THEORY OF CERTAIN NON-UNIVALENT ANALYTIC FUNCTIONS

We investigate the non-univalent function's properties reminiscent of the theory of univalent starlike functions. Let the analytic function psi(z) =Sigma(infinity)(i=1)A(i)z(i), A1 not equal 0 be univalent in the unit disk. Non-univalent functions may be found in the class F(psi) of analytic functionsfof the form f(z) =z+ Sigma(infinity)(k=2) a(k)z(k) satisfying (zf'(z)/f(z)-1) < psi(z). Such functions, like the Ma and Minda classesof starlike functions, also have nice geometric properties. For these functions, growth and distortiontheorems have been established. Further, we obtain bounds for some sharp coefficient functionals andestablish the Bohr and Rogosinki phenomenon for the class F(psi). Non-analytic functions that shareproperties of analytic functions are known as poly-analytic functions. Moreover, we compute Bohr andRogosinski's radius for poly-analytic functions with analytic counterparts in the class F(psi) or classesof Ma-Minda starlike and convex functions