Laplace type integral expressions for a certain three-parameter family of generalized Mittag-Leffler functions with applications involving complete monotonicity

In this paper, we derive a Laplace type integral expression for the function eα,βγ(t;λ) defined byeα,βγ(t;λ)≔tβ−1Eα,βγ(−λtα),where Eα,βγ(z) stands for the generalized three-parameter Mittag-Leffler function occurring in many interesting applied problems involving fractional differential equations. Our result is shown to enable us to extend certain findings by Mainardi (2010) [21] and others. As an application of the obtained Laplace type integral representation, we prove the complete monotonicity of the function eα,βγ(t;λ). We also establish several related positivity results and some uniform upper bounds for the function eα,βγ(t;λ).