For questions about proving and manipulating functional inequalities.

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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 ...
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13 views

What is the purpose of continuous and differentiable dependence

In learning Gronwall's inequality you also get to learn about continuous an differentiable dependence. I know the theorems but I have no idea about their application. What is the big idea of ...
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1answer
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Fractions in L^\infty

Let $f,g \in L^2(\Omega)$, where $\Omega$ is an open bounded subset of $R^n,\; n\ge 1$. We consider the two following properties: (1) $\frac{f}{g} {\bf 1}_{\{x\in \Omega;\; g(x)\ne 0\}} \in ...
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1answer
36 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 ...
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1answer
70 views

Inequalities help! [closed]

Let $a,b,c > 0$ and $a + b + c = 1$. Prove: $$\sqrt{\frac{ab}{c + ab}} + \sqrt{\frac{bc}{a + bc}} + \sqrt{\frac{ac}{b + ac}}\leq \frac32$$
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0answers
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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 ...
4
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1answer
159 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 ...
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1answer
64 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 $f\left ( x \right )$ ...
2
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1answer
85 views

Range of inner product of a sequence and its permutation

$a^n :=(a_i)_1^n$ is a finite sequence of real numbers of length $n$, where $\sum\limits_{i=1}^n a_i=0$ and $\sum\limits_{i=1}^n a_i^2=1$. Consider $s_n(a^n,\sigma):=\sum\limits_{i=1}^n ...
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2answers
69 views

Need help in solving inequality of this type: $|ax + b| > -c$

This is the equation: $|3x+6|>-12$. I solved it under two cases ($3x+6>-12$) and ($3x+6<12$). More than the answer, what I need to know is - Have I rightly constructed those $2$ cases, if ...
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0answers
55 views

If $\|f\|_X\le c_1\|f\|_Y$ then do we have $\langle f,g\rangle_X\le c_2\langle f,g\rangle_Y$?

Let $X$ and $Y$ be two inner product spaces with inner products $\langle \cdot,\cdot\rangle_X$ and $\langle \cdot,\cdot\rangle_Y$, respectively. Suppose we have $\|f\|_X\le c_1\|f\|_Y$ for any $f\in ...
4
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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 ...
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3answers
112 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.
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2answers
516 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 ...
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0answers
20 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
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2answers
43 views

Quadratic Absolute Value Inequality

Problem: Find all $x$ such that $|x^2-3x+1|<1$ I can't understand how to get started with this. I've never tried to solve quadratic Inequalities before. At first I thought of working with the ...
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2answers
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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 $x\le2.$ ...
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1answer
90 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 ...
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2answers
41 views

Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
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0answers
22 views

Inequalities for Laguerre polynomials

The following inequality holds, $$ \Big( 4\int_0^\infty rdr \big|\mathcal{L}_1(4r^2)\big|e^{-2r^2}\Big)^3 \geq 4\int_0^\infty rdr \big|\mathcal{L}_3(4r^2)\big|e^{-2r^2}, $$ where $\mathcal{L}_n(x)$ ...
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1answer
20 views

Bounds on functions via its derivatives

Suppose, we have a function $f$ where $f$ is: Contionuos. Non-Negative Has a derivative given by $f'$. Can we have a bound on $f$ in terms of its derivative $f'$? That is have an inequality that ...
2
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1answer
88 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This has been cross-posted to MO.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). ...
4
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1answer
76 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 ...
2
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1answer
73 views

Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=1+|a_{1}|+\cdots ...
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59 views

A question on (odd) perfect numbers

(Note: This has been cross-posted to MO.) Let $\sigma(x)$ be the (classical) sum of the divisors of $x$. A number $N \in \mathbb{N}$ is called perfect if $\sigma(N)=2N$. An even perfect number $U$ ...
2
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0answers
73 views

Near-Application of Cauchy-Schwarz Inequality

I have the following situation: I have two estimators of $\alpha$, both via maximum likelihood of the density: $$ f(x,y\mid \alpha,\beta) = f(y \mid x,\alpha,\beta)f(x \mid \alpha) $$ One uses only ...
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1answer
36 views

Implications of a Gronwall-type inequality

Assume that $$f(t) \le K\int_a^t f(s)\, ds, \qquad\text{for all $\,t \in [a,b]$.} $$ for some constant $K$, where $f:[a,b] \rightarrow [0,\infty)$. Let $U(t) = K\int_a^t f(s) ds$. If $U(a) = 0$ and ...
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1answer
46 views

Poincaré type inequality

Consider a sequence of functions $f_n \in H^1(M)$, the first Sobolev space of a complete (possibly noncompact) Riemannian manifold. If we have the normalization $\Vert \nabla f_n\Vert_{L^2} = 1$, ...
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1answer
18 views

Understanding the Derivation of Dual Geometric Programming Problem

Enthusiastic CS major interested in Optimization Theory here. Pardon me for overlooking something obvious. I'm referring to this nice tutorial/ebook: http://faculty.uml.edu/cbyrne/optfirst0.pdf In ...
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3answers
74 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
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2answers
608 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 ...
1
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0answers
32 views

help me to prove this

I would like to prove that $n/k!$ for $k$ that holds $k\leq \frac{\log n}{\log\log n}$ will always be bigger/equal to $1/2$. I tried to use stirling but got stuck. Any ideas? Thanks, Jonatan
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1answer
53 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 ...
1
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1answer
75 views

Numerical methods to solve nonlinear system of inequalities?

I know some methods to solve nonlinear system of equaltites? Relaxation Method, Newton method, nonlinear Jacobi method, nonlinear Seidel method. Is it exist some analogous method to solve nonlinear ...
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1answer
46 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
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1answer
90 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 $\int_{[0,1]} (f(x))^2 dx \ge n^2$.
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1answer
45 views

Estimating distance between two functions

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. I am trying to prove that $F$ is closed in $BC(\Bbb R, \Bbb R)$, where $BC$ is the space of ...
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1answer
31 views

A functional inequality

Suppose that $\varphi (t), \psi (t), w(t)$ are continuous functions on $[a,b]$ such that $w(t)>0$. If the inequality $$\varphi (t)\leq \psi (t) + \int_a^t w(s)\varphi (s)ds$$ holds on $[a,b]$, ...
4
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1answer
71 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 ...
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2answers
138 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 ...
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2answers
38 views

$ \int_{\mathbb R^d}\int_{\mathbb R^d}b(x,y)\,f(x)\,f(y)dx\,dy \leq ||b_+||_{L^2(\mathbb R^d\times \mathbb R^d)}||f||_{L^2(\mathbb R^d)}^2 $

We are given the following, $$ b:\mathbb R^d \times \mathbb R^d \rightarrow \mathbb R,\;\; f:\mathbb R^d\rightarrow \mathbb R $$ and $$ f\in L^2(\mathbb R^d)\; ,\;b\in L^2(\mathbb R^d\times \mathbb ...
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1answer
88 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
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1answer
62 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) = ...
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0answers
18 views

Non-constant solutions of the functional inequality

Does the functional inequality $$ F(|x|)+F(|y|) \geq \frac{1}{F(|x+y|)} $$ have non-constant solutions $F$? That is, $F(x)\neq \mathrm{const}$. $F$ is a real-valued function of a real variable.
6
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1answer
76 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 : ...
2
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1answer
30 views

Sketch $ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $

Sketch the following $$ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $$ I have considered this geometrically and ended up thinking that the complex numbers $z$ must satisfy $$0 < ...
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0answers
26 views

Hyperbolic trigonometric inequality

Is the following hyperbolic trigonometric inequality correct and if so, is there a simple derivation? $$\tanh A- \tanh B \geq (A-B)(\operatorname{sech}{^2}(A)), \qquad \forall A\ge B\ge 0.$$
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1answer
38 views

The functional inequality $f(|x|)+f(|y|) \geq 1/f(|x+y|)$

Please help me to solve the following problem. Does there exist a nonempty function $f: D_{f} \subset \mathbb{R} \to \mathbb{R}$ with $D_{f} \neq \emptyset$ such that $$ f(|x|)+f(|y|) \geq ...
2
votes
2answers
54 views

Trigonometric inequality $|\sin{a_1}|+|\sin{a_2}|+…+|\sin{a_n}|+|\cos{(a_1+a_2+…+a_n)}| \ge1$ for all real $a_i$

Prove that for all real numbers $a_1,a_2,...,a_n$ the following inequality holds: $$ |\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge 1 $$
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1answer
37 views

Simplifying Inequality Involving $\sigma$, $\beta$ and $x$

Given that $\sigma>0$, $\beta>0$, $x>0$ and $\sigma>\beta$, there are a couple of simplifications I cant derive: $$1.\,\,\sigma \geq x\,\,\,\,and\,\,\,\,\sigma\beta\geq ...