All Questions
30
questions
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
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:$$
\text{f}\left( \frac{1}{10} \right) +\text{f}\...
- 873
4
votes
4
answers
1k
views
Prove that $\forall x > 0, x - 1 \ge \ln(x)$
Prove that $\forall x > 0, x - 1 \ge \ln(x)$ .
Here is my proof:
We prove the inequality on two intervals, $(0,1]$ and $[1,+\infty)$.
First the easier one, $[1,+\infty)$.
Notice that at $x=1$, ...
- 1,527
4
votes
1
answer
300
views
Hypergeometric functions inequality
Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers.
From a simple plot it looks like
$_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} \,_2F_1(m+n,1,n+1,\...
- 594
4
votes
2
answers
300
views
Monotonicity of function averages
Please let me know if you know an answer to this problem. May be you could provide a reference to some publication on this topic?
Let $f(x)$ be a real-valued strictly convex function on $[0, 1]$. ...
- 41
3
votes
3
answers
131
views
$x-\sin(x) \geq \dfrac{x^3}{(x+\pi)^2}$
Let $x \geq 0.$ I need to prove that $x-\sin(x)\geq\dfrac{x^3}{(\pi+x)^2}.$
I tried the derivative, of $f(x)=x-\sin(x)-\dfrac{x^3}{(\pi+x)^2}$ which is $1-\cos(x)-\dfrac{x^2(x+3\pi)}{(\pi+x)^3},$ but ...
- 1,070
3
votes
1
answer
116
views
Proving or disproving: If $0<a<b<1$, then $(1-a)^b>(1-b)^a$
Prove or disprove
If $0<a<b<1$, then $$(1-a)^b>(1-b)^a$$
I think this looks true when evaluating the differential equation $\frac{dy}{dx}=-y$ with initial condition $y(0)=1$ using euler ...
- 305
3
votes
1
answer
136
views
Proving that $f(x)$ satisfying $2f(x)f(1)\leq f(x)f(1)+1\leq 2f(2x)$ for all $x>0$ is a constant function
How do I prove the following, where $f:[0,\infty)\to[0,\infty)$:
For $x>0$ if $f(x)$ satisfies
$$2f(x)f(1)\leq f(x)f(1)+1\leq 2f(2x)$$
then $f(x)$ is a constant function.
I have out found out ...
- 45
3
votes
1
answer
91
views
An inequality involving a quasiconvex function with binomial and power terms
The question is as follows.
Let us consider a positive integer number $x \in \{1,2,3,...\}$ and a positive real number $q \in [1,x]$. Show that
\begin{equation}
\sum_{m=0}^{x-1} \left( \frac{x+2-q}{m+...
- 109
3
votes
0
answers
222
views
Two-leg games in Elo rating for football teams
Do you know Elo rating for association football? It is a numerical estimation of strength of football clubs using simple mathematical formula based past results allowing predictions for the future.
...
- 777
2
votes
1
answer
81
views
Prove or find a counterexample $\forall x>0: f(2x)-f(x)<g(3x)-g(2x)$, given information about $f, g$.
Let $f,g:(0, \infty) \rightarrow \mathbb{R}$ be two functions that satisfy the following for all $x>0$: $g'(x)>f'(x)$ and $f''(x)>0$.
Prove or find a counterexample:
$$
\forall x>0: f(2x)-...
- 2,510
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
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
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
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