On higher moments of Fourier coefficients of holomorphic cusp forms II

Let Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let λf(n), λg(n), λh(n) be the nth normalized Fourier coefficients of three distinct holomorphic primitive cusp forms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f (z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2} (\Gamma), h(z) \in S_{k_3} (\Gamma)}$$\end{document} respectively. In this paper we are able to establish nontrivial estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{n{\leq}x} \lambda_f(n)^5{\lambda_g}(n), \quad \sum_{n{\leq}x} \lambda_f(n) \lambda_g(n)\lambda_{h}(n)^j$$\end{document}, where 1 ≤ j ≤ 4.