The Second Moment of Rankin-Selberg L-function and Hybrid Subconvexity Bound

Let $M,N$ be coprime square-free integers. Let $f$ be a holomorphic cusp form of level $N$ and $g$ be either a holomorphic or a Maa{\ss} form with level $M$. Using a large sieve inequality, we establish a bound of the form $\sum_{g}\left|L^{(j)}\left(1/2+it,f \otimes g\right)\right|^2 \ll_t M+M^{2/3-\beta}N^{4/3}$ where $\beta \approx 1/500$. As a consequence, we obtain subconvexity bounds for $L^{(j)}\left(1/2+it,f \otimes g\right)\ll (MN)^{1/2 - \alpha}$ for any $N<M$ satisfying the conditions above without using amplification methods. Moreover, by the symmetry, we establish a level aspect hybrid subconvexity bound for the full range when both forms are holomorphic.