Mathematica Aeterna, Vol. 3, 2013, no. 4, 249 - 252 Inequalities for the polygamma function Banyat Sroysang Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121 Thailand
[email protected] Abstract In this paper, we prersent some inequalities involving the polygamma function. Mathematics Subject Classification: 26D15 Keywords: Polygamma function, inequality 1 Introduction The Euler gamma function Γ is defined by ∞ Γ(x)= tx−1e−tdt, Z0 where x> 0. The polygamma function ψ of order n ∈ N is defined by dn+1 Ψ (x)= ln Γ(x), n dxn+1 where x> 0. The polygamma function may be represented as ∞ tne−xt Ψ (x)=(−1)n+1 dt, n 1 − e−t Z0 where n ∈ N and x> 0. In 2006, Laforgia and Natalini [1] proved that 2 Ψm(x)Ψn(x) ≥ Ψ m+n (x) 2 250 Banyat Sroysang m+n N where m, n, 2 ∈ and x> 0. In 2011, Sulaiman [2] gave the inequalities as follows. 1/p 1/q Ψ (x)Ψ (x) ≥ Ψ m + n (x) (1) m n p q m n ∈ N 1 1 where m, n, p + q , x> 0, p> 1 and p + q = 1. xp yq Ψ1/p(px)Ψ1/q(qx) ≥ Ψ + (2) n n n p q 1 1 ≤ 1 1 where n is a positive odd integer, x,y > 1, x + y 1, p> 1 and p + q = 1. xp yq −Ψ1/p(px)Ψ1/q(qx) ≤ Ψ + (3) n n n p q 1 1 ≤ 1 1 where n is a positive even integer, x,y > 1, x + y 1, p> 1 and p + q = 1.