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Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

17
votes
5answers
971 views

If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex

Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then how do we ...
1
vote
3answers
98 views

What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?

What is the maximum value of $$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$ if $x \in \mathbb{R}$ and $x > 1$? A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here. Lastly, note that ...
2
votes
2answers
252 views

Does this inequality hold true, in general?

Let $$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$ be the prime factorization of the positive integer $N$. Does the following inequality hold true in general? $$\prod_{i=1}^{\omega(N)}{\frac{...
2
votes
1answer
398 views

Improving the inequality $x\sigma_1(x) \leq \sigma_1(x^2)$ for $x \in \mathbb{N}$

Let $\mathbb{N}$ be the set of positive integers. For $x \in \mathbb{N}$, $\sigma_1(x)$ gives the sum of the divisors of $x$. (For example, $\sigma_1(3) = 1 + 3 = 4$.) We call the ratio $I(x) = \...
4
votes
1answer
119 views

Does this inequality have any solutions in $\mathbb{N}$?

Does this (number-theoretic) inequality have any solutions $x \in \mathbb{N}$? $$\frac{\sigma_1(x)}{x} < \frac{2x}{x + 1}$$ Notice that we necessarily have $x > 1$.
2
votes
2answers
232 views

Does the following inequality hold if and only if $N$ is an odd deficient number?

Let $N \in \mathbb{N}$. (That is, let $N$ be a positive integer.) This is in reference to two of my earlier questions here at MSE: Does the following inequality hold true, in general? Does this ...
11
votes
0answers
879 views

How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
9
votes
1answer
131 views

The Functional Inequality $f(x) \ge x+1$, $f(x)f(y)\le f(x+y)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function that satisfies the following conditons. $$f(x)f(y)\le f(x+y)$$ $$f(x)\ge x+1$$ What is $f(x)$? It is not to difficult to find ...
7
votes
2answers
2k views

Weighted Poincare Inequality

I'm trying to prove a result I found in a paper, and I think I'm being a bit silly. The paper claims the following: By the Poincare inequality on the unit square $\Omega \subset \mathbb{R}^2$ we have ...
10
votes
1answer
642 views

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
11
votes
1answer
422 views

Prove that $\int_0^1|f''(x)|dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$. $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1|f''(x)|dx\ge4.$ Also determine all possible $f$ when equality occurs.
2
votes
2answers
1k views

Inequality for Expected Value of Product

Let $(\Omega, \mathbb{P}, \mathcal{F})$ be a probability space, and let $\mathbb{E}$ denote the expected value operator. Consider the random variables $f: \Omega \rightarrow \{0,1,2\}$ and $g: \Omega ...
5
votes
1answer
131 views

Showing a function is in Holder space for some $a \in (0,1] $

Hi Im stuck on this exercise : for which $a \in (0,1]$ is $f(x)=x^{2}\sin(\frac{1}{x^{3}})$ in $C^{a}((0,1])$ This is my attempt so far : $|f(x)| \leq x^{2} $ $|f'(x)| = 2x\sin(1/x^{3})-\frac{3}{...
5
votes
0answers
154 views

Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
2
votes
0answers
860 views

Increasing rearrangement and Hardy-Littlewood inequality

Don't know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it. As you can see in Leoni (or in Lieb & Loss), the decreasing and the ...
18
votes
0answers
483 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
3
votes
0answers
151 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(...
6
votes
2answers
370 views

Is the function characterized by $f(\alpha x+(1-\alpha) y) \le f^{\alpha}(x/\alpha)f^{1-\alpha}(y)$ convex?

[Edit. The question lacks certain important conditions, as kindly pointed out by NeutralElement. Below is the amended version. I apologize for the omissions and many thanks to NeutralElement and ...
4
votes
1answer
981 views

Limit of a function satisfying an inequality

If $f(x)+f(y)\leq f(x+y)$ and $f:\mathbb{R}\to\mathbb{R}$, then can we find $\lim_{x\to 0} \frac {f(x)}{x}$? I am not sure whether the question is correct.Thank you.(I tried this idea: $f(x)=f(x+y-y)\...
3
votes
2answers
62 views

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 $g(2002)$. ...
3
votes
1answer
118 views

Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$

Let $f$ be a twice derivable function and $M_i =\sup_{x \in \mathbb{R}} |f^{(i)}(x)|$ and $|M_0|, |M_2|<\infty $. Preferably using the Taylor series on the interval $[x,x+h]$ show the following ...
2
votes
1answer
100 views

Are there (known) bounds to the following arithmetic / number-theoretic expression?

I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know): Are there (known) ...
2
votes
4answers
123 views

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?

Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2},$$ where $x \in {\mathbb{R} \setminus \{0\}}$, have a global minimum? I tried asking WolframAlpha, but it appears to give an inconsistent ...
-2
votes
2answers
112 views

If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Prove that for any set of three positive real $x, y, z$ such that $xy+yz+zx=3$ $x^2(y+z)+y^2(x+z)+z^2(x+y)+2\sqrt {xyz}\left(\sqrt{x^3+3x}+\sqrt{y^3+3x}+\sqrt{z^3+3x}\right)\ge$ $\ge 2xyz(x^2+y^2+z^2+...
4
votes
1answer
346 views

Is there a bound for Lipschitz constant in terms of second differences?

It is easy to show that if $f\colon[0,1]\to\mathbb R$ and $|f|\leq A$ and $|f''|\leq B$ then~$|f'|\leq 4A+B$. Indeed, by Taylor formula with remainder $f(x)=f(c)+(x-c)f'(c)+\frac12(x-c)^2f''(d)$ where ...
4
votes
5answers
106 views

Is the derivative of a function bigger or equal to $e^x$ will always be bigger or equal to the function ?!

It seems to be the case, but i don't have a proof. Given the function $f$ such that $f(x) \geq e^x$, is it true that $f'(x) \geq f(x)$?! I was experimenting with wolfram and it appears that $\frac{f'...
3
votes
1answer
116 views

$\frac{1}{2}(\frac{b_1}{a_1}-\frac{b_n}{a_n})^2(\sum_{1}^{n}{a_i^2 }) ^2 \ge (\sum_{1}^{n}{a_i^2 }) (\sum_{1}^{n}{b_i^2 })-(\sum_{1}^{n}{a_ib_i })^2$ [closed]

Let $a_1, a_2,....,a_n, b_1, b_2,...,b_n$, let $\frac{b_1}{a_1} = max \{\frac{b_i}{a_i}, i=1,2, \cdots n \}$ , $\frac{b_n}{a_n} = min \{\frac{b_i}{a_i}, i=1,2, \cdots n \}$ show that: $$\frac{1}{2}(...
2
votes
2answers
132 views

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I've recently been going over the mean value and intermediate value theorems, however I'm not sure where to start on this.
2
votes
1answer
66 views

Find function such that $\sum y_i\le\sum x_i\Rightarrow\sum f(y_i)\le\sum f(x_i)$

What kind of a function $f$ must be to satisfy the following? If $\sum_{i=1}^{n} y_i \leq \sum_{j=1}^{n} x_j$, where $x_j, y_i \in [0,1],\forall i,j$ then $$ \sum_{i=1}^{n} f(y_i) \leq \sum_{j=1}^{n} ...
1
vote
2answers
60 views

Find all $x$ to satisfy $|x-a|<|x-b|$.

Exercise: Using signs of inequality alone (not using signs of absolute value) specify the values of $x$ which satisfy the following relation. Discuss all cases. $$|x-a|<|x-b|$$ Solution: $|x-...
0
votes
1answer
53 views

Finding a function such that $(n-2)/2 + f(n-1) \leq f(n)$

By bounding a certain quantity defined on real numbers by $f(n)$ I derived the following inequality arising from an inductive argument. $ (n-2)/2 + f(n-1) \leq f(n).$ A solution to the above ...
-1
votes
1answer
51 views

Matries inequality with norms

Let $P$ and $C \neq0$ a $q \times q$ matrices. I want to prove that there exists a positive constants $\alpha$ such under some assumptions under $P$ we have the inequality $${\left\| {P\left( {I - C} ...