Motivated by the recent investigations of several authors, the main aim of this article is to derive several functional inequalities for a class of functions related to the incomplete Fox-Wright functions that were introduced and studied recently. Moreover, new functional bounds for functions related to the Fox-Wright function are deduced. Furthermore, a class of completely monotonic functions related to the Fox-Wright function is given. The main mathematical tools used to obtain some of the main results are the monotonicity patterns and the Mellin transform for certain functions related to the two-parameter Mittag-Leffler function. Several potential applications for this incomplete special function are mentioned.