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