ech T Press Science Computer Modeling in Engineering & Sciences DOI: 10.32604/cmes.2022.016927 ARTICLE Complete Monotonicity of Functions Related to Trigamma and Tetragamma Functions Mona Anis 1 , Hanan Almuashi 2 and Mansour Mahmoud 3,* 1 Faculty of Science, Mathematics Department, Mansoura University, Mansoura, 35516, Egypt 2 Faculty of Science, Mathematics Department, Jeddah University, Jeddah, 21589, Saudi Arabia 3 Faculty of Science, Mathematics Department, King Abdulaziz University, Jeddah, 21589, Saudi Arabia * Corresponding Author: Mansour Mahmoud. Email: mansour@mans.edu.eg Received: 11 April 2021 Accepted: 20 October 2021 ABSTRACT In this paper, we study the completely monotonic property of two functions involving the function (x) = [ψ ′ (x)] 2 + ψ ′′ (x) and deduce the double inequality x 2 + 3x + 3 3x 4 (2x + 1) 2 < (x)< 625x 2 + 2275x + 5043 3x 4 (50x + 41) 2 , x > 0 which improve some recent results, where ψ(x) is the logarithmic derivative of the Gamma function. Also, we deduce the completely monotonic degree of a function involving ψ ′ (x). KEYWORDS Trigamma function; tetragamma function; completely monotonic function; completely monotonic degree; inequality 1 Introduction The Euler’s gamma function is defned [1] by the improper integral Ŵ(x) =  ∞ 0 e −v v x−1 dv, x > 0 and the psi or digamma function is defned by the logarithmic derivative of gamma function, that is ψ(x) = Ŵ ′ (x) Ŵ(x) . The two derivatives ψ ′ (x) and ψ ′′ (x) are respectively called the trigamma and tetragamma functions. This work is licensed under a Creative Commons Attribution 4.0 International License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.