Integral representation and functional inequalities involving generalized polylogarithm

The purpose of this manuscript is to derive two distinct integral representations of the generalized polylogarithm using two different techniques. The first approach involves the Dirichlet series and its Laplace representation, which leads to a single integral representation. The second approach utilizes the Hadamard convolution, resulting in a double integral representation. As a consequence, an integral representation of the Lerch transcendent function is obtained. Furthermore, we establish properties such as complete monotonicity, Turan inequality, convexity, and bounds of the generalized polylogarithm. Finally, we provide an alternative proof of an existing integral representation of the generalized polylogarithm using the Hadamard convolution.