Starlikeness of integral transforms and duality

For λ satisfying a certain admissibility criteria, sufficient conditions are obtained that ensure the integral transformVλ(f)(z):=∫01λ(t)f(tz)tdt maps normalized analytic functions f satisfyingReeiϕ((1−α+2γ)f(z)z+(α−2γ)f′(z)+γzf″(z)−β)>0 into the class of starlike functions. Several interesting examples of λ   are considered. Connections with various earlier works are made, and the results obtained not only reduce to those earlier works, but indeed improved certain known results. As a consequence, the smallest value β<1β<1 is obtained that ensures a function f   satisfying Re(f′(z)+αzf″(z)+γz2f‴(z))>βRe(f′(z)+αzf″(z)+γz2f‴(z))>β is starlike.