Suppose $\displaystyle f$ is an analytic function in $\displaystyle D = \{z:|z|<1\}$ for which $\displaystyle \lVert f\rVert=\sup_{z\in F}\{(1-|z|^2)|f'(z)|\}$ is finite.

**(a)** Prove for $\displaystyle z\in D$ that $\displaystyle |f(z)-f(0)|\leq\frac{1}{2}\lVert f\rVert \log\frac{1+|z|}{1-|z|}$.

**(b)** If $\displaystyle \{f_n\}$ is a sequence of analytic functions in $\displaystyle D$ for which

$\displaystyle \lim_{|z|\to 1^-}[(1-|z|^2)|f'_n(z)|]=0$ for $\displaystyle n=1,2,3,\cdots$, and

$\displaystyle \lVert f_n-f\rVert \to 0$ as $\displaystyle n\to\infty$, prove $\displaystyle \{f_n\}$ converges to $\displaystyle f$ uniformly on compact subsets of $\displaystyle D$.