Analytic functions with some univalent Gelfond-Leontev derivatives. II (Q1321637)

For part I see \textit{G. P. Kapoor}, \textit{O. P. Juneja} and \textit{J. Patel} [Demonstr. Math. 23, No. 4, 863-877 (1990; Zbl 0747.30008).] Let \(f(z) = \sum^ \infty_{n=0} a_ nz^ n\) be analytic in \(| z | < R\), \(0 < R \leq \infty\). For a non-decreasing sequence \(\{d_ n\}^ \infty_{n=1}\) of positive numbers the Gelfond-Leontev (GL) derivative of \(f\) is defined as \(Df(z) = \sum^ \infty_{n=1} d_ n a_ nz^{n-1}\). The higher order GL derivatives \(D^ pf\), \(p = 2,3, \dots\) of \(f\) is given by \(D^ pf(z) = \sum^ \infty_{n=p} (d_ n \dots d_{n - p + 1}) a_ nz^{n - p}\). A relation between the radii of convexity \(\rho_ n (c)\) of the GL derivatives \(D^ nf\) of the function \(f\) and \(R\) is proved. Also an upper bound for \(\psi\)-type of an entire function assuming some of its GL derivatives to be analytic and univalent in the unit disk, is found. The results obtained in this paper are in the nature of extension and improvements of earlier results.