On higher moments of Hecke eigenvalues attached to cusp forms

Let f, g and h be three distinct primitive holomorphic cusp forms of even integral weights k1, k2 and k3 for the full modular group Γ = SL(2, ℤ), respectively, and let λf(n), λg(n) and λh(n) denote the nth normalized Fourier coefficients of f, g and h, respectively. We consider the cancellations of sums related to arithmetic functions λg(n), λh(n) twisted by λf(n) and establish the following results: ∑n≼xλf(n)λg(n)iλh(n)j≪f,g,h,εx1−1/2i+j+ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{n \leqslant x} {{\lambda _f}\left( n \right)} {\lambda _g}{\left( n \right)^i}{\lambda _h}{\left( n \right)^j}{ \ll _{f,g,h,\varepsilon }}{x^{1 - 1/{2^{i + j}} + \varepsilon }}$$\end{document} for any ε > 0, where 1 ≼ i ≼ 2, j ≼ 5 are any fixed positive integers.