Upper bounds for the moments of automorphic L-functions for GL(2)

Let f be a holomorphic Hecke eigenform of even weight κ for the full modular group, and let L(s, f) be the associated automorphic L-function of f. We prove that ∫T2TL1/2+it,f2rdt≪fTlogTr2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int }_{T}^{2T}{\left|L\left(1/2+\text{i}t,f\right)\right|}^{2r}\text{d}t{\ll }_{f}T{\left(\text{log}T\right)}^{{r}^{2}}$$\end{document} for 0≤r≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le r\le 1$$\end{document} unconditionally.