Multiplicative congruences with variables from short intervals

Recently, several bounds have been obtained on the number of solutions of congruences of the type (formula presented) where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval.