On a class of convex functions of bounded \(\alpha\)-type (Q1281333)

The author introduces a class \(CV^{\alpha}(R_1, R_2)\) consisting of \(\alpha\)-convex functions and generalizing the class \(CV(R_1, R_2)\) studied by \textit{A. W. Goodman} [Proc. Am. Math. Soc. 92, 541-546 (1984; Zbl 0559.30009)]. Perhaps such functions should be called \(\alpha\)-convex of bounded type because they are not necessarily convex. It is shown that the integral operator \(g(z) = \int_0^z (f(t)/t)^{1 - \alpha} (f'(t))^{\alpha} dt\) sets a 1-1 correspondence between the two classes (Theorem 1). The author also determines the order of \(\alpha\)-convexity and the estimates of \(| f(z)| \) on the class \(CV^{\alpha}(R_1, R_2)\). Several corrections are to be made in the paper. One should read \(q^{\alpha}(z)\) instead of \(q(z)\) in the relations determining the function both in the statement and the proof of Lemma 1. The proof of Theorem 2 is wrong, however, the replacement of \(\beta\) by \(\alpha\) makes it obvious. In the estimates of Theorem 3 \(F^{\alpha}\) is to be substituted for \(F\).