# Tagged Questions

For questions about proving and manipulating functional inequalities.

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.
23 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 $\Gamma(2p,qt)$ ...
822 views

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 ...
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 $x\le2.$ ...
53 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: $$\Phi(x)=\frac{1}{2}(Tx,x)$$ My exercise says that $\Phi$ is ...
30 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 $\mathcal{L}_n(x)$ ...
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 ...