In 1927, Hardy and Littlewood showed that Lp norm of functions on circle can be estimated only by information of Fourier coefficients in suitable sense. There have been some progress in this direction, in particular, the inequality was studied on compact homogeneous spaces recently. In this seminar, I will talk about the Fourier analysis on compact quantum groups and new such inequalities on concrete examples. Also, sharpness of the inequalities and some applications will be dealt with.