For questions about proving and manipulating functional inequalities.

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15
votes
0answers
294 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 ...
14
votes
11answers
1k 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
211 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
385 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
1answer
477 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
107 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 ...
9
votes
0answers
833 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. ...
7
votes
2answers
1k 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 ...
7
votes
2answers
99 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
3answers
182 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 ...
7
votes
2answers
91 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 $n$?
6
votes
1answer
538 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
2answers
188 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
84 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
1answer
80 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
1answer
79 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 ...
5
votes
1answer
60 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 ...
5
votes
1answer
74 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
2answers
64 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 $x\le2.$ ...
5
votes
1answer
69 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
0answers
119 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
114 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
1answer
85 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
3answers
77 views

If $f(x_{1}+x_{2})+2\ge f(x_{1})+f(x_{2})$, then $f(2^{-x})>2^{-x}+2$?

Let $f(x)$ be monotonic increasing on $[0,1]$ and such that $f(0)=2,f(1)=3$, and for any $x_{1},x_{2},x_{1}+x_{2}\in[0,1]$: $$f(x_{1}+x_{2})+2\ge f(x_{1})+f(x_{2})$$ Question: between $\...
4
votes
1answer
66 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
49 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
2answers
57 views

A sufficient condition on $C^1$ positive functions for $f(x+y)<f(x)+f(y)$

I am trying to show that if $f:(0,+\infty)\rightarrow\mathbb R$ is a $C^1$ function such that $$f'(x)<\frac{f(x)}{x}\quad \forall x\in (0,+\infty) \tag{$\star$}$$ then $$f(x+y)<f(x)+f(y)$$ ...
4
votes
1answer
94 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
1answer
754 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)\...
4
votes
1answer
198 views

Inequality $\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y-1)$

I need to prove $$\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y),$$ for $x>\alpha$ and $y>\beta$ with $0<\alpha,\beta \leq \frac{1}{2}$ constants, and $C(\alpha, \beta)$ is a ...
4
votes
1answer
167 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
4
votes
1answer
61 views

E(X)*E(1/X) <= $(a+b)^2$/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 ...
4
votes
2answers
144 views

Proving $\frac{p_{n+1}^2-p_{n+1}^{-C}}{p_{n+1}^2-1}>\frac{f(n) p_n \log p_n }{p_{n+1} \log p_{n+1}},$ where $f(n)\to1$, for some constant $C>1$

Are there two constants $C_1$, $C_2>1$ such that for large enough $n$ $$\frac{p_{n+1}^2-p_{n+1}^{-C_1}}{p_{n+1}^2-1}>\frac{2 C_2 p_{n+1} \log p_{n+1}-1}{2 C_2 p_{n} \log p_{n}-1} \left(\frac{p_n ...
4
votes
1answer
78 views

Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R $ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
4
votes
1answer
43 views

Finding appropriate function

In order to obtain an estimate I'm wondering if there is a positive function $f:\mathbb{R}\to \mathbb{R}_+$ such that $f(x) < |x|$ for $|x|$ large enough (say $|x|\ge C_0>0$) so that the ...
4
votes
0answers
42 views

Find how many such complex numbers exist

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\text{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers ...
4
votes
0answers
62 views

Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and there are $x_1, x_2, \ldots, x_n \in R^d$ such that $$\sum_{i=1}^n (x_i - x') = x - x'.$$ ...
3
votes
2answers
2k views

Inequality involving Square Root

$\sqrt{3x-x^2}<4-x$ I know I can't simply square both sides of an inequality. I have narrowed down the possible values of x => x belongs to [0,3] because the expression inside the square root ...
3
votes
2answers
193 views

Finding a function satisfying a certain inequality

This is a continuation of this post where I tried to find a function $f(n)$ that would satisfy the induction step of an inductive argument and it was shown that such function does not exist. Trying ...
3
votes
1answer
95 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 ...
3
votes
1answer
230 views

Why are so many inequalities and estimate in PDEs?

I want to study PDE, but when I read some PDE books, I notice that there are so many inequalities there and always lots of estimates. I really want to know why before I continue my study. I mean, ...
3
votes
1answer
111 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 $x > 1$.
3
votes
1answer
56 views

Functional inequality $\sum_{1\le i<j\le n}f(x_i+x_j)\ge \frac{n(n-1)}{2}f(a_1x_1+a_2x_2+…+a_nx_n)$

Let $n\in\mathbb N, n\ge 2$. Does there exist a set of non-zero real numbers $a_1, a_2,..,a_n$ with this condition: If function $f: \mathbb R \rightarrow \mathbb R$ satisfies the inequality $$...
3
votes
2answers
96 views

Questioning a Proof of Khinchine Inequality

[Khinchine Inequality] Let $a_1,\ldots,a_n\in R$, $\varepsilon_1,\ldots,\varepsilon_n$ be i.i.d. Rademacher random variables: $P(\varepsilon_i=1)=P(\varepsilon_i=-1)=0.5$, and $0<p<\infty$. Then ...
3
votes
1answer
229 views

Use of cut-off functions and partitions of unity

This is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold, $$\int e^w \sqrt{g}\, dx \leq C \exp \...
3
votes
0answers
60 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
3
votes
1answer
535 views

Exponential Inequality

I was working on a problem and reduced it to showing the following inequality: $$2x e^{x^2/6} \ge e^x - e^{-x} \text{ for $x \ge 0$}$$ I tried expanding everything in Taylor series to no avail. I ...
3
votes
0answers
233 views

A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and $\Omega(N)$...
2
votes
4answers
117 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
61 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.