Abstract
In this paper, we show that the functions x m | β ( m ) ( x ) | are not completely monotonic on ( 0 , ∞ ) for all m ∈ N , where β ( x ) is the Nielsen’s β -function and we prove the functions x m − 1 | β ( m ) ( x ) | and x m − 1 | ψ ( m ) ( x ) | are completely monotonic on ( 0 , ∞ ) for all m ∈ N , m > 2 , where ψ ( x ) denotes the logarithmic derivative of Euler’s gamma function.
Highlights
IntroductionMonotonic functions have attracted the attention of many authors. Mathematicians have proved many interesting results on this topic
(0, ∞) for all m ∈ N, where β( x ) is the Nielsen’s β-function and we prove the functions x m−1 | β(m) ( x )|
Monotonic functions have attracted the attention of many authors
Summary
Monotonic functions have attracted the attention of many authors. Mathematicians have proved many interesting results on this topic. Disproved Conjecture 1 by showing that there exists an integer m0 such that for all m ≥ m0 the functions Φm ( x ) are not completely monotonic on (0, ∞). ∆α,m ( x ) = x α | ψ(m) ( x ) | f or x > 0, m ∈ N, α ∈ R, and noted that it remains an open problem to determine all (α, m) ∈ R+ × N such that ∆α,m is completely monotonic. Matejicka [31] s showed that the function x m β(m) ( x ) is completely monotonic on (0, ∞) for m = 1, 2, 3. ∂x when the first integral is convergent and the second is uniformly convergent on [ a, b]
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