engleski

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 $e(t) = t^{; ; \beta-1}; ; ; E_{; ; \alpha, \beta}; ; ^\gamma(-\lambda t^\alpha)$ where $E$ 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 $e(t)$. We also establish several related positivity results and some uniform upper bounds for the function $e(t)$.

Three parameter Mittag-Leffler function; Laplace integral representation; Complete monotonicity; Bounding inequality

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