Abstract
Let \(\psi_{n}= ( -1 ) ^{n-1}\)\(\psi^{ ( n ) }\) (\(n=0,1,2,\ldots \)), where \(\psi^{ ( n ) }\) denotes the psi and polygamma functions. We prove that for \(n\geq0\) and two different real numbers a and b, the function $$ x\mapsto\psi_{n}^{-1} \biggl( \frac{\int_{a}^{b}\psi_{n}(x+t)\,dt}{b-a} \biggr) -x $$ is strictly increasing from \(( -\min ( a,b ) ,\infty ) \) onto \(( \min ( a,b ) , ( a+b ) /2 ) \), which generalizes a well-known result. As an application, the complete monotonicity for a ratio of gamma functions is improved.
Highlights
The classical Euler’s gamma and psi functions are defined for x > by ∞(x) = e–ttx– dt, (x) ψ(x) =,(x) respectively
We prove that for n ≥ 0 and two different real numbers a and b, the function x → ψn–1 b a ψn
Let us give the following assertion, which is an improvement of Theorem in [ ]
Summary
We prove that for n ≥ 0 and two different real numbers a and b, the function x → ψn–1 b a ψn –x b–a is strictly increasing from (– min(a, b), ∞) onto (min(a, b), (a + b)/2), which generalizes a well-known result. Theorem EP For x, a, b > , the digamma function ψ has the following properties: (i) Iψ (a, b) ≤ Iψ (a, b); namely, ψ – If ψn– is strictly decreasing with respected to x, the function x → Aψn (x) with
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