Average results on the order of a modulo p

Let a>1a>1 be an integer. Denote by la(p)la(p) the multiplicative order of a modulo primes p  . We prove that if xlog⁡xlog⁡log⁡x=o(y), then1y∑a≤y∑p≤x1la(p)=log⁡x+Clog⁡log⁡x+O(1)+O(xylog⁡log⁡x), which is an improvement over a theorem by Felix [Fe]. Additionally, we also prove two other related average results.