Monotonicity and convexity properties of the Nielsen's $\beta$-function

The Nielsen's $\beta$-function provides a powerful tool for evaluating and estimating certain integrals, series and mathematical constants. It is related to other special functions such as the digamma function, the Euler's beta function and the Gauss' hypergeometric function. In this work, we prove some monotonicity and convexity properties of the function by employing largely the convolution theorem for Laplace transforms.