AIMS Mathematics, 2020, 5(4): 3391-3407. doi: 10.3934/math.2020219

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Completely monotonic degree of a function involving trigamma and tetragamma functions

1 Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China
2 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
3 School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China

Let $\psi(x)$ be the digamma function. In the paper, the author reviews backgrounds and motivations to compute complete monotonic degree of the function $\Psi(x)=[\psi'(x)]^2+\psi''(x)$ with respect to $x\in(0,\infty)$, confirms that completely monotonic degree of the function $\Psi(x)$ is $4$, finds a relation between strongly completely monotonic functions and completely monotonic degrees, provides a proof for the relation between strongly completely monotonic functions and completely monotonic degrees, proves a property of logarithmically concave functions, and poses two open problems on lower bound for convolution of logarithmically concave functions and on completely monotonic degree of a function involving $\Psi(x)$.
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