Hausdorff-summability of power series II

Let H _x a regular Hausdorff method and P(w)=∑ a_k w^k a power series with positive radius of convergence. A theorem of Okada states that P(w) is summable (H _x ) for w in a certain starshaped region G(H _x ,P). We call G=G(H _x ,P) the exact region of summability for P if summability cannot hold for any w \( \in \bar G\) Okada's theorem is said to be sharp for H_x if G(H_x,P) is the exact region of summability for any P. Three items are treated: 1. Criteria for Okada's theorem to be sharp are given in terms of the distribution function X (t) and the Mellin transform \(D(z) = \int\limits_0^1 {t^z d\chi (t)} \) . 2. When is Okada's theorem sharp for product methods? 3. Special classes of functions P(w) are indicated such that G(H_x, P) is the exact region of summability for any H_x. We use the notations of “Hausdorff-Summability of Power Series I” referred as “I”.