Note on the second derivative of bounded analytic functions

Assume z₀ lies in the open unit disk D and g is an analytic self-map of D. We will determine the region of values of g''(z₀) in terms of z₀, g(z₀) and the hyperbolic derivative of g at z₀, and give the form of all the extremal functions. In particular, we obtain a smaller sharp upper bound for |g''(z₀)| than Ruscheweyh's inequality for the case of the second derivative. Moreover, we use a different method to obtain Sz{\'a}sz's inequality, which provides a sharp upper bound for |g''(z₀)| depending only on |z₀|.