For questions about proving and manipulating functional inequalities.
Questions tagged [functional-inequalities]
596 questions
4
votes
1 answer
Proving or disproving the functional inequality
I'm trying to prove or disprove the inequality
$$\int_{\mathbb{R}}f(x)^{p-2}f'(x)^4dx\leq \frac{3}{p(p-1)}\int_{\mathbb{R}}f(x)^pf''(x)^2dx$$
for any real-valued function $f$ (provided the RHS is finite) where $p\geq 6$ is an even integer.
I tried…
bellcircle
- 2,939
4
votes
2 answers
Find $g(2002)$ given $f(1)$ and two inequalities
It is given that $f(x)$ is a function defined on $\mathbb{R}$, satisfying $f(1)=1$ and for any $x\in \mathbb{R}$,
$$f(x+5)\geq f(x)+5,$$
$$f(x+1)\leq f(x)+1.$$
If $g(x)=f(x)+1-x$ then find $g(2002)$.
Here,
$$f(x+5)\leq f(x+4) +1,$$
I didn't get any…
pi-π
- 7,416
3
votes
1 answer
Does there exist a bounded real function with some property?
Does there exist bounded $f:\mathbb{R} \to \mathbb{R}$ such that $f(1)>0$ and for every $x,y\in \mathbb{R}$ we have $$f(x+y)^2\geq f(x)^2+2f(xy)+f(y)^2$$
I can't solve it. I've put $y=-x$ and similar stuff but doesn't lead anywhere (at least I don't…
nonuser
- 90,026
2
votes
0 answers
Functional inequality $f(x)≥f(x+yf(x))(y+1)$
Show that if function $f$ satisfies $f(x)\geq f(x+yf(x))(y+1)$ for $x,y>0$ then $f(x)>0$ is false. It is clear that $f$ is non-increasing but I can't show that it will pass OX and not assuming $f$ is differentiable.
Rhegf2wffw
- 21
2
votes
0 answers
Functional inequality $f(x)\cdot \cos( x-y)\le f(y)$.
I am doing special manual of functional equations. While i was searching problems for my book, i have found this: Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x) \cdot \cos(x-y)\le f(y)$.
The answer is easy, but i don't…
Dmitry Ch
- 21
- 3
2
votes
3 answers
Inequality: Tell me I'm algebraically wrong
I want to prove that, $$\left| x-y\right|\geq\left| x\right| -\left| y\right|$$ I approached the problem like this af:
Since $-\left|z\right|\leq z\leq\left|z\right|$ is true for any real number $z$, we can write the same for reals $x$ and $y$:…
HERO
- 469
1
vote
0 answers
I want to solve the inequality $z′(t)+1≥0$ for all $t≥0$
Let $z(t)$ be a differentiable function for all $t≥0$. I want to solve the inequality:
$$z′(t)+1≥0$$ for all $t≥0$.
where $z′(t)$ is the derivative of $z(t)$.
DER
- 3,011
1
vote
1 answer
When $f(x)f(y) \le f(xy)$ holds?
This may be a well-known question (however, I could not find a good answer).
Let $f$ be a positive function on $(0,1).$ My question is the following.
Under what conditions on $f$, the inequality $f(x)f(y) \le f(xy)$
holds for any $x,y \in…
sharpe
- 928
1
vote
1 answer
Inequalities between arbitrary numbers
I was solving an excercise on calculus when i came up with this problem:
Let $a,b\in\mathbb{R}$ if $a>8-b$ and $a>3$ can it be shown that $b<5$? I've given it some tries but it got me nowhere, could someone propose an idea?
Nikos127
- 129
1
vote
3 answers
RATIONAL INEQUALITY - Find the values of a such that range of $f(x)=\frac {x+1}{a+x^2}$ contains $[0,1]$
Find the values of $\text{“}a\text{''}$ such that range of $f(x)=\frac{x+1}{a+x^2}$ contains $[0,1]$
where am i wrong ??:
I took $2$ cases :
Case $1$ : $a+x^2 \gt 0$
, I solved the inequality $f(x) \ge 0$ , $f(x) \le 1$,
Case $2$: $a+x^2 \lt 0$
…
ABC123
- 105
0
votes
0 answers
application of Gronwalls inequality
My question comes from a proof in Daniel Stroock's book 'An introduction to the Analysis of Paths on a Riemannian Manifold' (lemma 3.60, page 86). He proves that a function F satisfies the integral inequality
$F(t) \le U + \int_0^t g(\tau)d\tau +…
quarague
- 5,921
0
votes
2 answers
If $f(x + 19) \leq f(x) + 19$ and $f(x + 94) \geq f(x) + 94$ how to prove that $f(x +1) = f(x) + 1$
We know that
$f(x + 19) \leq f(x) + 19$
and $f(x + 94) \geq f(x) + 94$ $\forall x \in \mathbb{R} $
And I have to prove that $f(x + 1) = f(x) + 1$.
I know that this question has been asked already, but I couldn't understand how to use the hints…
harold jefferson
- 37
- 3
0
votes
0 answers
Sufficient Conditions for $f(x_1)g(y_1)+f(x_2)g(y_2)\le f(x_1+x_2)g(x_1+x_2)$
Let $f$ and $g$ be two strictly increasing functions such that $%
f:[0,1]\rightarrow \mathbb{R}_{+}$ (with $f(0)=0$) and $g:[2,\infty )\rightarrow \mathbb{R}_{+}$.
Can anyone come up with sufficient conditions that will ensure (i) and (ii)
always…
MathGuy
- 98
0
votes
1 answer
Cyclic inequality with fractions
Let $x,y,z\geq0$. Show that $x^4(y+z)+y^4(z+x)+z^4(x+y)\leq\frac{1}{12}(x+y+z)^5$
Adele
- 87
0
votes
3 answers
Find a continuous function such that$f:\mathbb{R}\to \mathbb{R}$, $f(0)=0$ and $f(2x)\geq x+f(x)$ and $f(3x)\leq 2x+f(x)$.
Could you give me a hint for this problem?
I have to find all continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $f(0)=0$ and $f(2x)\geq x+f(x)$ and $f(3x)\leq 2x+f(x)$.
I think this might be all the $f(x)=|x|$, but can it be any other…
Julia
- 11