The starlikeness and convexity of multivalent functions involving certain inequalities. (Q1432780)

Let \({\mathcal T}(p)\) denote the class of functions of the form \[ f(z)= z^p+ a_{p+1} z^{p+1}+\cdots+ a_{p+n} z^{p+n}+\cdots, \] which are analytic and multivalent in \(U= \{z:|z|<1\}\). \(f\in{\mathcal T}(p)\) is said to be in the subclass \({\mathcal T}_\lambda(p; \alpha)\) if it satisfies the inequality \[ \text{Re}\{zf'(z)+\lambda z^2f''(z)\}/\{(1- \lambda) f(z)+\lambda zf'(z)\}> \alpha\quad (z\in U,\,p\in N,\, 0\leq\alpha< p). \] Many subclasses in the geometric function theory like multivalently convex and starlike functions of order \(\alpha\) are particular cases of the class \({\mathcal T}_\lambda(p; \alpha)\). Involving certain inequalities, in this note the authors find conditions for \(f\in{\mathcal T}(p)\), such that \(f\in{\mathcal T}_\lambda(p;\alpha)\).