Certain conditions for starlikeness of analytic functions of Koebe type (Q638084)

Summary: For \(\alpha \geq 0, \lambda > 0\), we consider the \(M(\alpha, \lambda)_b\) of normalized analytic \(\alpha - \lambda\) convex functions defined in the open unit disc \(\mathbb U\). In this paper, we investigate the class \(M(\alpha, \lambda)_b\), that is, \(\text{Re}\{(zf'_b(z)/f_b(z))[1 - \alpha + \alpha (1 - \lambda)(zf'_b(z)/f_b(z)) + \alpha \lambda(1 + (zf''_b(z)/f'_b(z)))]\} > 0\), with \(f_b\) is Koebe type, that is, \(f_b(z) := z/(1 - z^n)^b\). The subordination result for the aforementioned class is given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, we derive sufficient conditions for starlikeness of the class \(M(\alpha, \lambda)_b\) of \(n\)-fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.