Questions on proving, manipulating and applying inequalities.

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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! ...
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
1answer
67 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 ...
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 ...
-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 ...
-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 ...
1
vote
1answer
55 views

Geometric inequality $180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}$

Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}\le 180^{\circ}$$ My work so far: ...
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
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
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 ...
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
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
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) ...
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 $ ...
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 ...
2
votes
1answer
50 views

inequality involving heights and bisectors

Let $a,b,c,a \le b \le c$ be the sides of the triangle $ABC$, $l_a,l_b,l_c$ the lengths of its bisectors and $h_a,h_b,h_c$ the lengths of its heights. Prove that: $$\frac {h_a+h_c} {h_b} \ge \frac ...
1
vote
1answer
38 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
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
54 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 ...
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 ...
7
votes
3answers
159 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) ...
1
vote
2answers
28 views

Given $\int _{-1}^{1}g(x)= 1$ show that $\int _{-1}^{1}f(x)g(x)\geq 1$ for certain $f,g$.

Let $f$ and $g$ be two positive valued functions defined on $[-1,1]$, such that $f(x)f(-x)=1$, and $g$ is an even function with $\int _{-1}^{1}g(x)= 1$. Show that $\int _{-1}^{1}f(x)g(x)\geq 1$. I ...
0
votes
2answers
25 views

Solving inequality with a recursive formula without its closed form?

I have the following problem: $$a_{1}=1$$ $$a_{2}=3$$ $$a_{n+2}=a_{n+1}+5a_{n}$$ I have to prove this inequality: $$a_{n}<1+3^{n-1}$$ So my question, is there a way to solve this inequality without ...
0
votes
0answers
19 views

Proof of an inequality in C ,(2)

Let $n\ge 2$is a integer,$z_{1},z_{2},\cdots,z_{n}$ are $n$ complex numbers Prove that $$\sum_{k=1}^{n}|1+z_{k}|+\dfrac{1}{n-1}\sum_{1\le i<j\le n}|1+z_{i}z_{j}|\ge\sum_{k=1}^{n}|z_{k}|$$ for ...
0
votes
1answer
77 views

How to solve $2^x < x^2$

How do you solve : $$2^x < x^2$$ My math years are behind me, so I can't wrap my head around how to continue after this step : $$2^x - x^2 < 0$$ I think there's a trick since it's a 0 ...
0
votes
1answer
11 views

System of linear inequalities - mixture

I have a problem and I couldn't find a solution yet (and the method which way I can achieve the solution). Example: We have 4 types bottles of water. Bottle 1 - with capacity 750ml - we have 2 for ...
1
vote
2answers
62 views

Prove that $a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$ for $a, b, c > 0$

Prove for $a, b, c > 0$ that $$a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$$ Could you give me some hints on this? I thought that Jensen's inequality might ...
0
votes
0answers
9 views

Preserving the positive correlations of two functions with a mixed measure

Let $V_1,V_2:\mathbb{R}\rightarrow \mathbb{R}$, be two convex functions which admit a minimum (which implies $\lim_{x\rightarrow \pm\infty}V_i(x)= \infty $). Let ...
-1
votes
1answer
51 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 ...
9
votes
1answer
149 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 ...
1
vote
3answers
51 views

Prove this exp and log inequality?

show that $$e^x-\ln{(x+2)}>\dfrac{1}{6}\tag{(1)}$$ I know $$e^x>x+1,\ln{(x+2)}<x+1$$ so I have only prove $$e^x-\ln{(x+2)}>0$$ But How to prove $(1)$?
-3
votes
2answers
39 views

proving some identities about $xy=0$ [closed]

How do I prove that if $x\cdot y=0$ then $x=0$ or $y=0$? also a similar question if $x,y\in\Bbb R$ and $x^2+y^2=0$ then $x=0$ and $y=0$. How do I prove it? is it unprovable? Also how do I prove ...
0
votes
1answer
36 views

A binomial-related inequality

For integer $m\geq 1$, show that: $$\sum_{|k|<\sqrt{m}}{2m \choose m+k}\geq 2^{2m-1}.$$ What I have tried: I tried binomial expansion of $2^{2m}$ but it was unsuccessful. Any other idea?
0
votes
8answers
43 views

How can the following inequality hold for any real $x$?

$|1-x| + |1 + x| \ge 2$? I've tried proving this as: $|1-x| + |1+ x| \ge 1 - |x| + 1 - |x| = 2 - 2|x|$, and here, of course, I can't say $2- 2|x| \ge 2.$ Edit: Thanks so much, everyone.
2
votes
3answers
43 views

Proof help: Prove that $x^2+y^2+z^2 \geq xy+xz+yz$ [duplicate]

$x^2+y^2+z^2 \geq xy+xz+yz $ for all real numbers, x, y, and z. I'm not very good with working inequality proofs. Can someone help me prove this? The technique doesn't really matter.
1
vote
0answers
35 views

Prove the inequality about $Re(z)$

Consider three different vectors $x$,$y$ and $z$ in $\mathbb{C}^{n}$. So $x = (x_{1} \dots x_{n})$ and this is the same for $y$ and $z$. Now we have $\langle x,y\rangle = ...
2
votes
1answer
37 views

Find smallest k for which the inequality holds

The smallest positive number $K$ for which the inequality $|\sin^2 x - \sin^2 y| \le K|x-y|$ holds for all $x,y$ is
1
vote
1answer
33 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (v) and (vi) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
1
vote
1answer
19 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (iii) and (iv) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
0
votes
3answers
34 views

Is there an inequality for $\sinh(x)$ which is similar to this inequality $\cosh(x)\leq e^{x^2/2}$

Is there an inequality for $\sinh(x)$ which is similar to this cosh x inequality?
-4
votes
2answers
85 views

Trying to derive a contradiction with this simple inequality, [closed]

I am stuck at $$(a+d)^2 - 4(ad-bc) < 0$$ $$\implies (a+d)^2<4(ad-bc)$$ $$\implies (a+d)<2\sqrt{ad-bc}$$ where $a,b,c,d \ge 0$. Is there a contradiction to derive here? Also, the square ...
1
vote
1answer
25 views

A bound (dominated function) for $\cosh^2\left(t\sqrt{1-\gamma^2}\right)$

I would like to bound $$\cosh^2\left(t\sqrt{1-\gamma^2}\right)$$ for all $t\in\mathbb{R}$, where $\gamma^2\leq1$. How can I do such thing? This inequality maybe useful cosh x inequality
2
votes
3answers
32 views

Inequality, only one solution from algebra

I recently came along the following problem: $$f(x) = {4-x^2 \over 4-\sqrt{x}}$$ Solve for:$$f(x) ≥ 1$$ My Attempt Now I know that one of the restrictions on the domain is $x≥0$, thus one of the ...
0
votes
1answer
22 views

on Matrix Inequality

Let $A=(a_{ij})$, and $B=(b_{ij})$ be two $n$ by $n$ real symmetric matrices such that $$ a_{ij}\leq b_{ij}+\alpha, \quad \alpha>0. $$ Can we conclude that $A\leq B +\textbf{1}\alpha$? Note ...
1
vote
3answers
45 views

Prove that $0.5x^2 -3x ≥ -4.5$ for all real numbers x.

I'm not familiar at all with inequality proofs. How do I approach this problem?
2
votes
0answers
87 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 exist 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, ...