All Questions
30
questions
2
votes
0
answers
49
views
Prove that $(f+g)(x)<t$ ($t\in\mathbb{R}$) holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I got stuck on this question:
Prove that $(f+g)(x)<t$, $t\in\mathbb{R}$, holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I think one direction is ...
- 2,493
3
votes
1
answer
136
views
Proving that $f(x)$ satisfying $2f(x)f(1)\leq f(x)f(1)+1\leq 2f(2x)$ for all $x>0$ is a constant function
How do I prove the following, where $f:[0,\infty)\to[0,\infty)$:
For $x>0$ if $f(x)$ satisfies
$$2f(x)f(1)\leq f(x)f(1)+1\leq 2f(2x)$$
then $f(x)$ is a constant function.
I have out found out ...
- 45
2
votes
2
answers
94
views
Alternative proofs of this Inequality
So I was reading a paper which made the claim "It is easy to see that $\frac{1-e^{-\alpha}}{\alpha} > 1-\frac{\alpha}{2} > \frac{1}{1+\alpha}$ when $0 < \alpha < 1$."
Verifying ...
- 5,880
2
votes
0
answers
94
views
Which assumptions on $f$ and $T$ make $\int_0^T f(x) dx \le \int_0^T |f(x)|^2 dx\implies 1\le \int_0^T |f(x)| dx$ true?
Let $T>0$ be fixed and let $f$ be a real valued function such that
$$\|f\|_{\infty}\le T,$$
where $\|\cdot\|_{\infty}$ denotes the sup-norm.
My question is the following: if
$$\int_0^T f(x) dx \le \...
user603537
0
votes
0
answers
51
views
Gravitational potential energy (explanation)
Hello to you dear person reading, I need your help to please explain me this :
$${\displaystyle \Phi (P)={\frac {-GmM}{\|OP\|}}+K}{\displaystyle \Phi (P)={\frac {-GmM}{\|OP\|}}+K}$$
(I haven't learned ...
- 1
1
vote
1
answer
57
views
Property on the infimum
Let $f: I \to \mathbb{R}$ be a function, where $I \subset \mathbb{R}^+.$ Let $u>0,E=\{x \in I,f(x)>u\}$ . Define $y \in \overline{\mathbb{R}}^+:y=\inf E$ if $E \neq \emptyset$ and $y=+\infty$ ...
- 11
0
votes
1
answer
52
views
An inequality for an increasing, concave function
Let $f:[0,\infty)\rightarrow[0,\infty)$ be a smooth function such that $f(0)=0$,
$f''<0$, $f'(x)>0$ for $x\in(0,N)$ and $f'(N)=0$. I would like to show that
$$
\max_{x\in(0,\infty)}\bigg(\sum_{...
- 797
0
votes
1
answer
128
views
Alternative proof of $x^x \geq \sin x$ if $x>0$
My lecturer gave me this exercise: "Show that for $x>0$ it is $x^x \geq \sin x$."
This is my approach: since $x \geq \sin x$ for all $x \geq 0$, it is sufficient to show that $x^x \geq x$;...
- 2,162
3
votes
1
answer
116
views
Proving or disproving: If $0<a<b<1$, then $(1-a)^b>(1-b)^a$
Prove or disprove
If $0<a<b<1$, then $$(1-a)^b>(1-b)^a$$
I think this looks true when evaluating the differential equation $\frac{dy}{dx}=-y$ with initial condition $y(0)=1$ using euler ...
- 305
0
votes
1
answer
61
views
Show that the function $f(x)=\frac{1}{x}-\left(\frac{d}{2x+2}\right)^d$ is always positive for $x\geq 2$ and $d \in \{2,3,4,5\}$?
My first instinct was to show that the derivative is always negative for the conditions above and with the limit $=0$ it should follow that it is always positive but it is way to complicated for me. ...
- 1
1
vote
2
answers
197
views
A function inequality about $e^x$ and $\ln x$
If for any $x \in (1,+\infty)$，there is the inequality:
$$x^{-3} e^{x}-a \ln x \geq x+1$$
Find the value range of $a$ .
And I tried constructing the function $f(x)=x^{-3} e^{x}-a \ln x - x-1$ and ...
- 125
3
votes
3
answers
131
views
$x-\sin(x) \geq \dfrac{x^3}{(x+\pi)^2}$
Let $x \geq 0.$ I need to prove that $x-\sin(x)\geq\dfrac{x^3}{(\pi+x)^2}.$
I tried the derivative, of $f(x)=x-\sin(x)-\dfrac{x^3}{(\pi+x)^2}$ which is $1-\cos(x)-\dfrac{x^2(x+3\pi)}{(\pi+x)^3},$ but ...
- 1,070
4
votes
2
answers
300
views
Monotonicity of function averages
Please let me know if you know an answer to this problem. May be you could provide a reference to some publication on this topic?
Let $f(x)$ be a real-valued strictly convex function on $[0, 1]$. ...
- 41
5
votes
2
answers
235
views
For continuous, monotonically-increasing $f$ with $f(0)=0$ and $f(1)=1$, prove $\sum_{k=1}^{10}f(k/10)+f^{-1}(k/10)\leq 99/10$
A question from Leningrad Mathematical Olympiad 1991:
Let $f$ be continuous and monotonically increasing, with $f(0)=0$ and $f(1)=1$.
Prove that:$$
\text{f}\left( \frac{1}{10} \right) +\text{f}\...
- 873
2
votes
1
answer
100
views
Connection between relationships of the form $\|f \|_p \leq \|g\|_p$
Let us define the $L^p$ norm of a function $f:\mathbb{R} \to \mathbb{R}$ as
$$
\|f\|_p:= \left(\int \lvert f \rvert^p\right)^{1/p}.
$$
Let us label the following statement by $S\left(p\right)$:
$$
\|...
- 400
1
vote
1
answer
38
views
Bound of an Integrable function (Analysis)
For non-negative Riemann integrable function f in [a,b], and dissection $\mathcal D= {x_0,x_1,...,x_n } $, if $p(f,\mathcal D) $ is defined as $$p(f,\mathcal D)=\prod_{k=1}^n [1+(x_k-x_{k-1}) \inf_{x\...
- 165
2
votes
1
answer
81
views
Prove or find a counterexample $\forall x>0: f(2x)-f(x)<g(3x)-g(2x)$, given information about $f, g$.
Let $f,g:(0, \infty) \rightarrow \mathbb{R}$ be two functions that satisfy the following for all $x>0$: $g'(x)>f'(x)$ and $f''(x)>0$.
Prove or find a counterexample:
$$
\forall x>0: f(2x)-...
- 2,510
3
votes
1
answer
91
views
An inequality involving a quasiconvex function with binomial and power terms
The question is as follows.
Let us consider a positive integer number $x \in \{1,2,3,...\}$ and a positive real number $q \in [1,x]$. Show that
\begin{equation}
\sum_{m=0}^{x-1} \left( \frac{x+2-q}{m+...
- 109
8
votes
2
answers
171
views
Proving that $\frac{f(b)-f(a)}{b-a}+ \left(\frac{g(b)-g(a)}{b-a}\right)^2\le \max_{t\in [a,b]}\{f'(t)+(g'(t))^2\}$
Let $f,g\in C^1([a,b])$ with $a<b$ then prove that
$$\frac{f(b)-f(a)}{b-a}+ \left(\frac{g(b)-g(a)}{b-a}\right)^2\le \max_{t\in [a,b]}\{f'(t)+(g'(t))^2\}$$
It smells like there is some mean ...
- 24.2k
2
votes
2
answers
155
views
Is there an easy way to decide the sign of $\sqrt x \ln (x + 1) - \sqrt {x + 1} \ln x$?
Is there an easy way to decide the sign of $\sqrt x \ln (x + 1) - \sqrt {x + 1} \ln x$?
The original problem is to decide when ${(n + 1)^{\sqrt n }}$ is greater than ${n^{\sqrt {n + 1} }}$.
It seems ...
- 1,519
2
votes
1
answer
552
views
Inequalities about arctan and tanh.
Prove that for all real numbers $x\geq 0$ and $y \geq 0$ the following inequalities are true:
$$\arctan(x+y)\leq\arctan(x)+\arctan(y) \qquad \tanh(x+y)\leq \tanh(x)+\tanh(y)$$
I tried to use both ...
- 2,158
4
votes
4
answers
1k
views
Prove that $\forall x > 0, x - 1 \ge \ln(x)$
Prove that $\forall x > 0, x - 1 \ge \ln(x)$ .
Here is my proof:
We prove the inequality on two intervals, $(0,1]$ and $[1,+\infty)$.
First the easier one, $[1,+\infty)$.
Notice that at $x=1$, ...
- 1,527
0
votes
1
answer
32
views
If $\frac{d}{d t} (f(t)^2) \le 2f(t)g(t)$ then $\frac{d}{d t} \sqrt{K+f(t)^2} \le g(t)?$
Let $f(t), g(t)$ be a real-valued functions of $t\in(0,\infty)$ and $f(t), g(t)\ge 0$.
If $$\frac{d}{d t} (f(t)^2) \le 2f(t)g(t)$$
then how can I conclude that
$$
\frac{d}{d t} \sqrt{K+f(t)^2} \le g(t)...
0
votes
1
answer
370
views
Exponential type of $\sin z$
An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$.
The exponential type of $f$ is ...
- 680
3
votes
0
answers
222
views
Two-leg games in Elo rating for football teams
Do you know Elo rating for association football? It is a numerical estimation of strength of football clubs using simple mathematical formula based past results allowing predictions for the future.
...
- 777
4
votes
1
answer
300
views
Hypergeometric functions inequality
Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers.
From a simple plot it looks like
$_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} \,_2F_1(m+n,1,n+1,\...
- 594
1
vote
1
answer
140
views
Inequality holds?
Can anyone prove that
$$
\frac{\sum\limits_{i=1}^{k*} a_i i (x+\epsilon)^{(i-1)}}{\sum\limits_{i=1}^{k*} a_i (x+\epsilon)^{i}+k^*-1}>\frac{\sum\limits_{i=1}^{k*} a_i i x^{(i-1)}}{\sum\limits_{i=1}^{...
- 7,892
1
vote
4
answers
123
views
Does this inequality hold
Let $f(x)$ be a positive function on $[0,\infty)$ such that $f(x) \leq 100 x^2$. I want to bound $f(x) - f(x-1)$ from above. Of course, we have $$f(x) - f(x-1) \leq f(x) \leq 100 x^2.$$ This is not ...
- 95
0
votes
1
answer
141
views
Is the hypergeometric function $F(5/4,3/4; 2, z)$ bounded on $(0,1]$
Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)?
I'm aware of Euler's formula:
$$F(5/4,3/4; 2, z) = \frac{...
- 526
1
vote
0
answers
78
views
Lower bounds for holomorphic functions on annuli with explicit bounds on their power series
Let $f$ be a holomorphic function on $\mathbf{C}$ and consider its restriction to the annulus $X=B(0,1) - B(0,3/4)$ in the complex plane. (Here $B(0,r)$ is the open disc with radius $r$ around $0$.) ...
- 221