Let the function \(f(z)=z+a_ 2z^ 2+..\). be analytic univalent in \(| z| 0,\quad\lambda >0,\) and \(\gamma >\frac{1}{2}.\) the authors prove the following very interesting and useful results: \[ (1)\quad\int^{2\pi}_{0}|\log | f'(re^{i\theta})||^{\lambda}d\theta