The Sharp Distortion Estimate Concerning Julia's Lemma

For alpha > 0, let J(alpha) be the class of all analytic functions f in the unit disk D := {z is an element of C : vertical bar z vertical bar < 1} satisfying f (D) subset of D with the the angular derivative angle lim(z -> 1) f(z) - 1/z - 1 = alpha. For a, z is an element of D, let k(z) = vertical bar 1 - z vertical bar(2)/1 - vertical bar z vertical bar(2) and sigma(a)(z) = 1 - <(a)over bar>/1 - a z - a/1 - (a) over barz. Let z(0) is an element of D be fixed. For f is an element of J(alpha), we obtain the sharp estimate vertical bar f'(z(0))vertical bar <= 4 alpha k(z(0))(2)/(alpha k(z(0)) + 1)(2)vertical bar 1 - z(0)vertical bar(2) when alpha k(z(0)) <= 1. with equality if and only if f = sigma(-1)(w0) omicron sigma(z0). Here w(0) = (1- alpha k(z(0)))/(alpha k(z(0))+ 1). In case of alpha k(z(0)) > 1 we derive the estimate vertical bar f'(z(0))vertical bar <= k(z(0))/vertical bar 1 - z(0)vertical bar(2). It is also sharp, however in contrast to the former case, there are no extremal functions in J(alpha). The lack of extremal functions is caused by the fact that J(alpha) is not closed in the topology of local uniform convergence in D. Thus we consider the closure (J) over bar (alpha) a of J(alpha) and study (V) over bar (1)(z(0), alpha) := {f' (z(0)) : f. (J) over bar (alpha)} which is the variability region of f' (z(0)) when f ranges over (J) over bar (alpha). We shall show that partial derivative(V) over bar (z(0), alpha) is a simple closed curve and (V) over bar (z(0), alpha) is a convex and closed Jordan domain enclosed by partial derivative(V) over bar (z(0), alpha). Moreover, we shall give a parametric representation of partial derivative(V) over bar (z(0), alpha) and determine all extremal functions.