Fungrim entry: 54be3e

\left|{T}^{(r)}_{n - 1}(x)\right| \le \left|{T}^{(r)}_{n}(x)\right|

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| \ge 1

References:

V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.10.

TeX:

\left|{T}^{(r)}_{n - 1}(x)\right| \le \left|{T}^{(r)}_{n}(x)\right|

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| \ge 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("54be3e"),
    Formula(LessEqual(Abs(ComplexDerivative(ChebyshevT(Sub(n, 1), x), For(x, x, r))), Abs(ComplexDerivative(ChebyshevT(n, x), For(x, x, r))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, RR), GreaterEqual(Abs(x), 1))),
    References("V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.10."))

Topics using this entry

Chebyshev polynomials