Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

3
votes
1answer
114 views

A determinant inequality

Let $A,B$ be two $m\times n$ real matrices. Then $$|AA'|\cdot |BB'|\geq |AB'|^2.$$ For square matrices, it is the equality. How to prove this inequality then?
3
votes
6answers
76 views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without ...
3
votes
1answer
80 views

$2(1+abc)+\sqrt2(1+a^2)(1+b^2)(1+c^2)\ge(1+a)(1+b)(1+c)$ for real numbers $a,b,c$

$a,b,c$ reel sayılar için ; $$2(1+abc)+\sqrt2(1+a^2)(1+b^2)(1+c^2)\ge(1+a)(1+b)(1+c)$$ Olduğunu gösteriniz. Translation:1 For real numbers $a,b,c$, show that: ...
3
votes
2answers
67 views

Prove through induction that $3^n > n^3$ for $n \geq 4$

I'm new to induction and have not done induction with inequalities before, so I get stuck at proving after the 3rd step. The question is: Use induction to show that $3^n > n^3$ for $n \geq ...
3
votes
3answers
218 views

Bernoulli's Inequality

I'm asked to used induction to prove Bernoulli's Inequality: If $1+x>0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far: Let $n=1$. Then $1+x\geq 1+x$. This is true. Now ...
3
votes
1answer
83 views

An inequality with radicals

If $s_{1}\ge t_{1}\ge t_{2}\ge s_{2}\ge0$, does one always have $(s_{1}-t_{1}+s_{2}+t_{2})^{1/2}\ge\sqrt{s_{1}}-\sqrt{t_{1}}+\sqrt{t_{2}}-\sqrt{s_{2}}$? Thanks a lot!
2
votes
2answers
176 views

Solving $x\; \leq \; \sqrt{20\; -\; x}$

This is how I tried to solve it: By squaring both sides: $x^{2}\; \leq \; 20\; -\; x$ $x^{2}\; +\; x\; -\; 20\; \leq \; 0$ Thus $-5\; \leq \; x\; \leq \; 4$ However, it seems that values less ...
2
votes
3answers
160 views

Proving that $n!≤((n+1)/2)^n$ by induction

I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this: $V(1): 1≤1 \text{ true}$ $V(n): n!≤((n+1)/2)^n$ $V(n+1): ...
2
votes
5answers
165 views

Prove the triangle inequality [duplicate]

I want to porve the triangle inequality: $x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$ I figured out that probably the cases: $x\geq0$ and $y \geq 0$ $x<0$ and $y < 0$ $x\geq0$ ...
2
votes
3answers
88 views

How do I solve inequalities of the form $\frac{ax}{b}\geq0$?

I need to find the maximum domain for $f(x) = \sqrt{\frac{4x+13}{(x+5)(2-x)}}$ Therefore, I should solve the inequality $$\frac{4x+13}{(x+5)(2-x)} \ge 0$$ I don't remember how to solve inequalities ...
2
votes
4answers
140 views

Solving a quadratic Inequality

My question is: Solve $$9x-14-x^2>0$$ My answer is: $2 < x < 7$ Though I know my answer is right, I want to know in what ways I can solve it and how it can be graphically represented. ...
2
votes
2answers
373 views

How to prove $n < n!$ if $n > 2$ by induction?

I am stuck with the question below, Prove by mathematical induction that $n<n!$ for $n>2$.
2
votes
1answer
593 views

Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
2
votes
3answers
584 views

Squaring across an inequality with an unknown number

This should be something relatively simple. I know there's a trick to this, I just can't remember it. I have an equation $$\frac{3x}{x-3}\geq 4.$$ I remember being shown at some point in my life that ...
1
vote
4answers
178 views

Induction proof of $n^{(n+1) }> n(n+1)^{(n-1)}$

The question statement from my homework booklet goes: Prove by mathematical induction that $n^{n+1} > n(n+1)^{n-1}$ is true for all integers $n \geq 2$. I've managed to come up with this ...
1
vote
4answers
104 views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq ...
1
vote
1answer
64 views

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds:

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds: $P(Z\leq -u \space \text{or} \space Z\geq v) \leq \frac{4\sigma^{2} + ...
1
vote
1answer
51 views

Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality?

Robin's theorem says that if $$\sigma(n)<e^\gamma n\log\log n$$ holds for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any ...
1
vote
2answers
154 views

Does the following inequality hold if and only if $N$ is an odd deficient number?

Let $N \in \mathbb{N}$. (That is, let $N$ be a positive integer.) This is in reference to two of my earlier questions here at MSE: Does the following inequality hold true, in general? Does this ...
1
vote
1answer
133 views

Find the minimum of this expression

This is a problem in my exam and I can't find the solution using elementary inequality knowledge. Can anyone here help me solve this. Thanks $a,b,c $ are positive real numbers which satisfy ...
1
vote
1answer
112 views

A question on mean value inequality

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
1
vote
0answers
40 views

$ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq C ( |z_1 - z_2 |^2 + |z_2| ^2 )|z_1 - z_2|$?

I questioned about this inequality before, but how about weaker one: For $\alpha = 0,1,2,3$, does this inequality always hold for any complex number $z_1, z_2$? $$ | \;\overline{z_1}^{3-\alpha} ...
1
vote
0answers
175 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
1
vote
2answers
70 views

Prove, formally that: $\log_2 n! \ge n$ , for all integers $n>3$.

Prove, formally that: $\log_2 n! \ge n$ for all integers $n>3$. Hint: first prove that $n! ≥2^n$, for all integers $n >3$. So far what I have: Base case, $n = 4$, $4! = 24$ $2^4 = 16$. ...
1
vote
1answer
210 views

Chernoff inequalities for the sum of Exponential RVs

These two well-known Chernoff bounds for the sum of RVs $X=\sum_{k=1}^{n}X_k$ in mulitplicative form, $\mathbf{P}(X \leq (1- \delta)\mathbf{E}X) \leq e^{-\frac{\delta^2 \mathbf{E}X}{2}}\\ ...
1
vote
3answers
211 views

Prove this inequality: $|a_1b_1+a_2b_2+\cdots+ a_nb_n|\leq 1$ for two normalised vectors

Help me prove this inequality: $$|a_1b_1+a_2b_2+\cdots+a_nb_n|\leq 1$$ if $$\begin{align*} a_1^2+a_2^2+\cdots+a_n^2=1, \\ b_1^2+b_2^2+\cdots+b_n^2=1.\end{align*}$$
1
vote
2answers
111 views

A hard proof of two matrix's elements

This is not duplicate of A matrix's element proof, but it is harder than that one. Given an constant $\alpha \in (0,1)$, and an $n \times n$ matrix $X$ whose all entries are between 0 and ...
1
vote
1answer
69 views

Implication of an inequality

We know that $$|l(x+u)-l(x)|<1 \text{ for } x\geq y>0 \text{ and } u\in[0,1]$$ Why does: $$|l(y+u)|<1+|l(y)|,x \in (y+1,y+2)$$ imply that $$|l(x)| \leq 1 + |l(y+1)| \leq 2+|l(y)|$$
1
vote
3answers
192 views

Trouble with an Inequality

In showing that if $f,g\in L^p$, then $f+g\in L^p$, one can use the fact that $$|f+g|^p\leq 2^p\left(|f|^p + |g|^p\right).$$ The result I'd like help in proving is this: Given that $1\lt p ...
1
vote
1answer
160 views

Prove $2^n > n^3$ [duplicate]

Let $P(n)$ be the property: $2^n > n^3$. Let's use mathematical induction to prove that $P(n)$ is true for $n\geq10$. Basis: $P(10): 2^{10} > 10^3 \Leftrightarrow 1024 > 1000$ which is true. ...
0
votes
1answer
46 views

Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$

Prove the following: $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$ Hello everyone, I am trying to create some sort of ...
0
votes
3answers
50 views

Inequality for small values of $t$

Suppose $x,(y > 0)$ are real numbers. I want to know if it is true that for small $t$, we have $$ (tx)^2 + (ty)^2 \leq 2ty $$
0
votes
1answer
53 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
0
votes
1answer
80 views

How $x^2$ increases by $x+\frac{1}{x}$?

I was going through one of the topic "Introduction to Formal proof".In one example while explaining "Hypothesis" and "conclusion" got confused. The example is as follows: If $x\geq 4$ then $2^x \geq ...
0
votes
1answer
52 views

Inequality, triangle, Law of cosines, integer

Prove $$|a^2+1-2a\cos{\theta}|^{\frac{1}{n}}\ge| a^{\frac{2}{n}}+1-2a^{\frac{1}{n}}\cos{\frac{\theta}{n}}|$$ where $a>0$ , $0<\theta<\pi$ and $n\ge2$ and $n\in N^+$.
0
votes
2answers
133 views

How to show that we can always choose a smaller number?

Suppose $A$ consists of all numbers $x \ge 0$ such that $x^2 \lt 2$. If the number $\sqrt{2}$ did not exist, there would not be a least number greater than all the numbers of $A$; for any $y > ...
0
votes
0answers
49 views

Do these inequalities regarding the gamma function and factorials work?

I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In a previous question, I asked whether the following inequality is ...
0
votes
1answer
88 views

How prove this inequlity $\sum_{k=n}^{2n-3}\frac{|\sin{k}|}{k}<\frac{1}{\sqrt{2}},n\ge 3$

pove that: $$\sum_{k=n}^{2n-3}\dfrac{|\sin{k}|}{k}<\dfrac{1}{\sqrt{2}},n\ge 3$$ This problem is my frend creat it today,Thank you someone can prove it.Thank you my idea,long ago I have prove it ...
0
votes
0answers
88 views

Proving a simple inequality

Can someone show that the inequality bellow holds? $$ f(n) \leq f(n+1) \ $$ Where $$ \frac{\sum\limits_{k=1}^n \Lambda(k) {k}/{n}\lceil{n}/{k}\rceil{}\{ n/k \}}{\sum\limits_{k=1}^n \Lambda(k)}=f(n)$$ ...
0
votes
2answers
71 views

A matrix's element proof

Thanks again for copper.hat and Robert Israel's quick immediate reply. While I am modifying the questions, they've already given the answer. Now in this thread, I've changed it back to the original ...
-1
votes
3answers
619 views

$n^8 \lt 8^n-1$

How can I prove that $n^8 \lt 8^n-1$ for all $n$? I'm searching for technical way to show that without using calculus theorems, using well known facts such as $n^2 \lt 2^n$ and so. Thanks!
-2
votes
2answers
95 views

$x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$

Show that the number of solutions in nonnegative integers of the inequality $$x_1+x_2+\cdots+x_n\leq M,$$ where $M$ is a nonnegative integer, is $C(M+n, n)$.
31
votes
2answers
1k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
40
votes
2answers
722 views

How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$

Nice Question: let $x\in [0,2\pi]$, show that: $$\sin{\sin{\sin{\sin{x}}}}\le\dfrac{4}{5}\cos{\cos{\cos{\cos{x}}}}?$$ I know this follow famous problem(1995 Russia Mathematical olympiad) ...
37
votes
7answers
1k views

Inequality: $(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27$

Let be $a,b,c \geq 0$ such that: $a^2+b^2+c^2=3$. Prove that: $$(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27.$$ I try to apply $GM \leq AM$ for $x=a^3+a+1$, $y=b^3+b+1,z=c^3+c+1$ and $$\displaystyle ...
15
votes
4answers
509 views

Prove $\sum_{i=1}^{n}\frac{a_{i}}{a_{i+1}}\ge\sum_{i=1}^{n}\frac{1-a_{i+1}}{1-a_{i}}$ if $a_{i}>0$ and $a_{1}+a_{2}+\cdots+a_{n}=1$

Let $a_{i}>0,i=1,2,\cdots,n$, and $a_{1}+a_{2}+\cdots+a_{n}=1$. How can we prove that $$\displaystyle\sum_{i=1}^{n}\dfrac{a_{i}}{a_{i+1}}\ge\displaystyle\sum_{i=1}^{n}\dfrac{1-a_{i+1}}{1-a_{i}}$$ ...
19
votes
8answers
809 views

Comparing $2013!$ and $1007^{2013}$

I have to compare the following two numbers: $$2013! \text{ and } 1007^{2013}$$ where $n! = 1 \times 2 \times \cdots \times (n-1) \times n$. I tried in different ways to group the $1 \times 2 ...
34
votes
2answers
611 views

This is stupid but I have a bad cold with cough

Can we have distinct positive real $x,y,z \neq 1$ with $$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutaion? It does not work well if any ...
33
votes
4answers
863 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 ...
29
votes
3answers
989 views

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

The statement is, of course, 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 ...