Decreasing properties of two ratios defined by three and four polygamma functions

In the paper, by virtue of the convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, and of two Laplace transforms, in terms of the majorization, and in the light of other analytic techniques, the author presents decreasing properties of two ratios defined by three and four polygamma functions