Questions on proving, manipulating and applying inequalities.

learn more… | top users | synonyms (1)

6
votes
1answer
248 views

(Tournament of towns 1994) Prove the inequality

Let $a_1,a_2,\ldots,a_n$ be real positive numbers. Prove that $$\left(1+\frac{a_1^2}{a_2}\right)\left(1+\frac{a_2^2}{a_3}\right) \cdots \left(1+\frac{a_n^2}{a_1}\right) \geq(1+a_1)(1+a_2) \cdots (1+...
1
vote
4answers
51 views

Solving the Inequality $\frac{14x}{x+1}<\frac{9x-30}{x-4}$

The question says to find all the integral values of x for which the inequality holds. the question is $$\frac{14x}{x+1}<\frac{9x-30}{x-4}$$ My Solution \begin{align} & \frac{14x}{x+1} < ...
1
vote
1answer
42 views

Is $\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}$ bounded independently of $a$?

Fix an $a\in \mathbb{R}$ and consider the sum $$\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}.$$ Is this sum bounded independent of $a$? I think the answer should be yes since for $...
1
vote
0answers
30 views

Calculate (or estimate) $S(t)=\sum_{j=0}^k \binom k j |t-j|^{k-1}$.

Let $t\in\mathbb R$. Calculate, or estimate from above and below, the following sum $$S(t)=\sum_{j=0}^k \binom k j |t-j|^{k-1}.$$ I have not any idea.
0
votes
0answers
32 views

Squaring both sides of an inequality

I found that I can square both sides of an inequality as long as both sides are non-negative. But if we do, for example, this inequality: ...
0
votes
2answers
42 views

Why does this inequality hold for $|y|\geq 1$?

My lecture notes use this inequality for a complex $z=x+iy$ with $|y|\geq 1$ $$|\cot(z)| \leq \frac{1+\exp(-2|y|)}{1-\exp(-2|y|)}.$$ How can I show it? My attempt: \begin{align*} |\cot(z)| &= \...
-1
votes
4answers
92 views

Prove $n^{n/2} < n!$ if $n \gt 2$ [duplicate]

Ive been stuck on this question for so long.How do i do it? $n^{n/2} < n!$ if $n \gt 2, n \in \mathbb{N}$. Please help guys.
3
votes
3answers
104 views

Solving the trig inequality $|\sin{x} + \cos{x}| > 1$

$|\sin{x} + \cos{x} |> 1$ How to solve this kind of question? Is there any websites to learn trigonometry inequalities? My teacher only taught us the simple question but not the complicated one. ...
1
vote
1answer
52 views

Number of divisors greater than a number [closed]

Given a number $x$, it is easy to count its total number of divisors by combinatorial method. Is there a way to efficiently determine the number of divisors of $x$ greater than a given number $y$?
1
vote
1answer
56 views

Give an example of an inequality with exactly 3 solutions. [closed]

I am helping a friend try and do their pre-algebra and cannot for the life of me figure this out. I'm pretty sure it's an error in the phrasing, but I'm not too sure. The only inequalities I can come ...
1
vote
1answer
51 views

Proving the inequality $\bigg|\int_a^b f(x) \, dx - \frac{b-a}{n} \sum_{k=1}^n f\big(a + \frac{2k-1}{2n}(b-a)\big) \bigg| < \frac{C}{n^2}$

Suppose $f$ is twice differentiable and $|f''(x)| < B,$ some constant. Using Taylor's theorem, it is easy to show that $$\big|2Af(0) - \int_{-A}^A f(x)\;dx\big| < \frac{A^3B}{3}.$$ I am ...
-2
votes
0answers
32 views

Verifying inequality in mathematica

I am trying to prove or disprove the flowing inequality : $0<a, b<1, a^2+b^2=1, 0<x<1/2, 0<y<1/2$ $\rightarrow$ $1-a^{2x-1}b^{2y-1}>0$?? For this I wrote ...
2
votes
1answer
24 views

If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$?

The title says it all. If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$? Here $I(x)$ is defined to be the ratio $$I(x) = \...
3
votes
1answer
49 views

Maximizing $\frac{x(1-f(x))}{3-f(x)}$

Let $f:[0,1]\rightarrow[0,1]$ be a nondecreasing function such that $f(0)=0$ and $f(1)=1$. Let $x_1\in[0,1]$ be the value maximizing $x(1-f(x))$. Let $x_2\in[0,1]$ be the value maximizing $\frac{x(...
0
votes
1answer
32 views

Using induction on modified inequalities.

Here's the original problem: Prove by induction that $\left(\frac{1}{2}\right) \left(\frac{3}{4}\right) \cdots \left(\frac{2n-1}{2n} \right) \leq \frac{1}{\sqrt{n+1}}$ for all $n \in \mathbb{N}$. ...
0
votes
2answers
54 views

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$

For every real positive n prove that $\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}$.Hence,or otherwise prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$. Wher [x] denotes the greatest integer not ...
6
votes
0answers
74 views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
0
votes
2answers
47 views

How to prove $a,b,c \in \mathbb{R} \mid a+b+c \geq abc \implies 3abc(a+b+c) \geq 3(abc)^2$?

I'm working through some proofs from Cvetkovski's "Inequalities", when I came across a more difficult one (for newbies like me). Given $a, b, c \in \mathbb{R} \mid a+b+c \geq abc$, how can we prove ...
2
votes
3answers
62 views

How to compare $\left(\sin \left(x\right)\right)^{\cos \left(x\right)}$ and $ \left(\cos \left(x\right)\right)^{\sin \left(x\right)}$

I am new here ,can anybody help to solve this problem: How to compare $\left(\sin \left(x\right)\right)^{\cos \left(x\right)}$ and $ \left(\cos \left(x\right)\right)^{\sin \left(x\right)}$ in the ...
-5
votes
5answers
91 views

The 1000th partial sum of the series $\sum 1/n^2$ is less than 2 [closed]

Can anyone help me with this problem: Prove: $$1/1^2 + 1/2^2 +1/3^2 +\dots +1/1000^2 <2$$
3
votes
3answers
103 views

Can this upper bound for $\sum_{i=1}^n \lfloor \sqrt{p_i} \rfloor, p_i \in \Bbb P$ be improved?

I would like to find the smallest possible upper bound for the following sum of prime radicals (OEIS A062048): $\sum_{i=1}^n \lfloor \sqrt{p_i} \rfloor, p_i \in \Bbb P$ This is my attempt. It ...
0
votes
2answers
29 views

$f(x)=x\ln x-\frac{k}{x}$ and $f(x_1)=f(x_2)=0$ $\Rightarrow$ $f'\left(\frac{x_1+x_2}{2}\right)\not=0$

If $f(x)=x\ln x-\frac{k}{x}$. And $x_1$, $x_2$ are two roots of $f(x)=0$. Then $f'\left(\frac{x_1+x_2}{2}\right)\not=0$ First, I determine the range of $k$. Because $f=0$ has two roots $\iff$ $x^2\...
1
vote
3answers
53 views

Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.

I'm stuck with the proof of the following: Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$. I've tried starting with $s_n \...
0
votes
2answers
47 views

Inequality with complex root and positive imaginary part

Let $z$ be a complex number with $\mathrm{Im}(z)>0$, and we consider $$w:=\frac{-z+\sqrt{z^2-4}}{2}.$$ It is written that "we take the square root so that $\mathrm{Im}(w)>0".$ I want to prove ...
1
vote
2answers
79 views

p can take any value in the interval?

If the equation $(\cos(p)-1)x^2+\cos(p)x+\sin(p)=0$ in the variable $x$ has real roots, then $p$ can take any value in what interval? I applied the discriminant $D>0$. I get $\cos^2( p)-4(\cos(...
0
votes
2answers
47 views

Let $f(x)$ be a quadratic with nonreal roots. Show that if the coefficient of $x^2$ in $f(x)$ is positive, then $f(x)>0$ for all $x$

I do not understand the solution the book gives: "Completing the square gives $f(x)=a(x-h)^2+k$ for some constants $h$ and $k$. The roots of $f(x)$ are the solutions to the equation $a(x-h)^2+k=0$. ...
0
votes
4answers
64 views

If you take the reciprocal in an inequality, would it change the $>/< $ signs?

Example:$$-16<\frac{1}{x}-\frac{1}{4}<16$$ In the example above, if you take the reciprocal of $$\frac{1}{x}-\frac{1}{4} = \frac{x}{1}-\frac{4}{1}$$ would that flip the $<$ to $>$ or ...
5
votes
2answers
73 views

If the entries of a positive semidefinite matrix shrink individually, will the operator norm always decrease?

Given a positive semidefinite matrix $P$, if we scale down its entries individually, will its operator norm always decrease? Put it another way: Suppose $P\in M_n(\mathbb R)$ is positive ...
6
votes
2answers
225 views

Constant such that $\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$

What is the greatest constant $k>0$ such that $$\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$$ for all $0\leq b\leq 1$ and $0\leq c\leq d\leq 1$? The right-hand ...
4
votes
2answers
43 views

Question on the inequality of sequences

Given two sequence $(a_n)_{n \geq 0}$ , $(b_n)_{n \geq 0}$ satisfing $a_n,b_n >0$ for all $n$ and $\sum_{n}a_n \gtrsim \sum_{n}b_n$. My question is that: For a sequence $(c_n)_{n \geq 0}$ be ...
0
votes
1answer
28 views

Limit including lower branch of Lambert function

I am trying to show that $\frac{1}{2\left(1-e^x\right)}-\frac{1}{x}W_{-1}\left[\frac{x}{2\left(1-e^x\right)}\exp\left(\frac{x}{2\left(1-e^x\right)}\right)\right]\geq 1,$ for $x>0$, where $W_{-1}$ ...
0
votes
3answers
53 views

Have $n$ real root then show that ${(n-1)\left(c_{n-1}\right)^2} \geq 2 n c_{n-2} c_n$

What should I do here? I don't even know where to start from. Please help me by giving me a hint. If $$x^{n } - c_{1} x^{n-1}+c_2 x^{n-2} -c_3 x^{n-3}+\cdots+(-1)^{n-1} c_{n-1} x+(-1)^n c_n=0(c_1,...
0
votes
0answers
26 views

How can I deduce this interval on this inequality?

Good morning to everyone. I have an inequality and I have to find the sign of it. At first I thought that it's positive on all intervals (excepting the number $2$), but after looking on my answer ...
2
votes
0answers
53 views

Generalization of Jensen's inequality to multivariate functions

Is there a generalization of Jensen's inequality for convex multivariate functions? By convex, let's say $f$ is a multivariate function defined on the convex set $A$, and for all $x,y \in A$ and $\...
0
votes
2answers
61 views

Solve exponential inequality

I've come across the following exponential inequality and, unfortunately, I encountered some difficulties trying to solve it. $$ \left | x \right |^{2x^2 - 3x + 1} \leq 1, x \in \mathbb{R} $$ I ...
0
votes
1answer
70 views

As a reviewer of a math manuscript do you accept graph of a function as a proof for an inequality?

Let's say I have a function $f(x)$ and an empirical approximation of the function $\tilde{f}(x)$. I cannot prove mathematically that errors is in certain bound. However, when I plot the error, the ...
2
votes
1answer
58 views

Polynomial identity in positive terms, including AM-GM bound

Consider $n$ nonnegative numbers $x_1 \cdots x_n$. An easy consequence of the AM-GM inequality $$ \frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n} $$ is a lower bound on a ...
2
votes
1answer
40 views

Value of $F(r)$ to maximize $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
1
vote
1answer
30 views

Find non trivial estimation for a Dirichlet series

I would like to estimate a Dirichlet series, for this I need a estimation for $\sup_{k \in \mathbb{N}}\left|e^{iu f(p^{k+1})}-e^{iu(1+f(p^{k}))}\right|$ where $f$ is real arithmetic addtive function ...
1
vote
0answers
49 views

Upper bound for prime counting function $\pi(x)$

Let $\pi(x)$ denote the number of primes less than or equal to $x$. I want to prove $$ \pi(x) \leq \frac{9x\log 2}{\log x} $$ for every integer $x\geq 2$. In the problem (from Murty's $\textit{...
6
votes
3answers
153 views

Comment on $x$, if $x=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$.

The main question goes like this: $$x=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$$ Comment on $x$. There are options given as well: A) $x$ is an irrational number B) $2<x<3$ C) $x=3$ D) None of ...
1
vote
1answer
60 views

Location of roots in a quadratic equation

If $a$ & 4$a$ + 3$b$ + 2$c$ have same sign. Then \begin{align} ax^2 + bx +c =0 \end{align} ($a$ is not equal to zero) can not have roots belonging to : (a) (-1,2) (b) (-1,1) (c) (1,...
-3
votes
1answer
49 views

Inequality of three variables

I read from the Internet that one can prove the following inequality using the rearrangement inequality: If  $x,y,z>0$ and $xyz=1$ then $$\frac{1}{x+y+1} + \frac{1}{y+z+1} + \frac{1}{x+z+1}\leq \...
1
vote
2answers
108 views

Prove $\frac{a}{b} + \frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b} + \frac{a+b}{a+c} + \frac{b+c}{b+a}$ [closed]

Prove that $\frac{a}{b} + \frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b} + \frac{a+b}{a+c} + \frac{b+c}{b+a}$ with a,b,c > 0
2
votes
4answers
64 views

How to solve this inequality with absolute value: $ \frac{\left|x-3\right|}{\left|x+2\right|}\le 3 $

Good morning to everyone. I have an inequality that I don't know how to solve: $$ \frac{\left|x-3\right|}{\left|x+2\right|}\le 3 $$ I tried to solve it in this way: $$ \frac{\left|x-3\right|}{\left|x+...
0
votes
1answer
19 views

Relationship of $L_1$ distance between CDFs and PDFs?

Let $F:(-\infty,\infty)\rightarrow[0,1]$ and $G:(-\infty,\infty)\rightarrow[0,1]$ two CDFs with PDFs $f$ and $g$, respectively. Is there a connection/inequality between: $$d_1 = \int_{-\infty}^{\...
1
vote
1answer
35 views

Upper limits for $s_n$ $\leq$ $t_n$ [duplicate]

Let $(s_n)$ and $(t_n)$ be two sequences of real numbers. Suppose there exists $N_0$ such that for all $n>N_0$, $s_n \leq t_n$. Two Questions: Suppose that $\lim s_n$ and $\lim t_n$ both exist. ...
1
vote
6answers
102 views

If $x\in \left(0,\frac{\pi}{4}\right)$ then $\frac{\cos x}{(\sin^2 x)(\cos x-\sin x)}>8$

If $\displaystyle x\in \left(0,\frac{\pi}{4}\right)\;,$ Then prove that $\displaystyle \frac{\cos x}{\sin^2 x(\cos x-\sin x)}>8$ $\bf{My\; Try::}$ Let $$f(x) = \frac{\cos x}{\sin^2 x(\cos x-\sin ...
3
votes
1answer
46 views

Operator norm under shrinkage

If I have a $n$-dim matrix $A=\{a_{ij}\}$, and I multiply each elements by a factor $w_{ij}$ in $[0,1]$, and get a new matrix $A_w=\{a_{ij}w_{ij}\}$. Do I have $$||A||\ge \lVert A_w\rVert$$ where the ...
2
votes
1answer
81 views

How to show that $ \left|\int_0^\infty \cos(ax)e^{-x^4} dx \right| \le \left|\int_0^\infty \cos(bx)e^{-x^4} dx \right|$

How to show that \begin{align} \left| \frac{\int_0^\infty \cos(ax)e^{-x^4} dx}{\int_0^\infty \cos(bx)e^{-x^4} dx} \right| \le 1 \end{align} if $a\ge b \ge 0$. This is what I did. One has to show ...