Tagged Questions

Questions on proving and manipulating inequalities.

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4
votes
1answer
107 views

Inequality for the integral $\frac{\ln x}{x^n}$

Define the integral $I_{n}$ as follows for $n$ an integer greater than $1$: $I_{n}:=\int_{1}^{e}\frac{\ln x}{x^n}dx$ Is it true that $$I_{n}\leq \frac{1}{n-1}\left(1-\frac{1}{e^{n-1}}\right)?$$ ...
2
votes
1answer
119 views

Inequality for integral

Define the following integral with $n$ an integer greater than $1$: $$I_{n}=\int_{0}^{1}\frac{e^t}{(1+t)^n}dt.$$ Is it true that for all $n \geq 2$, $$ ...
4
votes
2answers
259 views

Geometric inequality: $2r^2+8Rr \leq \frac{a^2+b^2+c^2}{2}$

Suppose $a$, $b$, and $c$ are the lengths of the sides of a triangle, and $R$ and $r$ are its circumradius and inradius respectively. How can one prove the following inequality? $$2r^2+8Rr \leq ...
11
votes
1answer
582 views

Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$

For $a,b \in \mathbb R$, $p\geq2$ I try to show $$\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p.$$ Is this a popular inequality (At least I could not ...
3
votes
2answers
170 views

Does anyone know how to prove this inequality

Does anyone know how to prove the following inequality ...
4
votes
2answers
683 views

mean value theorem

Prove using mean-value theorem that $x/(1+x^2)<\arctan x<x$ for $x>0$ I got the first part but how do I prove $\arctan x< x$ using the MVT? The first part was done easily by ...
1
vote
2answers
195 views

Proof $\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$

I try to proof the following $$\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$$ with $r \geq 3$ and $r \in \mathbb{P}$. Do I have to make in induction over $r$ or any better ideas? Any help is ...
6
votes
1answer
335 views

Sum of divisor ratio inequality

Consider the divisors of $n$, $$d_1 = 1, d_2, d_3, ..., d_r=n$$ in ascending order and $r \equiv r(n)$ is the number of divisors of $n$. Is there any expression $f(n) < r(n)$ such that, ...
4
votes
0answers
167 views

Paradox running times of Reduce for polynomial inequalities in Mathematica 8 [closed]

I found something odd about the time needed by Reduce to solve a polynomial inequality (under Wolfram Mathematica 8). Consider the following: ...
1
vote
2answers
110 views

Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $

I would like to show that: $$ \prod_{i=0}^{n} {n \choose i}\leq \left(\frac{2^n-2}{n-1}\right)^{n-1} $$ $$n=2p $$ $$ \prod_{i=0}^{2p} {2p \choose i}=\prod_{i=1}^{2p-1} {2p \choose i}\leq {2p\choose ...
4
votes
1answer
170 views

Complex number inequality?

Suppose $|z|>1$ for $z$ a complex number. I'm trying to build a certain comparison test to test convergence. I'm wondering, is it true that $$ \frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1}? $$
3
votes
1answer
204 views

Proof of an inequality under Equivalence of Weierstrass and Euler's Definitions of Gamma Function

I am using my lecturer's notes on Special Functions. When dealing with Gamma Functions under the title Equivalence of Wierstrass and Euler's Definitions, he has used a lemma (with no proof). I tried ...
-2
votes
1answer
317 views

Logarithmic form of Young's inequality

If $\frac {1}{p}+\frac {1}{q}=1$, Prove: $\log (\frac {x^p}{p}+\frac {y^q}{q})$ bigger or equal to $\frac {1}{p} \log (x^p) +\frac {1}{q} \log (y^q)$ Insightful proof would be appreciated! How can ...
3
votes
7answers
15k views

Why do we reverse inequality sign when dividing by negative number?

We all learned in our early years that when dividing both sides by a negative number, we reverse the inequality sign. Take $-3x < 9$ To solve for x, we divide both sides by -3 and get $x > ...
9
votes
2answers
286 views

Proof of a (simple?) inequality

I have a feeling that the following inequality should be very easy to prove: $$ x^n \geq \prod_{i=1}^n{(x+k_i)},\quad\text{where } \sum_{i=1}^{n}{k_i}=0,\quad \text{and } x+k_i>0\text{ for all } ...
1
vote
1answer
253 views

Theorem 191 from the book Inequalities of Hardy, Littlewood and Pólya.

In the pic, in the second proof of Thm 191 (the one that starts at the paragraph:"We can prove theorem 191 without appealing to the more difficult theorem 190..."), I don't understand why: $$\int ...
3
votes
3answers
376 views

Inequality for logarithms

I conjecture the following inequality is true $$\ln x \le (x - 1)\ln\frac{x}{x-1}$$ for all $x > 1$, but I cannot give a proof. I will appreciate if someone can provide one.
4
votes
1answer
460 views

Where can I find the paper by Guy Robin?

\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation} In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ...
4
votes
1answer
246 views

Find number of roots in some area (Rouché's theorem)

The task is to find number of $ {z^4} + {z^3} - 4z + 1 = 0$ in the area $1 < \left| z \right| < 2$. (this task is in the Rouché's theorem paragraph) I used this theorem many times, but I ...
6
votes
1answer
556 views

Least-squares left-inverse having smallest Frobenius norm

While trying to prove that the left-inverse of $A$ provided by the least-squares solution to $y=Ax$ has the smallest Frobenius norm, I am stuck at a point which I describe below: Let $B$ be any ...
0
votes
4answers
134 views

An easy inequality

I am trying to prove (or disprove) this inequality for more than one hour without success. $$[1-(b+c)]^2+[1-2c]^2\ge 4bc$$ where $b,c>0$ and satisfies $b+c<1$. Frustratingly, I failed to find ...
1
vote
1answer
97 views

Linear inequalities question: $k+5 > 0$

A text book that I'm reading has $k < -5$ as the solution for $k+5 > 0$. What I want to know is how this can be - why is it not $k > -5$? Edit: looks like the text book's made a mistake.
37
votes
4answers
1k views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
1
vote
1answer
79 views

$(\log_23)^x-(\log_53)^x\geq(\log_23)^{-y}-(\log_53)^{-y}$

$(\log_23)^x-(\log_53)^x\geq(\log_23)^{-y}-(\log_53)^{-y}$ I guess the function $f(x)=(\log_23)^x-(\log_53)^x$ monotonically increasing, so I get the answer $x\geq-y$,but how to prove it not using ...
7
votes
4answers
421 views

Inequality with central binomial coefficients

For every even positive number $N$ we have: $$ {2N \choose N } < 2^N {N \choose N/2 } < 2 {2N \choose N } $$ (Furthermore, $\frac{2^N {N \choose N/2 }}{{2N \choose N }} \to \sqrt{2} $ for ...
2
votes
1answer
253 views

Solving a recurrence inequality

I am not sure if "recurrence inequality" is the correct term or whether it is possible to actually find an answer to this problem but anyways. Let $n$ be a fixed natural number. Let $R(x,y)$ be a ...
14
votes
1answer
363 views

How to prove this inequality in Euclidean space?

Prove that $$\begin{align*}&|a+b||a+c|+|a+b||b+c|+|a+c||b+c|\\ \leq &(|a|+|b|+|c|) \cdot |a+b+c|+|a||b|+|a||c|+|b||c|\end{align*}$$ in Euclidean space $\mathbb{R}^n$. I have been ...
0
votes
2answers
85 views

Domains and setbuilder notation

We're learning about domains and setbuilder notation in school at the moment, and I want to make sure what I did was right. My thought process: \begin{align*} -\frac12|4x - 8| - 1 &< -1 \\ ...
1
vote
1answer
122 views

Find the maximum of this binary entropy look-alike function

Let $b,c \in (0,1)$ be such that $b+c<1.$ Define the following function for $p \in (0,1) :$ $$ I(p;b,c):=(b+c)[p \log\frac{1}{p}+(1-p)\log\frac{1}{1-p}]-cH(p;b,c)-bH(p;c,b)$$ where ...
2
votes
1answer
185 views

How to prove the following inequality: $1+ac+ab+3a\leq b+c+abc+3bc$?

Show that $$1+ac+ab+3a\leq b+c+abc+3bc$$ if $1\leq a\leq bc,$ $1\leq b\leq ac,$ $1\leq c\leq ab.$
2
votes
2answers
58 views

Showing that $\left|x-1+\frac 1{1+x}-\left(y-1+\frac 1{1+y}\right)\right|\leq 2|x-y|$

I asked a question here the other day and one of the steps in the answer I encountered was: $$\left|x-1+\frac 1{1+x}-\left(y-1+\frac 1{1+y}\right)\right|\leq 2|x-y|$$ when $x,y\ge0$ I can't seem to ...
4
votes
1answer
329 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
1
vote
2answers
191 views

Inequality in Complex Plane

In continuation to my previous post : Inequality in Complex Plane I'm still having a small problem with a similar inequality : For $z$ such that: $|z|> 1$ I wish to prove: $$1+|z|+\dots+|z^{n-1}| ...
1
vote
2answers
178 views

Inequality in Complex Plane

I'm studying numerical analysis and in the book I'm reading there is a theorem thats find a raduis such that all the roots of a polynomial $P$ (with coefficient in $\mathbb{C}$) are in the open disk ...
7
votes
4answers
332 views

Sum of powers-of-integers bound

I am looking for a nice proof of this inequality: \begin{equation} \sum_{k=1}^{n-1} k^n < n^n, \quad n > 0. \end{equation} Example: $1^4 + 2^4 + 3^4 = 1+16+81 = 98 < 4^4 = 256.$
13
votes
4answers
656 views

Inequality with Complex Numbers

Consider the following problem: Prove that for every set of complex numbers $\{z_i\}$, with $i$ ranging from one to $n$, there is a subset $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge ...
2
votes
1answer
116 views

Maximise $L^q$ norm of a vector, for fixed $L^1$ and fixed $L^p$ norms

Consider a vector $x \in \mathbb R_+^n$ and $p,q \in \mathbb R$ such that $1<p<q$. We fix $\sum \limits_{i=1}^{n}|x_i| = 1$ and $ \left(\sum \limits_{i=1}^{n}|x_i|^p \right)^\frac{1}{p} = ...
2
votes
3answers
108 views

Verifying an Inequality: $\left(\frac{n-1}{n}\right)^{n-1} \leq \left(\frac{n-k-1}{n-k}\right)^{n-k-1}$ [duplicate]

Possible Duplicate: How to prove $(1+1/x)^x$ is increasing when $x&gt;0$? Let $k \in \mathbb{N}$ be fixed. For all $n \in \mathbb{N}$, is it true that ...
3
votes
3answers
145 views

Squeeze an integral

Would you have any idea about this problem ? Prove that for all nonnegative integers $n$, the following inequalities hold: $$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...
2
votes
1answer
696 views

Proof of Chebyshev's theorem

(a) Show that $\int_2^x\frac{\pi(t)}{t^2}dt=\sum_{p\leq x }\frac{1}{p}+o(1)\sim\log\log x.$ (b) Let $\rho(x)$ be the ratio of the two functions involved in the prime number theorem: ...
1
vote
3answers
477 views

Binomial inequality

Show that we have: $$ \binom{n}{s}\leq n^n $$
2
votes
2answers
108 views

Chromatic number in relation to the amount of vertices

We have a finite undirected graph $G := (V, E)$ and its complementary graph $\overline G := (\overline V, \overline E)$. How do we show that $\chi (G) \cdot \chi (\overline G) \ge |V|$? We know ...
4
votes
1answer
18k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
0
votes
2answers
317 views

Maxima of bivariate function

[1] Is there an easy way to formally prove that, $$ 2xy^{2} +2x^{2} y-2x^{2} y^{2} -4xy+x+y\ge -x^{4} -y^{4} +2x^{3} +2y^{3} -2x^{2} -2y^{2} +x+y$$ $${0<x,y<1}$$ without resorting to checking ...
2
votes
1answer
108 views

Find all real numbers $x$ such that $||x+1| - |3x - 1|| < 1$

This is on my final exam review. I have the solution, but I do not understand it. When looking at $|x+1| - |3x - 1|$, I see four cases: a) $x + 1 > 0$ and $3x - 1 > 0$ $x > -1$ and $x ...
3
votes
1answer
1k views

Show that a function $f(x)$ is always greater than function $g(x)$ in a interval $(a,b)$

I would like to learn pointers to how to prove "a function $f(x)$ always greater than function $g(x)$ in a interval $(a,b)$". In other words show that $f(x) > g(x)$ in $(a,b)$. Their end point ...
29
votes
3answers
1k views

A combinatorial proof of $n^n(n+2)^{n+1}>(n+1)^{2n+1}$?

The statement is simply that the sequence $\left(1+\frac{1}{n}\right)^n$ is increasing. Since the numbers $n^m$ have quite natural combinatorial interpretations, it makes me wonder if a ...
1
vote
1answer
153 views

Proof of an inequality with expectation of random variables

I'm solving some exercises and got stuck. The setting: let $\varepsilon > 0 $ given, and $D(\varepsilon):=\{(x,y) \in \mathbb{R_+}^2\mid |x-y| \ge \varepsilon\mbox{ and }\min(x,y) \le ...
1
vote
1answer
84 views

Simple inequality question

This is probably very simple, but for some reason I don't seem to see why if $\forall x \in \mathbb R^n, \|x-\phi(x)\|>c$ for some $c>0$ then $\|x-y\|>\|\phi(x)-\phi(y)\|$, where $\phi$ is ...
1
vote
1answer
140 views

Why is this true for large enough n?

$$ \begin{align*} \Pr[\text{bin } i \text{ has at least } k \text{ balls}] &\leqslant \left( \frac{e}{k} \right)^k = \left( \frac{e \ln \ln n}{3 \ln n} \right)^{\frac{3 \ln n}{\ln \ln n}} ...