closed]

Let $x,y,z>0$ such $$2xy+2yz+2xz=1+x^2+y^2+z^2$$ show that $$xy+yz+xz\ge xyz$$
6
votes
0answers
100 views

How prove this ineqlity [closed]

Let $x,y,z,w>0$ and such that $xyzw=1$. Show that $$ \dfrac{1+x}{1+x^2}+\dfrac{1+y}{1+y^2}+\dfrac{1+z}{1+z^2}+\dfrac{1+w}{1+w^2}\le 4. $$
3
votes
1answer
37 views

How can I prove this inequality involving logarithm?

$n^2 \geq n \log_{2}n$ I tried like this: $n^2 \geq n \log_{2}n$ $n^2-n \log_{2}n \geq 0$ $n(n-\log_2 n) \geq 0$ I don't know what to do after this?
0
votes
2answers
95 views

Minimum value of $4a+b$

Let $ax^2+bx+8=0$ be an equation which has no distinct real roots then what is the least value of $4a+b$ where $a,b\in \Bbb R$. My Try: I differentiated the given function to get $f'(x)=2ax+b$ now ...
0
votes
2answers
60 views

Solve $|x^2-5|\geq 4$

$$|x^2-5|\geq 4$$ $|x^2-5|\geq 4\Rightarrow$ $x^2-5\geq 4 $ or $x^2-5\leq -4$ Case: $x^2-5\geq 4\Rightarrow x^2-9\geq0\Rightarrow (x-3)(x+3)\geq 0$ so the answer is $x\geq 3$ or $x\leq -3$ case: $...
1
vote
5answers
121 views

Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$

$$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$ I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$ enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$ But I can find how to continue from here
1
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0answers
13 views

A false identity involving $2^{\frac{1}{\zeta(s)}}$ for $\Re s>1$, from these particular values of the Riemann Zeta function and its alternating

Yesterday when I was exploring symbolic calculations $\dagger$ about specializations in $z=\frac{1}{n}$ with $n>1$ an integer, of $$\zeta(z)=(1-2^{1-z})^{-1}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^z}:=...
1
vote
2answers
82 views

Prove that: $\forall x\in (0,\frac{ \pi}{4(n-1)})$ $\tan(nx)> n \tan(x)$

Let $n\in \mathbb{N} , n> 1$ Prove that : $\forall x\in (0,\frac{ \pi}{4(n-1)})$ $\tan(nx)> n \tan(x)$ I know: $f(x) = \tan x$ is convex function $f(a x + b y) < a f(x) + b f(y), a+b=...
5
votes
1answer
185 views

Inequality with a rational polynomial

Let $$P(x)=x^{n-1}+a_{n-2}\,x^{n-2}+a_{n-3}\,x^{n-3}+\cdots+a_0\in\mathbb{Q}[x]$$ be a monic rational polynomial of degree $n-1$. I want to show that, for every set of $n$ distinct integers $\{x_1,...
1
vote
4answers
102 views

Help in proving an inequality

Show that $$a^4 + b^4\ge\frac{1}{8}$$ if $a+b=1.$
0
votes
1answer
25 views

Energy inequality heat equation

Consider $u \in C_1^2(\Omega \times [0,T]), \Omega\subset\mathbb{R}^n$ as a solution of the problem $ u_t - \Delta u = f, \text{ in } \Omega \times (0, T]$, $u = 0, \text{ on } \partial\Omega \...
9
votes
5answers
289 views

Which number is greater, $11^{11}$ or $9^{12}$?

Which number is greater than $11^{11}$ or $9^{12}$? My work so far: $11^{11}=285311670611>9^{12}=282429536481$. But to verify the validity of equality should be in the range of easily ...
-1
votes
1answer
25 views

Existence of a set function

Consider the set S of all subsets of {1, 2, 3, 4}. Show that there exists a function h : S → [0, 1] that satisfies all following conditions: Condition 1: h(∅) = 0 Condition 2: h({1, 2, 3, 4}) = 1 ...
0
votes
0answers
44 views

Jensen's inequality for two random variable

Prove: Let $X$ and $Y$ be two random variables in probability space $\left ( \Omega ,\mathcal{F},\mathbb{P} \right )$ , and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a convex function, then $$f\left ( ...
2
votes
2answers
53 views

How to find the restrictions of side length on an obtuse triangle

Question: In Triangle ABC, the angle ∠ABC is an obtuse angle. The Side AB is 1cm, and the side BC is 3cm. Side AC is (3x+10)/(x+3) cm Find the restriction(s) on x. I have tried a few different ...