Turan´ s inequality type for polygamma functions and some new inequalities - A.P. Farajzadeh (Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series, 2019)
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.