Let \(j_ n(\alpha,\beta,x)\) \((\alpha,\beta >-1)\) be the orthonormal system of Jacobi polynomials and let \(\sum^{\infty}_{k=0}a_ k(f)j_ k(\alpha,\beta,\cos \Theta)\) be the Jacobi-Fourier series of a function \(f(\Theta)\in L^ P(\rho)\), where \[ \rho (\Theta)=2^{(\alpha +\beta +1)/2}(\sin \Theta /2)^{\alpha +1/2}(\cos \Theta /2)^{\beta +1/2},\quad 0<\Theta <\pi. \] The author considers the Fejér means \(\sigma_ n(f,\Theta)\) of the above series and gives a necessary and sufficient condition for the validity of \[ \| \rho (\Theta)[\sigma_ n(f,\Theta)-f(\Theta)]\|_{L^ p(0,\pi)}=O(1/n), \] provided \(1<p<\infty\) and \(\alpha,\beta >-1/2\).