Fungrim home page

Fungrim entry: 54be3e

Tn1(r)(x)Tn(r)(x)\left|{T}^{(r)}_{n - 1}(x)\right| \le \left|{T}^{(r)}_{n}(x)\right|
Assumptions:nZ1  and  xR  and  x1n \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
Absz\left|z\right| Absolute value
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
RRR\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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC