On the convergence of solutions of functional differential equations with infinite delay

Let $\text{C(β), S1(β),}\text{and}\text{K(β, λ)}$ be the classes of univalent functions defined in $\text{E = {z: ¦z¦< 1}}$, which are convex of order β, starlike of order β and close-to-convex of order β type λ. Let $\text{f(z) = (}\text{1}\text{α}\text{)z}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>\alpha </mtext>}}\text{∝}{}^{\mathrm{z}}{}_0\text{z}{}^{\mathrm{<mtext>1</mtext><mtext>x?2</mtext>}}\text{F(z)dz, 0 ? α < 1}$. We discuss the properties of the function f when this function F belongs to the class K(β, λ) and its various subclasses.