Tagged Questions
3
votes
0answers
46 views
looking for reference or nice proof of trig lemma
Math people:
I am looking for a reference or a nice proof of the following fact. I have proven it myself, but my proof is messy: let $\theta \in (0,1]$ and $\alpha \in (0, \frac{1}{2}\theta^2]$. ...
3
votes
1answer
48 views
looking for reference for integral inequality
Math people:
I would like a reference for the following fact (?), which I proved myself (I am 99% sure the proof is valid) but which has probably been done before. My proof was a little messy. If ...
1
vote
2answers
35 views
Spivak problem on Schwarz inequality
I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality:
$$
x_1y_1 + x_2y_2 \leq ...
7
votes
1answer
136 views
Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$
Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that
$$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$
Prove that
$$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$
Where should ...
12
votes
2answers
118 views
Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$
How am I supposed to prove this inequality?
$$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$
Jordan inequality might be an option but led me ...
0
votes
0answers
61 views
Geometrical Inequality
Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals
$AC$ and $BD$ intersects at $E$. If the shortest height of the
triangle $ACD$ equals the radius of the incircle of the triangle ...
1
vote
1answer
35 views
How does one justify this inequality?
Let $f \in C[0,1]$ and let $\| f \|$ be the max of $f$ on $[0,1]$. Then we have
$|f + g|(x) = |f(x) + g(x)| \leq |f(x)| + |g(x)| \leq |f|(x) + |g|(x)$.
If $f$ has its max at $x_0$, then
...
1
vote
1answer
78 views
9
votes
2answers
172 views
Inequality $\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0$
Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$,
$$
\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0.
$$
I have this question a very long time ago from a book or magazine but I ...
1
vote
1answer
57 views
An inequality for a recursive relation
Define
$$
f\left( n+1\right) =f\left( n\right)\cdot e^{f\left( n\right) }
$$
for all $n\in \left\{ 1,2,...\right\} $, with $f\left( 1\right) =-1$.
Show that for all $n$
$$
f\left( n\right) \geq ...
1
vote
1answer
82 views
Does there exist such a pentagon that can be covered by a circle?
Does there exist a pentagon in which every two nonadjacent vertices
is connected by a diagonal and the minimal height of every triangle formed by the
sides and diagonals of the pentagon whose two ...
3
votes
2answers
70 views
May I use the triangle inequality for infinite series?
I have to prove the following statement:
Let $\lim_{n\to \infty}r_n=0$. Show that $\forall\varepsilon>0 \ \ \ \exists \, n_0 \in \mathbb N \ \ \ \forall x \in(-1,1):$
$$\left\lvert ...
2
votes
3answers
36 views
Question about absolute value in inequalities
My book presents the following: $$7 \le x \le 9 $$ so $$ -1 \le x - 8 \le 1 $$ and $$ |x-8| \le 1$$
I usually get confused with the way that taking the absolute value of an expression works. Could ...
0
votes
2answers
55 views
lower limit of $ \frac {|x+y\space|}{|x\space|+|y\space|} + \frac {|y+z\space|}{|y\space|+|z\space|} + \frac {|z+x\space|}{|z\space|+|x\space|}$
Let $x,y,z$ be non-zero real numbers . Then what is the minimum value (if exists) of $$ \frac {|x+y\space|}{|x\space|+|y\space|} + \frac {|y+z\space|}{|y\space|+|z\space|} + \frac ...
1
vote
0answers
24 views
Inequality of Partial Taylor Series
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}{k!} ...
1
vote
3answers
76 views
Infinite Series
How can you show that
$$\left(1-\frac{2}{n^2}\right)^{n^2/2} \le \frac{1}{e}\:\: \qquad\forall n \ge 2$$
Any ideas? Infinite series have never really been my thing. Thanks
3
votes
2answers
55 views
Prove the inequality $x^\alpha \le y^\alpha + z^\alpha$.
Given a triplet of non-negative numbers $x$, $y$, $z$ for which holds $x \le y + z$
one needs to prove that the inequality $x^\alpha \le y^\alpha + z^\alpha$ is also correct for all $\alpha \in ...
0
votes
1answer
19 views
Range of values for optimization?
Example 1: A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the dimensions of the window be to let in the ...
3
votes
4answers
58 views
Hyperbolic cosine
I have an A level exam question I'm not too sure how to approach:
a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$
b) Deduce $ \cosh x > x$
c) Find the point P such that it lies on ...
8
votes
3answers
188 views
Funny integral inequality
Assume $f(x) \in C^1([0,1])$,and $\int_0^{\frac{1}{2}}f(x)\text{d}x=0$,show that:
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq \frac{1}{12}\int_0^1[f'(x)]^2\text{d}x$$
and how to find the smallest ...
1
vote
2answers
38 views
In search for the domain in which the inequality holds
I wrote this simple inequality and raised the question what is the maximal domain such that the inequality holds, and the inequality is:
$\dfrac{1}{x+y}>{\dfrac{1}{x}}+{\dfrac{1}{y}}$. ...
8
votes
4answers
166 views
prove that $\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} , \ x>0$,
Can you show very easy methods? I hope I'll see many methods. Thank you everyone.
Prove that:
$$\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} \ \ \ \qquad \forall x>0.$$
2
votes
1answer
63 views
looking for upper bound on quantity with “erf”
Math people:
I am looking for an upper bound on
$$g(x,t) = \frac{-\operatorname{erf}(t)+2\operatorname{erf}(\frac{1}{2}t(x+1))-\operatorname{erf}(xt)}{(x-1)^2},$$
where $x > 1$ and $t > 0$. ...
1
vote
1answer
54 views
Is this inequality true for integrals?
$$\left | \int_ {-a}^{0} f - \int_{0}^{a} f \right | \geq \left | \left | \int_ {-a}^{0} f \right |- \left | \int_{0}^{a} f \right| \right | \geq \left | \int_ {-a}^{0} |f| - \int_{0}^{a}| f ...
6
votes
1answer
113 views
With $xy+yz+zx=-1$, proving: $x^2+2y^2+2z^2 …$
Assuming $xy+yz+zx=-1$, prove that :
$$x^2+2y^2+2z^2 \geq \frac{1+\sqrt{17}}{2}$$
1
vote
1answer
50 views
How to establish the estimate?
I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
4
votes
1answer
70 views
About inequalities (general)
I wonder if there is a sharper inequality than Hölder's inequality. I mean, we have
$$\int_A |fg|d\mu \leq \left( \int_A |f|^p d\mu \right)^{1/p} \left( \int_A |g|^q d\mu \right)^{1/q}$$
for ...
1
vote
0answers
65 views
reverse triangle inequality
I am trying to find a prove or disprove for the following inequality:
$$\left|\left|a\right|^{p}-\left|b\right|^{p}\right|\leqslant ...
1
vote
1answer
75 views
Proof of limit inequality
Prove that for any sequence $\{x_n\}$ of positive real numbers
$$\lim\text{sup}\sqrt[n]{x_n}\leq \lim\text{sup}\frac{x_{n+1}}{x_n}.$$
My attempt:
Let $A = \lim\text{sup}\frac{x_{n+1}}{x_n}$. ...
9
votes
3answers
177 views
Prove that $(e+x)^{e-x}>(e-x)^{e+x}$
I get stuck with proving that $$(e+x)^{e-x}>(e-x)^{e+x}$$ for $x \in (0, e)$. All I know, is that it is doable with Jensen inequality, and I started with defining $$f(x)=(e+x)^{e-x}$$ and further ...
10
votes
4answers
322 views
Proving the inequality $\arctan\frac{\pi}{2}\ge1$
Do you see any nice way to prove that
$$\arctan\frac{\pi}{2}\ge1 ?$$
Thanks!
Sis.
0
votes
0answers
48 views
Inequality : Partial Sum of $N$th root of unities
Let $\hat{\lambda}_{B}(l)=\sum_{n=0}^{B-1}e^{j\frac{2\pi}{N}nl}$, where $N$ is any positive integer and $1\leq B\leq N$ and $0 \leq l \leq N-1$. $\hat{\lambda}_{B}(l)$ is the partial sum of the ...
3
votes
2answers
83 views
Prove that $\frac{\int_0^1xf^2(x) \mathrm{d}x}{\int_0^1 xf(x) \mathrm{d}x}\le\frac{\int_0^1 f^2(x) \mathrm{d}x}{\int_0^1 f(x) \mathrm{d}x}$ [duplicate]
Let $f:[0,1]\rightarrow\mathbb{R_+}$ be a monotone decreasing function. We want to prove that
$$\frac{\int_0^1x(f(x))^2 \,\mathrm{d}x}{\int_0^1 xf(x) \,\mathrm{d}x}\le\frac{\int_0^1 (f(x))^2 ...
0
votes
3answers
83 views
A not too simple complex number inequality
Prove the following inequality $\forall n>0$ $\forall z \in \mathbb{C}$ such that $|z|=1$:
$$\vert z+\frac{1}{z} \vert <\vert z^{n} + i \vert + \vert \overline{z}^{n} + i \vert \leq 2\sqrt{2} ...
7
votes
3answers
135 views
Prove $\left(\frac{n+1}{\text{e}}\right)^n<n!<\text{e}\left(\frac{n+1}{\text{e}}\right)^{n+1}$
How to prove this inequality?
$$\left(\frac{n+1}{\text{e}}\right)^n<n!<\text{e}\left(\frac{n+1}{\text{e}}\right)^{n+1}$$
1
vote
1answer
66 views
How to prove: for some $c>0,x>2 , c,x\in \mathbb R , \, \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$
How to prove:
for some $c>0,x>2 , c,x\in \mathbb R$
$$ \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$$
I have tried my textbook, notes and also tried to find ...
4
votes
2answers
86 views
Does $1 + \frac{1}{x} + \sqrt{\frac{2x}{x + 1}},$ have a global minimum?
Does the following function have a global minimum:
$$1 + \frac{1}{x} + \sqrt{\frac{2x}{x + 1}},$$
where $x \in \mathbb{N}$?
I tried using WolframAlpha, but it appears to give an inconsistent ...
3
votes
2answers
161 views
Find the maximum area of the regular pentagon
Find the maximum area of the regular pentagon that inscribed a unit square.
2
votes
1answer
61 views
How to prove this inequality : $(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}} \, ; x>0,\ y>0;\ 0<a<b$
Prove that when $\displaystyle x>0,\ y>0;\ 0<a<b$
$$\displaystyle(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}}$$
1
vote
0answers
88 views
Proof of an inequality problem
Wise men or women over the world!! I badly ask you to help me.
Let $N$ and $B$ be two positive integers such that $1\le B\le \frac{N}{2}$ and $N=ug$ (for convenience, assume that $N$ is even.)
For ...
8
votes
1answer
143 views
Prove that $\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$
Prove that
$$\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$$
EDIT: inspired by Michael Hardy's suggestion I got that
$$\arcsin ...
3
votes
2answers
178 views
How to prove $f(x,y) = \frac{2 \log(y) \sqrt{y ( x-y)}}{x \log(x)}$ is bounded by $1$ for $(x,y) \to (0,0) $?
Let
$$f(x,y) = \frac{2 \log(y) \sqrt{y ( x-y)}}{x \log(x)} $$
with $(x,y) \in D = \{(x,y)\mid 0 < y \leq x \le 1 \}$.
How would one show (or disprove) that
$$
\forall \epsilon > 0\ \ \exists ...
0
votes
1answer
21 views
Positivity of mixed polynomial
I have posted a related question before (link), but I didn't really get a completely satisfactory answer, and also I believe that I was able to simplify the problem a little. Therefore, I hope that it ...
6
votes
4answers
121 views
Proving ${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2}$
While simplifying an inequality, this inequality was derived:
$${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2},\quad\quad\quad\quad n\in \mathbb{N}$$
Do you have any idea to prove it? It is ...
7
votes
1answer
200 views
On Magnitudes of Sums of Roots of Unity and a Simple Trigonometric Inequality
The Problem
Let $r,q,m$ be positive integers such that $4 \leq r$ and $1<m,q\leq r/2$. Is it the case that
$$\left | \sum_{k=0}^{q-1} \zeta^{km}\right | < \left |\sum_{k=0}^{q-1} ...
3
votes
2answers
133 views
Mean value theorem help?
So I am learning this chapter "mean value theorem", and there is an exercise.
Prove the inequality of $e^{x} \gt x+1$ for $x$ different from zero and the inequality of $2x \arctan x \ge \ln ...
2
votes
2answers
103 views
About the use of Stirling approximation
How to prove this inequality:
$$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$
Sry I forgot to mention that $x>300$
3
votes
3answers
132 views
How to show $\sqrt{f(x)} \geq \sqrt{f(1)} +\frac{1}{2}(x-1)$?
Suppose $f$ is a function such that $f(x) > 0$ and $f'(x)$ is continuous at every real number $x$. If $f'(t) \geq \sqrt{f(t)}$ for all $t$, then show that
$$\sqrt{f(x)} \geq \sqrt{f(1)} ...
2
votes
1answer
99 views
A trigonometric integral inequality
$$\displaystyle\frac{4\sin 1}{\pi }<\int_{0}^{1}{\frac{\cos x}{\sqrt{1-{{x}^{2}}}}}\text{d}x\le \frac{\pi }{2}\ln \left( \sec 1+\tan 1 \right)$$
I've got no ideas for this one.
2
votes
1answer
71 views
Integral inequality with sin exp
For $\displaystyle f(x)=\int_x^{x+1}\sin (\text{e}^t)\text{d}t$
Prove that :
$\displaystyle \text{e}^x\left|f(x)\right|\le 2$
