Fungrim entry: 2a4b9d

\left|U_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\left|U_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("2a4b9d"),
    Formula(LessEqual(Abs(ChebyshevU(n, z)), Pow(Add(Abs(z), Sqrt(Add(Pow(Abs(z), 2), 1))), n))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

Chebyshev polynomials