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Let f be a fixed self-dual Hecke–Maass form for S L (3 , ℤ) , and let u be an even Hecke–Maass form for S L (2 , ℤ) with Laplace eigenvalue 1 / 4 + k 2 , k > 0. A subconvexity bound for L (1 / 2 , f × u) is improved to O (k 2 1 / 1 6 + 𝜀) , and a subconvexity bound for L (1 / 2 + i t , f) is improved to O ((1 + | t |) 2 1 / 3 2 + 𝜀). New techniques employed include an application of an asymptotic formula by Salazar and Ye [Spectral square moments of a resonance sum for Maass forms, Front. Math. China 12(5) (2017) 1183–1200] to make error terms negligible, an iterative algorithm to locate stationary point, and a non-trivial estimation of Kloosterman sums. [ABSTRACT FROM AUTHOR]