Sums of absolute values of cusp form coefficients and their application

Let Sk(Γ)Sk(Γ) be the space of holomorphic cusp forms of even integral weight k   for the full modular group. Let λf(n)λf(n), λg(n)λg(n) and λh(n)λh(n) be the n  th normalized Fourier coefficients of three holomorphic Hecke cusp forms f(z),g(z),h(z)∈Sk(Γ)f(z),g(z),h(z)∈Sk(Γ), Sk1(Γ)Sk1(Γ) and Sk2(Γ)Sk2(Γ) respectively. In this paper we consider several averages of absolute values of Fourier coefficients of holomorphic Hecke cusp forms. In particular, we have (under suitable conditions)∑n⩽x|λf(n)λg(n)|≪x(logx)−2θ3,∑n⩽x|λf(n)λg(n)λh(n)|≪x(logx)−θ3, where and throughout this paperθ=1−83π=0.1512…. As an application, we consider sums of coefficients of the Rankin–Selberg L-function and the triple product L-function, and prove that∑n⩽xλf(n)λg(n)≪x35(logx)−2θ3,∑n⩽xλf(n)λg(n)λh(n)≪x79(logx)−θ3, and other similar results.