For questions about proving and manipulating functional inequalities.

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-4
votes
1answer
36 views

Numbers with variable power is poitive. [on hold]

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a<0$.
2
votes
0answers
36 views
+50

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 ...
0
votes
1answer
31 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 ...
0
votes
1answer
44 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 $\Vert \nabla f_n\Vert_{L^2} = 1$, ...
0
votes
1answer
13 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 ...
2
votes
0answers
51 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 $U$ ...
3
votes
2answers
326 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
vote
0answers
30 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
-1
votes
0answers
21 views

solving system of inequalities and maximizing, with 4 variables

I am keenly interested in the steps used to solve the problem described below, I want to build my knowledge so I can apply it to more complex problems. ... Given $x$, $y$, $A$, $B$ such that $0 < x ...
4
votes
1answer
48 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
vote
1answer
26 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 ...
4
votes
1answer
38 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 ...
3
votes
1answer
71 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 $\,f:[a,b]\to \mathbb{R},\, b-a\ge 4,\,$ is differentiable in $(a,b)$ and continuous in $[a,b]$. Prove that there is $x_0\in (a,b)$, such that $$f'(x_0)<1+\big(\,f(x_0)\big)^2\!.$$ ...
4
votes
1answer
87 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 $\int_{[0,1]} (f(x))^2 dx \ge n^2$.
1
vote
1answer
41 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 ...
0
votes
1answer
28 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 $[a,b]$, ...
2
votes
2answers
32 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 ...
3
votes
2answers
132 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 ...
1
vote
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
votes
1answer
73 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
votes
1answer
28 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 < ...
1
vote
0answers
22 views

Hyperbolic trigonometric inequality

Is the following hyperbolic trigonometric inequality correct and if so, is there a simple derivation? $$\tanh A- \tanh B \geq (A-B)(\operatorname{sech}{^2}(A)), \qquad \forall A\ge B\ge 0.$$
2
votes
1answer
57 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) = ...
0
votes
1answer
34 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
50 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: $$ |\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge 1 $$
0
votes
1answer
33 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 ...
10
votes
2answers
190 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 ...
1
vote
0answers
30 views

Inequality to bound $\sum_i a_i b_i - \sum_i c_i d_i$ (harmonic eigenfunction/graph) type sum with constraints

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
2
votes
1answer
66 views

Proving inequality $(x^2+y^2)(y-1)+yx-y^2<0$

I have an inequality which came out of Lyapunov function for system of ODE's: $$(x^2+y^2)(y-1)+yx-y^2<0.$$ To prove stability of my solution, I have to prove that the inequalty is true in area ...
1
vote
2answers
260 views

L2 norm and L1 norm inequality

In the vector space, we have the following inequality $$ ||x||_2 \leq ||x||_1 $$ where x is a vector. I am wondering that we have similar inequality for function's norm. L1 norm of function f is ...
1
vote
1answer
39 views

Upper bound for integral over boundary in terms of integral over interior

I've encountered quite some papers in which it is simply assumed that $\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq ...
1
vote
2answers
72 views

Difficult but Interesting Inequalities Problems

1.) Consider the identity $$(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.$$ Find the minimum of $\max(A,B,C)$ over $0 \leq p \leq 1$. 2.) Let $n$ be a positive integer. Show that the smallest integer ...
6
votes
1answer
502 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 ...
0
votes
1answer
46 views

A question on logic and some functional inequalities

Suppose that I have a (generic) function $g$ and arguments $a, b \in \mathbb{N}$. I know that $g$ satisfies the inequalities $$1 < \frac{g(b)}{b} < \frac{g(a)}{a} < 2.$$ I also know that ...
1
vote
0answers
24 views

An integral inequality

Suppose $K\in C[0,1]^2$, $G\in C[0,1]$ are arbitrary and given. The question is that does there exists $H\in B[0,1]$ continuous a.e. with possibly finitely many discontinuities such that $$ ...
4
votes
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 ...
1
vote
1answer
77 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 $\gcd(q, n) = 1$). ...
1
vote
2answers
43 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.
6
votes
3answers
125 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 ...
0
votes
1answer
43 views

Is it correct to approach this with Holder Inequality? What am I doing wrong?

I know that $ \forall n\in N, a_0 + a_1 + ... + a_n = 1$, with $a_0, a_1, ... a_n > 0$ and $f(t) = a_0t^n+a_1t^{n-1}+...+a_n, \forall t\in R$ I have to prove that for every $x > 0$ $$ ...
1
vote
1answer
31 views

$ {\|f\|}_p = \sqrt[p]{\int_{a}^{b} |f(x)|^p {\rm d}x}$ is a norm

Consider the space $C([a,b])$ of all continuous functions $f\colon [a,b]\rightarrow \mathbb{R}.$ Show that the function $\|\cdot\|_p\colon C([a,b]) \rightarrow [0,\infty),p>1$, given by $$ ...
0
votes
0answers
37 views

hint for investigating positivity/negativity of a function

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function ...
0
votes
1answer
26 views

Find the size of squares cut from a box.?

This has been taking me days to do and I really want to do it for test practice. I actually have absolutely no idea how to even start this, so if I can get a hint, advice, or something to start me ...
0
votes
0answers
43 views

bounding supremum with Sobolev embedding theorem

I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am ...
1
vote
0answers
27 views

On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
1
vote
1answer
38 views

Inequality regarding exponential function

For every positive $x$ and for every $n$ show that $(1+\sqrt{\frac{x}{n}}+\sqrt[3]{\frac{x}{n}})e^{n\arctan{\frac{x}{n}}}>e^x$. I plotted it, it seems that the inequality holds. Any ideas how to ...
1
vote
0answers
50 views

application of Gronwalls inequality

My question comes from a proof in Daniel Stroock's book 'An introduction to the Analysis of Paths on a Riemannian Manifold' (lemma 3.60, page 86). He proves that a function F satisfies the integral ...
1
vote
1answer
32 views

$\| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le C\left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f g''\|_{L^2}\right)$ holds?

I want to know that the following inequality holds $$ \| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le ^\exists C_{>0} \left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f ...
5
votes
0answers
107 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 ...
2
votes
2answers
69 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 ...