Subclasses of certain analytic functions (Q1409291)

Let \(A\) denote the class of functions \(f(z)=z+ \sum^\infty+ {n=2}a_nz^n\) which are analytic in the open unit disc \(E\) of the complex plane. Let \(H(\lambda,\mu)\) \((\text{Re} \lambda\geq 0\), \(\mu\geq 0)\) denote the subclass of \(A\) consisting of functions \(f(z)\) with \(f(z)\neq 0\) for \(z\neq 0\) and \[ \left|{z^2f'(z)\over\bigl(f(z)\bigr)^2}-\lambda z^2\left({z\over f(z)} \right)''-1\right |<\mu\;(z\in E) \] and let \(H_0(\lambda,\mu)\) be the subclass of \(H(\lambda,\mu)\) consisting of functions \(f\in H(\lambda,\mu)\) with \(f''(0)=0\). In this paper the authors derive certain properties of the class \(H(\lambda,\mu)\) like distortion results, coefficient estimates and some subordination results. All the results involve only simple computations and are simple consequences of known results.