Turan´ s inequality type for polygamma functions and some new inequalities

Abstract

Turan’s inequality is known as
Pn−1(x)Pn+1(x) ≤ P
2
n
(x), x ∈ [−1, 1], n = 1, 2, ...,
where Pn denotes the Legendre polynomial of degree n . In this note, we show that Turan´
s
inequality is still true when the Legendre polynomial of degree n replace by
Ψn(x) = 1
(−1)n+1 Z ∞
0
e
−txt
n
1 − e−t
dt
polygamma function of order n. In fact, we prove more than Turn’s type inequality that zero
is the greatest lower bound of the sequence
{Ψ
2
n
(x) − Ψn−1(x)Ψn+1(x)}.
Moreover, some new inequalities for polygamma function of order n are established.