On higher moments of Fourier coefficients of holomorphic cusp forms II

Let S _k (Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let λ _f (n), λ _g (n), λ _h (n) be the nth normalized Fourier coefficients of three distinct holomorphic primitive cusp forms ${f (z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2} (\Gamma), h(z) \in S_{k_3} (\Gamma)}$ respectively. In this paper we are able to establish nontrivial estimates for $$\sum_{n{\leq}x} \lambda_f(n)^5{\lambda_g}(n), \quad \sum_{n{\leq}x} \lambda_f(n) \lambda_g(n)\lambda_{h}(n)^j$$ , where 1 ≤ j ≤ 4.