Norm estimates for convolution transforms of certain classes of analytic functions

For normalized analytic functions f in the unit disk Δ, we consider the classFβ={f:Re(1+zf″(z)f′(z))<32β,z∈Δ}, where 23<β⩽1, andS∗(A,B)={f:zf′(z)f(z)≺1+Az1+Bz,z∈Δ} where −1⩽B