Involves averaging arithmetic and integral partial functions over sparse set

Let p be a prime number, k≥ 0 and f be a class of arithmetic functions satisfying some simple conditions. In this short paper, we study the asymptotical behaviour of summation function $$ψf,k(x):=∑n≤ xΛ (n){f ( [ x/n ] ) }{ [ x/n ]ᵏ } ,~~~~~~~~~~~ πf,k(x):=∑p≤ x{f ( [ x/p ] ) }{ [ x/p ]ᵏ } $$ as $x→ ∞ $, where $ [ · ] $ is the integral part function, $Λ (n)$ is the von Mangoldt function.