For questions about proving and manipulating functional inequalities.
Questions tagged [functional-inequalities]
596 questions
0
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0 answers
Functional inequality from Art of problem solving.
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be real valued function such that $f(0)=0$ and
$f(x+y)\geq f(x)+y f(f(x))$. Find all such possible $f$.
When $ x=0$ then $ f(y)\ge 0$ for all $ y\in \mathbb R$
For all $ y\ge 0$ then choose $ x=-y$ we…
Conorlash
- 15
0
votes
1 answer
Range of values in inequalities
The function f and g are defined by
$$ f(x) = (x-2)(4-x), 0 \le x \le 4 $$ $$ g(x) = |x-2|, 0 \lt x \le A $$
(i) Find the range of values of A for which the composite function $fg$
is defined.
(ii) If $ A = 5 $, find the range of values of x for…
0
votes
1 answer
Information inequality
Information inequality
If $\theta_0$ is identified $[\theta \neq \theta_0,\implies f(z, \theta) \neq f(z, \theta_0)]$ and $E [\ln f(z, \theta) ] < \infty$
for all $\theta$ then $L(\theta) = E[\ln f(z,\theta)]$ has a unique maximum at…
alto de aitana
- 1
- 4
0
votes
3 answers
Prove the basic algebraic inequality used extensively everywhere.
How can I prove rigourously that if for any real numbers $x$ and $y$
(both are positive)
If $x \ge y$ then
This implies $x^n \ge y^n$ for any real $n$ as well.
(n is not negative)
This is used everywhere for example when n is half or two.
Is there a…
Matt
- 1,150
0
votes
4 answers
Prove that $e^x>x+1 \forall x\ne 0$
I need to prove that $e^x>x+1 \forall x\ne 0$.
Any ideas of hints about how to begin? I don't have any idea except the graphical way.
milan
- 13
0
votes
2 answers
Need help in solving inequality of this type: $|ax + b| > -c$
This is the equation:
$|3x+6|>-12$.
I solved it under two cases ($3x+6>-12$) and ($3x+6<12$).
More than the answer, what I need to know is -
Have I rightly constructed those $2$ cases, if wrong, please explain.
Thanks.
Romy
- 55
- 4
-1
votes
1 answer
The functional inequality $f(|x|)+f(|y|) \geq 1/f(|x+y|)$
Please help me to solve the following problem.
Does there exist a nonempty function $f: D_{f} \subset \mathbb{R} \to \mathbb{R}$ with $D_{f} \neq \emptyset$ such that
$$
f(|x|)+f(|y|) \geq \frac{1}{f(|x+y|)}
$$
for all $x, y \in \mathbb{R}$ such…
Amaqiff
- 1
-3
votes
2 answers
Inequality : $\sqrt {x} - 6 - \sqrt{10} -x \geqslant1$
I have solved it by squaring both sides and got inequality $x \geqslant 17/2$ but after that, the solution part have concluded on the equation
$4x^2 + 289 - 68x \geqslant4(10 - x)$
How this equation is formed.
