Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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 $x > 1$.
4
votes
1answer
986 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
40 views

Understanding the proof of an inequality

Basically the method applied is the following, we fix $a=\frac{a_1+a_2+...+a_n}{n}$, if: $$f(x)\ge f(a)+f'(a)(x-a) $$This inequality holds for all x, then summing up the inequality will give us the ...
4
votes
1answer
363 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
1answer
243 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
187 views

Convexity of a functional on a Sobolev space

Denote a specific Sobolev space by $$W^{2,2}(a,b) = \left\lbrace x\in L_2(a,b) : x'\in AC[a,b],\quad x''\in L_2(a,b) \right\rbrace $$ where $AC[a,b]$ is the class of absolutely continuous functions on ...
4
votes
1answer
243 views

Integral inequality involving the derivative

Let $f : [0, 1] \to \mathbb{R}$ a differentiable function with continuous derivative and $\int_0^1 f(x) dx=\int_0^1 x f(x) dx=1$. Prove that $\int_0^1 |f'(x)|^3 dx \ge \left(\frac{128}{3\pi}\right)^2$ ...
4
votes
1answer
451 views

Prove an integral inequality ($e^{\alpha x}u(x) \le e^{\alpha y}u(y) + \int_y^x e^{\alpha \xi} f(\xi) d\xi$) under certain hypotheses

Let $A\in \mathbb{R}$ and $A>0$, $\alpha \in \mathbb{R}$, and $u,f \in C([0,A])$. Suppose that for every $g \in C^\infty((0,A))$ we have $$g'(x_0) + \alpha u(x_0) \le f(x_0)$$ if $x_0$ is a ...
4
votes
2answers
150 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
85 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
315 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 \...
4
votes
2answers
341 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 $\Omega(N)$...
4
votes
1answer
46 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
67 views

Is there analytic solution to this functional problem?

Let $f(x)$ be a function on $[0,+\infty)$. I want to solve the following functional problem: $\min L(f)=\int_0^{\infty} (f'(x))^2\mathrm dx$ subject to $f(0)=a$ $f(x)\ge 0,\forall x$ $\int_0^\...
4
votes
0answers
63 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
95 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
3k 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
3answers
83 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 $\...
3
votes
2answers
63 views

$g : [0,1]\to\Bbb R$ is a concave function with $g(0) =0$ and $g(1)= \beta$. Show that $g(z) \geq \beta z$, $z \in [0,1]$.

I came across the following, let $g : [0,1] \to \Bbb R$ be a concave function with $g(0) =0$ and $g(1)= \beta$. It implies $g(z) \geq \beta z$, $z \in [0,1]$. Why is the statement $g(z) \geq \beta z$ ...
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
2answers
87 views

Show that $x^e\le e^x$ for all $x\gt 0$ and $x \in \mathbb {R}$

I understand that this question may seem quite simple, but although I can see different ways of showing this, I don't understand how it follows from the context I was given (i.e why the second part of ...
3
votes
2answers
202 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
5answers
140 views

Why does proving inductively $n < 2^n$ for $n \geq 1$ imply it is true for real values of $n$?

If we prove by induction that $2^n > n$ for $n \geq 1$ where $n \in N^+$, how can one know this inequality holds for real values of n like $2^{2.5} > 2.5$? Maybe a bit silly question but I can'...
3
votes
1answer
78 views

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 ...
3
votes
2answers
98 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 $$f(x+y)<f(x)+f(y)$$ ...
3
votes
1answer
119 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 ...
3
votes
1answer
186 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
1answer
47 views

Find all real value of k in the given inequality.

Find all the real values of k for which the following system of inequalities $-3<{x^2+kx-2\over x^2 -x +1} < 2$ is fullfilled for all real values of $x$ I have attempted this question but ...
3
votes
1answer
40 views

Derivative of ratio of exponential functions: $ \frac{d}{dk}\left(\frac{x_{i}^{k}}{\sum_{j}x_{j}^{k}}\right). $

Suppose we have a row vector $$X = [x_1\; \cdots \; x_n],$$ where $x_i>0$ for all $i$. I have the function \begin{equation} \frac{d}{dk}\left(\frac{x_{i}^{k}}{\sum_{j}x_{j}^{k}}\right).\quad (1) \...
3
votes
1answer
155 views

Prove that $W^{1,p}$-functions are “$L^{p}$-Lipschitz”.

Let $U$ be open in $\mathbb{R}^{n}$ and $V\subset\subset U$. Let $u\in W^{1,p}(U)$ for $p\in[1,\infty)$. Want to show for $x\in V$ and $h\in\mathbb{R}$, $$ \sup_{h}\int\left(\frac{u(x+he_{1})-u(x)}{...
3
votes
1answer
122 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}(...
3
votes
1answer
627 views

fraction power vector-norm inequality

If X is a Banach Space and $x,y\in X$. Then by the definition of a Banach algebra we know $$\|x.y\|\leq\|x\|\|y\|$$ and thats how we can have relation for any positive power. i.e. $n\in N$, $$\|x^n\|...
3
votes
1answer
101 views

Finding *-weak limit of functional

I´m trying to prove that the following functional sequence converges *-weakly, finding it´s limits: $$\lambda_n(x)=\int_{-1}^{1}x(t)\cos(n\pi t)dt,\;\;x\in L^2(-1,1),\;\; n\in\mathbb{N}$$ My idea is ...
3
votes
1answer
63 views

Anyone can help me prove an exponential inequality?

We know that $x_1\lambda_1=x_2\lambda_2$ and $x_1,x_2,\lambda_1,\lambda_2>0$. We also know that $x_1\lambda_1>1$ and $x_2\lambda_2>1$. We also have $\lambda_1 > \lambda_2$. Now we want to ...
3
votes
1answer
170 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 ...
3
votes
2answers
159 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
191 views

Help explain why the proof works - Gradient Estimate Using the Maximum Principle

I found the following proposition and proof in a textbook for one of my classes, and the end of the proof seems a little "unfinished" to me. In other words, as the author left it, I'm not entirely ...
3
votes
1answer
116 views

Interpolation inequalities

Let $\Omega$ be a regular domain of $\mathbb{R}^d$, $d=2,3$. Let $\mathcal{T}_h$ be a triangulation of $\Omega$ of size $h>0$. Assume we can prove \begin{equation*} \begin{aligned} \|v\|_{L^2(\...
3
votes
0answers
82 views

Can we find an estimate like this: $\left|\int_0^tf\,dg\right|\le C\|f\|_{\infty}\|g\|_{\gamma}$?

Let $f\in\mathcal C^{\lambda}([0,T])$ and $g\in\mathcal C^{\gamma}([0,T])$ (i.e. they are Holder continous), where $\lambda+\gamma>1$. Consider now the following estimate: $$ \left|\int_0^tf\,dg\...
3
votes
0answers
137 views

Question about transportation-entropy inequality (From Villani's book: Optimal Transport, Old and New)

I was reading Villani's book: Optimal Transportation, Old and New. From page 80-83, he introduced some results about dual formulation of transport inequality. Assume $C(\mu,\nu)$ is the optimal ...
3
votes
0answers
43 views

A special solution for a functional inequality from $\mathbb{R}\times \mathbb{R} $ onto $\mathbb{R}$ [duplicate]

Is there a bijection solution $\eta:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ (with an explicit formula, if it is possible) for the inequality $$\min\{x,y\}\leq \eta(x,y)\leq \max\{x,y\}\;\; ...
3
votes
1answer
188 views

What is Hoeffding's inequality in Hilbert space?

Suppose I have random variables $X_1, X_2,...,X_n \in \mathcal{H}$, where $ \mathcal{H}$ is some Hilbert space. How can I bound the following term - $ P(\| \sum_{i = 1}^n X_i - E[X_i] \|_{\mathcal{...
3
votes
0answers
155 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}(...
3
votes
0answers
65 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
611 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 ...
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
4answers
5k views

Find the range of values of $x$ which satisfies the inequality.

Find the range of values of $x$ which satisfies the inequality $(2x+1)(3x-1)<14$. I have done more similar sums and I know how to solve it. I tried this one too but my answer doesn't matches the ...
2
votes
2answers
145 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
5answers
66 views

Trig function bounded on interval (without calculus), prove that $x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$.

If $0 \lt x \lt \dfrac{\pi}{2}$, prove that $$x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$$ This question must be done without calculus. First, I tried splitting it into the intervals $(0,\pi/4)$ ...
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 ...