A new comprehensive class of analytic functions defined by multiplier transformation

Abstract

For functions belonging to the class S I m ( δ ) , δ ∈ [ 0 , 1 ) and m ∈ N , of normalized analytic functions in the open unit disc U , which is investigated in this paper, the author derives several interesting differential subordination results. These subordinations are established by means of a special case of the multiplier transformations I ( m , λ , l ) f ( z ) namely I ( m , λ , l ) f ( z ) = z + ∑ j = n + 1 ∞ ( λ ( j − 1 ) + l + 1 l + 1 ) m a j z j , where n ∈ N , m ∈ N ∪ { 0 } , λ , l ≥ 0 and f ∈ A n , A n = { f ∈ H ( U ) : f ( z ) = z + a n + 1 z n + 1 + ⋯ , z ∈ U } . A number of interesting consequences of some of these subordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided.