# Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}{\lVert#1\rVert}\newcommand{\abs}{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
5answers
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### 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 ...
11answers
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### If $f(x)\leq f(f(x))$ for all $x$, is $x\leq f(x)$?

If I have $f(x)\leq f(f(x))$ for all real $x$, can I deduce $x\leq f(x)$? Thank you.
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### Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
1answer
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### submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
1answer
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### Does any one-to-one function exist that satisfies this inequality for all real numbers?

Does there exist a one-to-one function $f: \Bbb R \to \Bbb R$ such that $f(x^2) - (f(x))^2 \geq \frac 1 4\ \ \forall x \in \Bbb R$ ? I've tested this with many one-to-one functions but the ...
5answers
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### How can I prove $\log_2{(1+x)} \geq {x}$ for $0<x<1$

How can I prove the following inequality for $0<x<1$ $$\log_2{(1+x)} \geq {x}$$ I tried to use $\ln(1+x) \geq x-\frac{x^2}{2}$ for $x\geq 0$ and convert the $\ln$ to $\log_2$ to prove that, ...
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### Function with Weird Property!

Does there exist a function $f: \mathbb{R} \to \mathbb{R}\setminus\{0\}$ such that $(x-y)^2 \geq 4f(x)f(y), \forall x \neq y \in \mathbb{R}$? Spoiler I know of a proof using countability. But is ...
1answer
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### Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
0answers
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### 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 ...
3answers
127 views

### How to show $\Big\vert \frac{\sin(x)}{x} \Big\vert$ is bounded by $1$?

This may be a silly question, but I cannot figure it out. I want to prove that $\Big\vert \frac{\sin(x)}{x} \Big\vert \leq 1$ for $x\in[-1,0)\cup(0,1]$, but I don't even know where to start.
5answers
109 views