arXiv:1711.04286v1 [math.AP] 12 Nov 2017 A p(x)-Laplacian Extension of the D´ ıaz-Saa Inequality and Some Applications Peter Tak ´ a ˇ c Institut f¨ ur Mathematik, Universit¨ at Rostock Ulmenstraße 69, Haus 3 D-18055 Rostock, Germany peter.takac@uni-rostock.de Jacques Giacomoni LMAP (UMR 5142) Universit´ e de Pau et des Pays de l’Adour Avenue de l’Universit´ e, F-64013 Pau cedex, France jacques.giacomoni@univ-pau.fr June 8, 2021 1