Starlikeness of integral transforms and duality

For λ satisfying a certain admissibility criteria, sufficient conditions are obtained that ensure the integral transform V λ ( f ) ( z ) : = ∫ 0 1 λ ( t ) f ( t z ) t d t maps normalized analytic functions f satisfying Re e i ϕ ( ( 1 − α + 2 γ ) f ( z ) z + ( α − 2 γ ) f ′ ( z ) + γ z f ″ ( 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 < 1 is obtained that ensures a function f satisfying Re ( f ′ ( z ) + α z f ″ ( z ) + γ z 2 f ‴ ( z ) ) > β is starlike.