Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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18
votes
5answers
1k 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 ...
18
votes
0answers
489 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 ...
15
votes
11answers
2k views

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.
12
votes
2answers
257 views

How prove this function inequality $xf(x)>\frac{1}{x}f\left(\frac{1}{x}\right)$

Let $f(x)$ be monotone decreasing on $(0,+\infty)$, such that $$0<f(x)<\lvert f'(x) \rvert,\qquad\forall x\in (0,+\infty).$$ Show that $$xf(x)>\dfrac{1}{x}f\left(\dfrac{1}{x}\right),\...
11
votes
1answer
2k views

Solve $f (x + y) + f (y + z) + f (z + x) \ge 3f (x + 2y + 3z)$

Find all functions $f : \mathbb{R} \to \mathbb{R}$ which satisfy : $f (x + y) + f (y + z) + f (z + x) ≥ 3f (x + 2y + 3z)$ for real $x,y,z$. Attempt at solution: I have tried plugging in $x = -...
11
votes
1answer
425 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.
11
votes
0answers
885 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. ...
10
votes
1answer
646 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 ...
9
votes
1answer
134 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 ...
8
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 ...
8
votes
2answers
121 views

A functional equation with inequality

Find all (at least one) functions $f\colon \mathbb{R}\to \mathbb{R}$ (or show there is none), such that $$ f(x^3+x)≤x≤f(x^3)+f(x), \quad \text{for all $x\in \mathbb{R}$}. $$ This is a problem asked ...
7
votes
2answers
320 views

Find the function such $f(x+f(y)+xf(y))\ge y+f(x)+yf(x),\forall x,y\in(-1,+\infty)$

Let $ f ( x ) : ( - 1 , + \infty ) \to ( - 1 , + \infty ) $ be a continuous monotonic function, such that $ f ( 0 ) = 0 $, and $$ f ( x + f ( y ) + x f ( y ) ) \ge y + f ( x ) + y f ( x ) \quad \...
7
votes
2answers
120 views

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 $n$?
7
votes
1answer
102 views

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 ...
6
votes
1answer
576 views

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 ...
6
votes
5answers
770 views

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, ...
6
votes
2answers
194 views

An inequality in positive real continuous function

I proposed my conjecture as follows: Let $f(x)$ is a positive real continuous function that is convex on $[m, M]$, let $m \le x_i \le M$, for $i=1,2,...,n$ then show that $$\frac{f(x_1)+f(x_2)+.....+...
6
votes
1answer
79 views

$P(x)+P'''(x)\geq P'(x) + P''(x)$ then $P(x)>0 \, \forall x\in \mathbb R $

Let $P(x)$ be a polynomial function of real coefficients with the following property: $$P(x)+P'''(x)\geq P'(x) + P''(x) $$ then, $$P(x)\geq0 \quad \forall x\in \mathbb R $$ I've tried writing the ...
6
votes
2answers
105 views

Find $f$ such that $(f(x))^2 \ge f(x + y)(f(x) + y), \forall x, y$

Find all $f : (0, \infty) \rightarrow (0, \infty)$ such that $(f(x))^2 \ge f(x + y)(f(x) + y), \forall x, y \gt 0$ My guess there is no such function but I cannot prove it. The most obvious idea ...
6
votes
2answers
389 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 ...
6
votes
1answer
108 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
6
votes
2answers
538 views

Nonlinear Grönwall inequality

Let $T>0$, $\alpha,\beta>0$ and consider a non-negative continuous function $x$ on $[0,T]$ such that for all $t \in [0,T]$ one has $$x(t) \leq \alpha+\beta\left(\int_0^t x(s)\,\mathrm ds \right)^...
6
votes
1answer
107 views

Inequality About $f(t)=\int_{0}^t \sqrt{\cos(x)} dx$

During my projet, I encountered the following function defined for all $\displaystyle t\in[0,\frac{\pi}{2}]$ by : $$f(t)=\int_{0}^t \sqrt{\cos(x)} dx$$ and I need to prove the inequality below : $$\...
6
votes
0answers
57 views

Inequality for convex function $f:(0,\infty)\to\mathbb{R}$ [duplicate]

I have been working through the Exercises at the end of Chapter $1$ of Bollobas' Linear Analysis. Chapter $1$ is on inequalities, and the text is fairly brief. I have found the problems unexpectedly ...
6
votes
0answers
376 views

Question about the proof of General Sobolev Inequality in P.D.E. by Evan

I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows: Theorem (General Sobolev ...
5
votes
2answers
546 views

Find all continuous functions such that $f(x)f(2x)\dots f(nx) \le an^k$

Find all continuous functions $f : \mathbb{R} \rightarrow [1,\infty)$ for which there exist $a \in \mathbb{R}$ and $k$ a positive integer such that $$f(x)f(2x)\dots f(nx) ≤ an^k,$$ for every real ...
5
votes
5answers
188 views

Prove that the range of $f$ is all of $\mathbb{R}$.

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continous function such that $|f (x)−f (y)| \geqslant |x −y|$, for all real $x$ and $y$. I need to prove that the range of $f$ is all of $\mathbb{R}$. I ...
5
votes
5answers
462 views

Find all function satisfying a given inequality

Find all functions $ F : R \rightarrow R $ having the property that for any $x_1$ and $x_2$ the following inequality holds: $ F(x_1) - F(x_2) \le (x_1 - x_2)^2 $ My attempt: Observe that $ -(x_1 -...
5
votes
2answers
1k views

An upper bound of binary entropy

Binary entropy is given by $$H_{\mathrm b}(p) = -p \log_2 p - (1 - p) \log_2 (1 - p), \hspace{6 mm} p \le \frac{1}{2}$$ How can I prove that $$H_{\mathrm b}(p) \le 2 \sqrt{p(1-p)}$$
5
votes
1answer
91 views

When does $f\left [g\left (x \right ) \right ]=g\left [ f\left ( x \right ) \right ]$ have roots.

I met a interesting equation: $$\sin\left [\cos\left (x \right ) \right ]=\cos\left [\sin\left (x \right ) \right ]$$ (And of course, the equation has no roots). So, Let $f\left ( x \right )$ ...
5
votes
1answer
204 views

How to find $f(x)$ when $f\left(\frac{x+y}{x-y}\right)\ge\frac{f(x)-f(y)}{f(x)+f(y)}$

We have the function $f(x)$ continuous on $ (-\infty,0)\cup (0,+\infty)$ with $f(1)=-1$ $\displaystyle\lim_{x\to 0}xf(x)=-1$ and $f'(1)$ exsits. For any $x,y\in\mathbb R$ we have $$f\...
5
votes
1answer
116 views

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 ...
5
votes
1answer
133 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
3answers
440 views

Prove inequality $\ln \left( \frac{e-e^x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$ for $0<x \le 1$

The first function could be called 'exponential mean' of $y$ and $x$: $$f(y,x)=\ln \left( \frac{e^y-e^x}{y-x} \right)$$ We can obtain it by Cauchy mean value theorem. What is interesting, it ...
5
votes
1answer
111 views

For a random variable $X$ such that $P(a<X<b)=1$, showing $E(X)E\left(\frac{1}{X}\right) \le\frac{(a+b)^2}{4ab}$

I've worked on the following problem and have a solution (included below), but I would like to know if there are any other solutions to this problem, particularly more elegant solutions that apply ...
5
votes
1answer
74 views

An estimate of $C^2$

Let $u$ be a function on a domain $\Omega\subset R^n$, and $D^2u=\left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right)_{n\times n}$ be the Hessian of $u$. If $D^2u$ is positively definite and $\...
5
votes
2answers
76 views

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 ...
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 ...
4
votes
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.
4
votes
5answers
112 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'...
4
votes
1answer
95 views

If $\,\,f:[a,b]\to \mathbb{R}, \,b-a\ge 4$, is differentiable, then $\,f'(x_0)<1+(\,f(x_0))^2$, for some $x_0\in (a,b)$.

Suppose that $\,f:[a,b]\to \mathbb{R}$, where $\,b-a\ge 4,\,$ is differentiable in $(a,b)$ and continuous in $[a,b]$. Prove that there is $x_0\in (a,b)$, such that $$f'(x_0)<1+\big(\,f(x_0)\big)^...
4
votes
1answer
82 views

Find $f$ if $f(x)\leq x$ and $f(x+y)\leq f(x)+f(y)$ for all $x,~y\in \mathbb{R}.$

Find the formula of function $f:\mathbb{R}\to \mathbb{R}$ if: $$f(x)\leq x$$ and $$f(x+y)\leq f(x)+f(y)$$ for all $x,~y\in \mathbb{R}.$ Attempt. Identity function $I(x)=x$ satisfies the needed ...
4
votes
1answer
89 views

If $f(0)=f(1)=1$ and $|f(a)-f(b)| < |a-b|$ then $|f(a)-f(b)| < \frac{1}{2}$

Problem: $f$ be a function on $[0,1]$ such that $f(0)=f(1)=1$ and $f(a)-f(b) < |a-b|$ for all $a$ not equal to $b$. Prove that $|f(a)-f(b)| < \frac{1}{2}$. My attempt: Things I observed are ...
4
votes
2answers
111 views

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$?

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$? Here $x!$ is the factorial of $x$. Sure, I do know differential calculus, but my problem is that I do not know how to compute for the ...
4
votes
1answer
62 views

Suppose that $g(x) = \frac{f(x+1)-f(1)}{f(x)-f(0)} \geq g(1)$, what do we know about $f$?

Suppose $f:\mathbb{R_+} \to \mathbb{R}$ is a continuous and strictly increasing function. Define $g(x) = \frac{f(x+1)-f(1)}{f(x)-f(0)}$. For which $f$ the function $g$ satisfies $g(x) \geq g(1)$ for ...
4
votes
1answer
274 views

Bounded almost-homomorphisms on the integers

Let $f : \mathbb{Z} \to \mathbb{Z}$ be an "almost-homomorphism": The set $\{f(n+m)-f(n)-f(m) : n,m \in \mathbb{Z}\}$ is bounded. We may assume that $f$ is odd, i.e. $f(-n)=-f(n)$ for all $n$. Assume ...
4
votes
1answer
379 views

Is there a bound on second derivative in terms of third derivative?

An argument similar to the one outlined here shows that the sup-norm of $f''$ on $[0,1]$ can be bounded in terms of the sup-norms of $f$, $f'$, and $f'''$. Can the sup-norm of $f''$ be bounded in ...
4
votes
2answers
132 views

Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
4
votes
1answer
97 views

If $\int_{[0,1]} x^k f(x) dx =1$ then $\int_{[0,1]} (f(x))^2 dx \ge n^2$

Let $k \in \{0,1,...,n-1 \}$ and $f:[0,1] \to \mathbb{R}$ be a continous function. If $\int_{[0,1]} x^k f(x) dx =1$ for all such $k$ then show that $\int_{[0,1]} (f(x))^2 dx \ge n^2$.
4
votes
2answers
2k views

Inequality between $L^2$- and $L^1$-norms for functions

In the vector space $\mathbb{R}^n$, we have the inequality $$ ||x||_2 \leq ||x||_1 $$ where $x$ is a vector. I am wondering that we have similar inequality for function's norm. The $L^1$-norm of ...