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Let \(\varGamma (x)\) denote the classical Euler gamma function. The logarithmic derivative \(\psi (x)=[\ln \varGamma (x)]'=\frac{\varGamma '(x)}{ \varGamma (x)}\), \(\psi '(x)\), and \(\psi ''(x)\) are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function \([\psi '(x)]^{2}+ \psi ''(x)\), its q-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.

On complete monotonicity for several classes of functions related to ratios of gamma functions