Questions on proving, manipulating and applying inequalities.

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0
votes
1answer
31 views

finding an absolute value inequality

The question asks, "find an absolute value inequality whose solution's are x>2 and x<-12". I have no idea where to start and was wondering if anyone could help
1
vote
1answer
60 views

Prove an inequality

for $x > -1$, Prove the following inequality: $$\left( {\ln (1 + x) + \sum\limits_{k = 1}^n {\frac{{{{( - 1)}^k}{x^k}}}{k}} } \right){( - x)^{n + 1}} \le 0$$ Following the advice to use ...
31
votes
10answers
2k views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
4
votes
1answer
633 views

Equality in Minkowski's inequality proof(no integrals)

So what I'm looking for is a proof for when does the equality hold in Minkowski's inequality? I'm talking about this form of inequality: $\left( \sum_{K=1}^n |x_k + y_k|^p \right)^{\frac{1}{p}} \leq ...
2
votes
2answers
569 views

Proof of Markov's inequality using alternate form of expectation

For nonnegative random variables $X$, there is an alternate expression for the expectation: $$E[X] = \int_0^\infty P(X \ge t) \mathop{dt}.$$ I am familiar with proofs of Markov's inequality $$P(X \ge ...
4
votes
0answers
63 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
5
votes
1answer
124 views

Inequality related with concave property

Assume that $f>0,f'<0$ and $f$ is logconcave(the log of $f$ is concave) and twice differentiable. Can we prove, or give a counter example to the following claim: there exists $\bar x>0$ such ...
12
votes
2answers
341 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
0
votes
1answer
45 views

Proving an inequality involving binomial coefficients

I have $3$ positive integers $i, k, q$, satisfying $1 \le k \le i \le q$. I am trying to prove the following inequality: $$\frac{{\left( {\begin{array}{*{20}{c}} {q - k}\\ {i - k} \end{array}} ...
3
votes
0answers
65 views

Howto prove that $\sum_{cyc}\cos\frac{A}{2}\cos\frac{B}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{7-4\sqrt{2}}{R}r$

let $ABC$ is a triangle with inradius $r$ and circumradius $R$. Show that ...
0
votes
2answers
67 views

Combinatorial Inequality

For any integer $n>1$ prove that, $$\large 2^n < {2n \choose n} < \frac{2^n}{\prod^{i=n-1}_{i=0}(1-\frac{i}{n})}$$ Now proving that the first term is smaller than the third term is ...
1
vote
2answers
55 views

Prove that $a^a\cdot b^b>\bigg(\dfrac{a+b}{2}\bigg)^{a+b}$ where $a\ne b$

Prove that $a^a\cdot b^b>\bigg(\dfrac{a+b}{2}\bigg)^{a+b}$ where $a\ne b$ My work: $a^a\cdot b^b>\bigg(\dfrac{a+b}{2}\bigg)^a\cdot\bigg(\dfrac{a+b}{2}\bigg)^b\implies ...
0
votes
1answer
66 views

Find an N satisfying this inequality

I am trying to catch up again. $$ \frac{1}{2^{n}} + \frac{1}{3^{n}} + \frac{1}{4^{n}} < \frac{1}{365} $$ Find an $N$ whereby all $n \geq N$ give correct outcomes I thought that as long as ...
2
votes
0answers
95 views

Apostol Limit Proof

This is an interesting proof of the product limit law. I can see the squeeze theorem but how do you work out the step when applying the triangle-CS inequality with norms? Will this work for any $n$ ...
1
vote
1answer
32 views

help with integral inequality

Let $P(R)=e^R\cdot\int_R^{\infty}F(z)e^{-z}dz=\int_0^{\infty}F(R+z)e^{-z}dz$. Is it true that $P(R) \geq 0$ for all $R$ implies $F(z) \geq 0$ for all $z$? In my case, $F(z)$ is a difference of CDF ...
5
votes
6answers
120 views

How do I prove $\sqrt{x^2 + y^2} \le |x| + |y|$?

Only a hint on how to prove this, if not a complete proof, would also be appreciated.
8
votes
4answers
391 views

Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\ldots+\frac{1}{3001}<\frac43$

Prove that $$1<\dfrac{1}{1001}+\dfrac{1}{1002}+\dfrac{1}{1003}+\ldots+\dfrac{1}{3001}<\dfrac43 \, .$$ My work: $$\begin{eqnarray*} ...
1
vote
2answers
54 views

Need guidance on a problem about oscillation from Spivak's Calculus on Manifolds

I've been stuck on this particular problem for a while now: Let $f: [a,b] \rightarrow \mathbb{R} $ be an increasing function. If $x_1, ... ,x_n \in [a,b]$ are distinct, show that $\sum\limits_{i=1}^n ...
0
votes
1answer
22 views

When is $\cos\frac{\pi}{x}<0$?

This seems really simple, but I'm trying to find a way to solve $\cos\frac{\pi}{x}<0$. I get $$x\geq \frac{\pi}{\arccos0}=\frac{\pi}{(2k+1)\frac{\pi}{2}} = \frac{2}{2k+1}$$ for $k\in\mathbb{Z}$. ...
0
votes
3answers
401 views

Using Cauchy-Schwarz inequality to prove that the mean of n real numbers is less than or equal to the root-mean-square of those numbers

Expressed mathematically, the question is to prove the that $\frac{1}{n}$ $\sum_{i=1}^{i=n}{a_i}\leqslant$ $\sqrt{\frac{1}{n}\sum_{i=1}^n{x_i}^2}.$ First of all, what form of Cauchy-Schwarz should I ...
2
votes
2answers
29 views

Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that..

Question : Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that $$\log \frac{x_0}{x_1}1993 + \log \frac{x_1}{x_2}1993 +\log ...
3
votes
5answers
132 views

Prove that, $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}\ge 4$ where we do not use AM-GM inequality on the given statement to prove it.

Prove that, $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}\ge 4$ where we do not use AM-GM inequality on the given statement to prove it. Typically, I am actually looking for a little advanced ...
0
votes
1answer
62 views

Inequality challenge

I was studying inequations when I encountered this problem here. How can I find a region of values for m where this inequation is true? $$-3<\frac{x^2+mx-2}{x^2-x+1}>2$$ Thanks
0
votes
2answers
72 views

Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
2
votes
1answer
177 views

Logarithms melting my brain

So I've got an inequality: $\ln(2x-5) > \ln(7-2x)$ and I attempt to solve by doing the following: $$\frac{\ln(2x)}{\ln(5)} > \frac{\ln(7)}{\ln(2x)}$$ $$\Rightarrow \ln(2x) \cdot \ln(2x) > ...
2
votes
1answer
90 views

Show that $\lim \inf a_n\le\lim\inf s_n.$

Let $\{a_n\}$ be a bounded sequence of real numbers. Let $s_n=\dfrac{a_1+a_2+\cdots+a_n}{n}~\forall~n\in\mathbb N.$ Show that $\lim \inf a_n\le\lim\inf s_n.$ The only definition I know of limit ...
4
votes
1answer
58 views

If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $(a+b)(a+c)=(b+c)^2$, prove that $(b-c)^2>8(b+c)$.

If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $$(a+b)(a+c)=(b+c)^2$$prove that $$(b-c)^2>8(b+c).$$ The first thing I did after I saw the problem was turning the inequality into this: ...
1
vote
1answer
36 views

Which one is valid: $ 2\cos((a+b)/2)<2$ or $ 2\cos((a+b)/2)\leq2$?

Which one is valid: $ 2\cos((a+b)/2)<2$ or $ 2\cos((a+b)/2)\leq2$? I need this to be true for my proof.
1
vote
1answer
52 views

If $x,y,z\in(0;1)$, prove that $(x+1)(y+1)(z+1)\ge \sqrt{8(x+y)(y+z)(z+x)}$.

If $x,y,z\in(0;1)$, prove that $$(x+1)(y+1)(z+1)\ge \sqrt{8(x+y)(y+z)(z+x)}$$ Both sides of the inequality are positive, so I could square them: $$(x+1)^2(y+1)^2(z+1)^2\ge8(x+y)(y+z)(z+x)$$ Even ...
3
votes
1answer
383 views

Exponential Inequality

I was working on a problem and reduced it to showing the following inequality: $$2x e^{x^2/6} \ge e^x - e^{-x} \text{ for $x \ge 0$}$$ I tried expanding everything in Taylor series to no avail. I ...
0
votes
1answer
761 views

Cauchy–Schwarz inequality on vector-valued L2 space

Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking ...
2
votes
1answer
255 views

Finding the regions of absolute stability for the theta method

I am trying to find for which choices of $\theta$ for which the theta method absolutely stable but I am having a lot of trouble solving the resulting inequality. It is straight forward to produce the ...
3
votes
2answers
63 views

Minimum value of the function $\sqrt{(1+1/m)(1+1/n)}$

If $m, n$ are positive real variables whose sum is a constant $k$, then what is the minimum value of $$\sqrt{\bigg(1 + \frac{1}{m}\bigg)\bigg(1 + \frac{1}{n}\bigg)}$$
18
votes
4answers
334 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
1
vote
0answers
60 views

Proving an inequality involving integral

Let $g: [a,b]\mapsto [0,1]$, with $\int_a^b|g'(t)|^2\,\mathrm{d}t\leq 1$. Suppose $b-a<\delta$, and define $$ \bar{g}=\frac{\int_{a}^{b}g\left(t\right)\,\mathrm{d}t}{b-a} $$ Show for ...
2
votes
1answer
91 views

How prove this inequality $\sum_{cyc}\sqrt{\frac{\cos{A}\cos{B}}{\cos{C}}}\ge\frac{3\sqrt{2}}{2}$

let $x,y,z>0$,and such $$x^2y^2+y^2z^2+x^2z^2+2x^2y^2z^2=1$$ show that $$x+y+z\ge\dfrac{3\sqrt{2}}{2}$$ My idea: in $\Delta ABC$,we have $$\cos^2{A}+\cos^2{B}+\cos^2{C}+2\cos{A}\cos{B}\cos{C}=1$$ ...
3
votes
2answers
68 views

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $a_1+a_2+\ldots+a_{100} >1$.

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $$a_1+a_2+\ldots+a_{100}>1$$ Prove the following statements: (i) Let $n_0$ be the smallest integer $n$ such that ...
1
vote
3answers
81 views

Find the minimum of : $P=(1+\frac{1}{a^3})(1+\frac{1}{b^3})(1+\frac{1}{c^3})$

$a;b;c\in \mathbb{R}^+$ such that $a+b+c=6$. Find the minimum of : $P=(1+\frac{1}{a^3})(1+\frac{1}{b^3})(1+\frac{1}{c^3})$ Thanks :) I have no ideas about this problem ! :(
0
votes
3answers
28 views

Let $x,y$ satisfy $x-3\sqrt{x+1}=3\sqrt{y+2}; x\ge-1;y\ge-2$. Find the min and max values of: $P=x+y$

Let $x,y$ satisfy $$x-3\sqrt{x+1}=3\sqrt{y+2}, \qquad x\ge-1,\quad y\ge-2.$$ Find the min and max values of: $P=x+y$
5
votes
2answers
86 views

An inequality from derivatives

Let $f:\;\mathbb{R}\to\mathbb{R}$ be differentiable with $f(0)=0$ and $f''(0)$ exists and is positive. I would like to show that there exists $x>0$ such that $f(2x)>2f(x)$. My try: ...
0
votes
1answer
125 views

Proving absolute value inequality by contradiction

Prove that for $|x|, |y|, |z| \geq 2$ the following holds: $|x^2 + y| + |y^2 + z| + |z^2 + x| \geq |x| + |y| + |z|$ So I thought about a simple proof by contradiction but am not sure whether it's a ...
0
votes
2answers
48 views

Having trouble solving an inequality

I'm a trying to prove a recurrence relation (by substitution) for an algorithm class and I'm shamefully stuck in a rather simple looking inequality. I need to solve this inequality for constant $c$: ...
1
vote
1answer
67 views

Prove this trigonometry inequality

I'm having difficulty proving that tan(26°) < 0.5 < tan(27°) . Any idea ? Thanks. p.s. 26 and 27 are in degrees.
4
votes
1answer
129 views

Prove: $\frac{2x^2+xy}{(y+\sqrt{zx}+z)^2}+\frac{2y^2+yz}{(z+\sqrt{xy}+x)^2}+\frac{2z^2+zx}{(x+\sqrt{yz}+y)^2}\ge1$

Prove: $$\frac{2x^2+xy}{(y+\sqrt{zx}+z)^2}+\frac{2y^2+yz}{(z+\sqrt{xy}+x)^2}+\frac{2z^2+zx}{(x+\sqrt{yz}+y)^2}\ge1$$ ($x,y,z>0$)
3
votes
4answers
176 views

Integral inequality for continuous function

Let $ f $ be a continuous, real-valued function on $[0, 1] $. Show that $$\int_0^1 \int_0^1 |f (x)+f (y)| dx dy \ge \int_0^1 |f (x)| dx $$ I tried to dissect the square in triangles and use ...
2
votes
1answer
619 views

How to shade the union and intersection of inequalities

How does one find the union and intersection of two inequalities by shading regions in a graph? For instance, find the union and intersection of $y \lt 3$ and $x \ge 2$?
5
votes
3answers
183 views

An inequality for sides of a triangle

Let $ a, b, c $ be sides of a triangle and $ ab+bc+ca=1 $. Show $$(a+1)(b+1)(c+1)<4 $$ I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4 $. I am ...
5
votes
4answers
189 views

How to find the minimum value of this function?

How to find the minimum value of $$\frac{x}{3y^2+3z^2+3yz+1}+\frac{y}{3x^2+3z^2+3xz+1}+\frac{z}{3x^2+3y^2+3xy+1}$$,where $x,y,z\geq 0$ and $x+y+z=1$. It seems to be hard if we use calculus methods. ...
3
votes
0answers
74 views

Prove this inequality for all integers $m>2$

Prove this inequality for all integers $m$: $$∑_{n=2}^{m} \frac{1-n^{2α-1}}{n^{\alpha}} > \frac{1-(m+1)^{2α-1}}{(m+1)^{\alpha}}$$ for all $0<α<1/2$ and $m>2$.
1
vote
1answer
216 views

Prove one case of the Reverse Triangle Inequality $|x-y|≥|x|-|y|$ for all reals $x$ and $y$ [duplicate]

Prove this inequality for all reals $x$ and $y$: $$|x-y|≥|x|-|y|$$