Questions on proving, manipulating and applying inequalities.

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1
vote
0answers
39 views

Prove that if $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ than $|f(x)| \leq c \left|\frac{1}{x}\right|^N$

The task is: Knowing that $\forall \lambda >0, x \neq 0$ $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ prove that: $|f(x)| \leq c \left|\frac{1}{x}\right|^N$ I would really appreciate ...
1
vote
4answers
35 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
2
votes
2answers
37 views

Prove by induction that $\sum_{k=1}^nk^p < (n+1)^{p+1}/(p+1), \quad n,p \in \mathbb{N}$

For $n=1$, we have at the left side $1^p$, and at the right side: $$ \frac{2^{p+1}}{p+1}\mathrm{~which~is } >1$$ so it holds for $n=1$, but how can we prove that $$ ...
1
vote
2answers
23 views

Matrix norm inequality proof - does this use Cauchy-Schwarz?

The matrix norm for $A : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (so $A$ is an $m \times n$ matrix) is given by $$\|A\| = \sup_{X \in \mathbb{R}^n \setminus \{0\}} \frac{|AX|}{|X|}$$ where $| \cdot |$ ...
2
votes
2answers
71 views

How to properly find supremum of a function $f(x,y,z)$ on a cube $[0,1]^3$?

Solving an applied problem I was faced with the need to find supremum of the following function $$f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3}$$ where $f\colon\ [0,1]^3\backslash\{(1,1,1)\} ...
7
votes
3answers
158 views
+100

(Elegant) proof of an inequality: $h(x) \geq 1- (1-\frac{x}{1-x})^2$, where $h$ is the binary entropy function

I am looking for the most concise and elegant proof of the following inequality: $$ h(x) \geq 1- \left(1-\frac{x}{1-x}\right)^2, \qquad \forall x\in(0,1) $$ where $h(x) = x \log_2\frac{1}{x}+(1-x) ...
2
votes
2answers
37 views

Help solving the inequality $2^n \leq (n+1)!$, n is integer

I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook: $2^n \leq (n+1)!$ Where $\mathbf{n}$ in an integer. I've tried applying lg to both ...
1
vote
0answers
27 views

Use Chebyshev’s inequality to choose $n$ such that $P(\bar{X} > 4) > 0.9$

Use Chebyshev’s inequality to choose n such that $$ P(\bar{X_n} > 4) > 0.9 $$ where $$ E[\bar{X_n}] = 5 \ \ \ \ \ Var[\bar{X_n}] = \frac{4}{n} $$ The problem I am having when using Chebyshev's ...
6
votes
6answers
201 views

Extreme of $\cos(A)\cos(B)\cos(C)$ in a triangle without calculus.

If $A,B,C$ angles of a triangle, show extreme value of $$\cos(A)\cos(B)\cos(C)$$ I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing helps. On this topic ...
7
votes
0answers
100 views

When might some a variable leave the basis?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...
0
votes
1answer
21 views

Does a sum of squares become smaller as the number of terms increases?

I am interested in the following question: Let $,kn$ be a positive integeres. Assume $\sum_{i=1}^{k} L_i=\sum_{i=1}^{k+1} \tilde L_i=n$, where $L_i,\tilde L_i$ are positive integers. Is it true ...
11
votes
5answers
1k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
2
votes
0answers
37 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! ...
7
votes
1answer
352 views

Inequality involving factorial $\binom nk<(en/k)^k$

I am trying to prove following inequality: $$\binom{n}{k}<(en/k)^k$$ I tried Stirling approximation but I could not get anything further. Then I get $$\binom{n}{k}\approx \frac{\sqrt{2\pi ...
3
votes
0answers
88 views

prove that $(\frac{n}{3})^n<n!<e\cdot(\frac{n}{2})^n$ [duplicate]

prove that $$(\frac{n}{3})^n<n!<e\cdot(\frac{n}{2})^n$$ I tried to prove by the induction that $(\frac{n}{3})^n<n!$ and $n!<e\cdot(\frac{n}{2})^n$, but I failed my assumption ...
0
votes
1answer
93 views

Proving an inequality involving factorials: $(\frac{n}{2})^n \ge n! \geq (\frac{n}{3})^n$ [duplicate]

For $n \geq 6$, where $n$ is a natural number, prove that $(\frac{n}{2})^n \ge n! \geq (\frac{n}{3})^n$. I tried using induction but could not do it.
14
votes
1answer
2k views

Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$

I met an inequality, I ask, do not mathematical induction to prove that: Prove \[ \left(\frac n2\right)^n > n! > \left(\frac n3\right)^n \] without using induction
0
votes
1answer
26 views

Algebraic Inequality

If a,b,c are positive real numbers and $z = \frac{b^2 + c^2}{b+c} + \frac{c^2 + a^2}{a+c} + \frac{a^2+b^2}{a+b}$ then only one of the following statements is always true , which on is it ? a) ...
-2
votes
0answers
46 views

Inequalities: $e^{-x}> 2-x$ [closed]

Tried to solve it through the ln method, but didn't know how to proceed from there. Here's what I have done: $$e^{-x} > 2-x$$ $$-x > \ln(2-x)$$ $$x < -\ln(2-x)$$ Really need help to solve ...
2
votes
1answer
66 views

Prove the inequality $\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\geq 4$

$a, b, c, d$ are positive reals. How would I prove the inequality $$\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a} \geq 4$$ I have tried using the rearrangement inequality with ...
0
votes
3answers
41 views

Inequality with a square root

If the inequality $ (x+2)^{\frac{1}{2}} > x $ is satisfied. what is the range of x ? My approach - I squared both the sides and proceeded on to solve the quadratic obtained in order to solve the ...
1
vote
2answers
50 views

Problem in Solving an Inequality

The problem is: $Prove$ $that$ $|\sin^2 (x)-\sin^2 (y)|\le |x-y|$ $ for $ $ all $ $ x,y>0$. $My$ $work$ : $$\sin^2 (x)\le|\sin x|\le|x|\le|x-y|+|y|$$ and so is $$|\sin^ 2 (x)-\sin^2 (y)|\le ...
3
votes
1answer
27 views

Rational function between a constant and a third root

Is there a rational function $f(x)\in{\mathbb Q}(x)$ such that $\sqrt{2} \leq f(x) \leq \sqrt[3]{2x}$ for all $x\geq\sqrt{2}$ ? My thoughts : it is easy to find such an $f$ if we relax the ...
9
votes
0answers
129 views
+50

Proof of an inequality in $\mathbb{C}$

Let $z\in \mathbb{C}, n \geq 2$. Show this complex inequality $$|z^n-1|^2\le |z-1|^2\left(1+|z|^2+\dfrac{2}{n-1}\Re{(z)}\right)^{n-1}$$ For $n=2$ the inequality is easy to prove: $$|z^2-1|^2\le ...
-3
votes
2answers
29 views

Fraction with power denominator [closed]

I'm very confused as to how you're supposed to solve an inequality in which there is a fraction with a power as a denominator. Example: $$2^x + 8/{2^x} > 6$$ Thank you in advance!
0
votes
2answers
19 views

Proof the inequalitiy for the two Matrices $A, B$

Let $ A,B \in C^{nxn}$ then $$ 1 \le || (\lambda I-B)^{-1})(A-B)||\le ||(\lambda I-B)^{-1}||*||(A-B)||$$ for any eigenvalue $\lambda $ of $ A $ which is not an eigenvalue of $ B$ and any operator ...
0
votes
1answer
11 views

On the relationship between $\max(p_i)$ and $\omega(b)$, if $\sigma(b^2)/b^2$ is bounded above by a specific number $U$

Let $\omega(x)$ denote the number of distinct prime factors of $x$, and let $\sigma(x)$ be the sum of the divisors of $x$. Denote the abundancy index of $x$ by $I(x) = \sigma(x)/x$. Let the number ...
1
vote
2answers
47 views

Proving $\sqrt x\ge\log(x+1)$

What is a simple proof that $\sqrt x\ge\log(x+1)$ for $x\ge 0$? I'm trying to prove that $\sum_n\frac{\log n}{n(n-1)}$ converges, and my idea is to upper bound this with the telescoping sum ...
0
votes
2answers
174 views

A Very Hard Inequality

Find the smallest constant $c$ such that for any positive integers $a_1,a_2,\ldots,a_n$ for $n \geq 3$, the following inequality holds: \begin{align} ...
0
votes
2answers
23 views

Application of A.M. -G.M. inequality

Let x, y,z be positive numbers. The least value of $ \frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{.5}}$ is a) $\frac{9}{2^{.5}}$ b) 6 c) $\frac{1}{6^{.5}}$ d.) None of the above I tried applying the A.M. ...
1
vote
3answers
52 views

Prove when $abc=1$: $ \frac{a}{2+bc} + \frac{b}{2+ca}+\frac{c}{2+ab} \geq 1$

Question: Prove the following inequality which holds for all positive reals $a$, $b$ and $c$ such that $abc=1$: $$ \frac{a}{2+bc} + \frac{b}{2+ca}+\frac{c}{2+ab} \geq 1$$ My thoughts were ...
1
vote
1answer
34 views

Geometric inequality involving sides of the triangle

I was triying to learn geometric inequalities and I got into this problem: Let $a,b,c$ be the sides of the $\Delta ABC$. Show that: $$ \left (\frac S R \right)^2 \le \frac 3 8 \left (\frac {ab \sqrt ...
1
vote
1answer
13 views

Question about the validity of a proof involving the abundancy index

Let $\sigma(x)$ be the sum of divisors of $x$, and denote the abundancy index of $x$ by $I(x) = \sigma(x)/x$. Consider the number $y^2 \in \mathbb{N}$, and suppose that I know that $I(y^2) < 4/3$. ...
1
vote
1answer
19 views

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$?

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$? $$p^2 q^2 \geq 3 p^2 q + 3p^2 + 3pq^2 + 3pq + 3p + 3q^2 + 3q + 3$$ I tried to use Wolfram Alpha, and it says that ...
0
votes
2answers
19 views

Quadratic inequality (Sign Reversal?)

I have the following inequality $\ (2x-3)^2-9>7$ I can reduce it down to $\ 2x-3>±4$ Now here is where I encounter a problem. Apparently the next step is $\ 2x-3>4 ~OR~ 2x-3<-4 $ ...
0
votes
0answers
25 views

upper bound on derivatives of a function defined on an arc

Given a smooth arc on the complex plane by $z=\cos t + 0.5 i \sin t,\; t\in[\pi/10,\pi/5] $ , and a non-analytic function $f(z) = \text{Re } z $ defined on the arc. Obviously, $f(z) = g(t) ...
19
votes
3answers
674 views

Inequality with five variables

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$ Easy to show ...
0
votes
1answer
111 views

Cyclic Inequality in n (at least 4) variables

Proof of a cyclic inequality. Let $a_i$ , $i=1..n$, $n \geq 3$ be real numbers with $a_i > 0 $ and $\prod_{i=1}^n a_i= 1$. Denote $S = \sum_{i=1}^n a_i$ (unrestricted). Prove (or disprove) the ...
1
vote
1answer
37 views

Determine the maximum and the minimum of an expression

Let $x,y,z \in \Bbb R, x,y,z \gt 0$ such that $x^2+y^2+z^2=1$. Determine tha maximum and the minimum possible values of the expression $$\frac {x^3+y^3+z^3} {x+y+z}.$$
1
vote
1answer
22 views

Use contraction mapping theorem to show that the integral equation has a unique continuous solution on $t \in [0,3]$

I have to use the contraction mapping theorem to prove that the integral equation with continuous functions $K(t,s)$ and $f(t)$, $\begin{equation*} x(t) = \lambda \int_{0}^{3} K(t,s) x(s)ds + f(t), ...
3
votes
1answer
55 views

Simple inequality with a,b,c

I'm looking for proof of $$\sqrt{a(b+c)}+\sqrt{b(c+a)}+\sqrt{c(a+b)} \leq \sqrt{2}(a+b+c)$$ I tried using $m_g \leq m_a$, generating the permutations of $\sqrt{a(b+c)} \leq \frac{a+(b+c)}{2}$ and ...
0
votes
1answer
78 views

Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$ [duplicate]

Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$ I'm not quite sure how to prove this. I want to try ...
2
votes
0answers
74 views
+300

When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there some some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, ...
2
votes
2answers
43 views

Find the minimum $k$

Find the minimum $k$, which $\exists a,b,c>0$, satisfies $$ \frac{kabc}{a+b+c}\geq (a+b)^2+(a+b+4c)^2$$ My Progress With the help of Mathematica, I found that when $k=100$, we can take ...
1
vote
1answer
29 views

Inequality with floored squareroots

$(\lfloor \sqrt{n}\rfloor +1)^2\ge n+1$, for all $n\in \mathbb{N}$. I have convinced myself that this is true, but would like to see a formal proof.
1
vote
4answers
53 views

If $a,A,b,B,c,C$ are non negative reals such that $a+A=b+B=c+C=k$ Prove that $aB+bC+cA \le k^2$

If $a,A,b,B,c,C$ are non negative reals such that $a+A=b+B=c+C=k$ Prove that $aB+bC+cA \le k^2$ I substituted $B=k-b,C=k-c,A=k-a$ and plugged them to get a quadratic of $k$ which I had to show ...
4
votes
2answers
120 views

Property of elements of a positive definite matrix

I am stuck in proving a property of a positive definite matrix. Let $A$ be a positive definite matrix. Then: $$|A_{ij}| \le \sqrt{A_{ii}A_{jj}} \le \frac{A_{ii}+A_{jj}}{2}.$$ My work: As for ...
3
votes
4answers
343 views

Does $|x|^p$ with $0<p<1$ satisfy the triangle inequality on $\mathbb{R}$?

I am curious about whether $|x|^p$ with $0<p<1$ satisfy $|x+y|^p\leq|x|^p+|y|^p$ for $x,y\in\mathbb{R}$. So far my trials show that this seems to be right... So can anybody confirm whether ...
-1
votes
1answer
49 views

Inequality of complex numbers involving modules [duplicate]

Let $z \in \Bbb C$ such that $|z| \ge 1$. Show that $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$ My try: I wrote $|z|^2$ as $z\times \bar z$, but I didn't get to any result. Can ...
0
votes
1answer
61 views

Let $a_1,a_2,a_3,…a_n\geq0$ and $a_1+a_2+a_3+…+a_n=2m$,then prove that the greatest possible value of $\sum_{i=1}^{n}a_i.a_{i+1}$.

Let $a_1,a_2,a_3,....a_n\geq0$ and $a_{n+1}=a_1$ with $a_1+a_2+a_3+....+a_n=2m$,then find the greatest possible value of $$\sum_{i=1}^{n}\ a_i\cdot a_{i+1}$$ I tried Cauchy-Schwarz but did not ...