Let \(H^{1}(\mathbb{D})\) be the usual Hardy space on the unit disc \(\mathbb{D}\), consisting of analytic functions \(F(z)\) on \(\mathbb{D}\) such that \[ \| F \| _{H^{1}}=\sup_{0-1\), and \(t_{n}^{(\alpha,\beta)}=2^{(\alpha +\beta+1)/2}\left(\int^{1}_{-1}\left(P_{n}^{(\alpha,\beta)}(x)\right)^{2}(1-x)^{\alpha}(1+x)^{\beta}dx \right)^{-1/2}.\) The system \(\{R_{n}^{(\alpha,\beta)}(\theta)\}_{n=0}^{\infty}\) is com\-plete and orthonormal in \(L^{2}(0,\pi)\) with respect to the ordinary Lebesgue measure \(d\theta\), and the Fourier-Jacobi coefficients of a function \(f\) are defined by \(c_{n}^{(\alpha,\beta)}=\int_{0}^{\pi} f(\theta) R_{n}^{(\alpha,\beta)}(\theta) d\theta \) . The space \(H^{1}(0,\pi)\) is defined by \(H^{1}(0,\pi)=\{h| _{(0,\pi)}: h\in \Re H^{1}\), even\} with the norm \(\| f \| _{H^{1}(0,\pi)}=\| h \| _{\Re H^{1}}\), where \(f=h| _{(0,\pi)}\). Then the authors prove the following analogue of Hardy's inequality (1): Let \(\alpha, \beta\geq -1/2\). Then, the Fourier-Jacobi coefficients \(c_{n}^{(\alpha,\beta)}\) of a function \(f\in H^{1}(0,\pi)\) satisfy \[ \sum_{n=0}^{\infty} \frac{| c_{n}^{(\alpha,\beta)} | }{n+1} \leq C \| f \| _{H^{1}(0,\pi)} , \] where \(C\) is a constant independent of \(f\).