I need it as soon as possible.. please do anything you know about it . will give a like

Problem 1.5. Recall that for any q1, the q norm of a vector xRn is defined by xq=(i=1nxiq)q1. Let X=(X1,,Xn) be a vector with independent entries such that Xi is sub-Gaussian with variance proxy 2 and E(Xi)=0. (a) Show that for any q2, and any xRd, x2xqn21q1, and prove that the above inequality cannot be improved (b) Show that for for any q>1, EXq4nq1q (c) Recover from this bound that Emax1inXi4elogn Problem 1.5. Recall that for any q1, the q norm of a vector xRn is defined by xq=(i=1nxiq)q1. Let X=(X1,,Xn) be a vector with independent entries such that Xi is sub-Gaussian with variance proxy 2 and E(Xi)=0. (a) Show that for any q2, and any xRd, x2xqn21q1, and prove that the above inequality cannot be improved (b) Show that for for any q>1, EXq4nq1q (c) Recover from this bound that Emax1inXi4elogn