A class of completely monotonic functions involving the polygamma functions

Abstract

Abstract Let Γ ( x ) $\Gamma (x)$ denote the classical Euler gamma function. We set ψ n ( x ) = ( − 1 ) n − 1 ψ ( n ) ( x ) $\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$ ( n ∈ N $n\in \mathbb{N}$ ), where ψ ( n ) ( x ) $\psi ^{(n)}(x)$ denotes the nth derivative of the psi function ψ ( x ) = Γ ′ ( x ) / Γ ( x ) $\psi (x)=\Gamma '(x)/\Gamma (x)$ . For λ, α, β ∈ R $\beta \in \mathbb{R}$ and m , n ∈ N $m,n\in \mathbb{N}$ , we establish necessary and sufficient conditions for the functions L ( x ; λ , α , β ) = ψ m + n ( x ) − λ ψ m ( x + α ) ψ n ( x + β ) $$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$ and − L ( x ; λ , α , β ) $-L(x;\lambda ,\alpha ,\beta )$ to be completely monotonic on ( − min ( α , β , 0 ) , ∞ ) $(-\min (\alpha ,\beta ,0),\infty )$ . As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

Keywords

• Polygamma functions
• Inequalities
• Psi function
• Complete monotonicity