ABSTRACT
Ultraspherical Polynomial Inequalities (a) [Lorch] If C(α)n is an ultraspherical and if 0 < α < 1, then
sinα θ|C(α)n (cos θ)| ≤ 1
(α− 1)! ( 2
n
)1−α , 0 ≤ θ ≤ π.
The constant is best possible.
(b) If x2 ≤ 1− ( 1− α n+ α
)2 then
( C (α) n+1(x)
)2 ≥ C(α)n (x)C(α)n+2(x). (c) [Nikolov] If x ≥ 1, n ∈ N then:
( C (α) n
)′ (x)
C (α) n (x)
≥ n(n+ 2α)
(2α+ 1)x+ (n− 1)√x2 − 1 if α > − 1 2 ;
≤ n 2(n+ α)
α(n+ 1)x(n2 − 1)√x2 − 1 if 0 ≤ α ≤ 1.
