On special strong differential subordinations using multiplier transformation

For functions belonging to the class S I m ( δ , λ , l , ζ ) , δ ∈ [ 0 , 1 ) , λ , l ≥ 0 and m ∈ N ∪ { 0 } , of analytic functions in U × U ¯ , which are investigated in this paper, the author derives several interesting strong differential subordination results. These strong differential subordinations are established by means of a special case of the extended multiplier transformations I ( m , λ , l ) f ( z , ζ ) namely I ( m , λ , l ) f ( z , ζ ) : = z + ∑ j = 2 ∞ ( 1 + λ ( j − 1 ) + l l + 1 ) m a j ( ζ ) z j , where m ∈ N ∪ { 0 } , λ , l ≥ 0 and f ∈ A ζ ∗ , A ζ ∗ = { f ∈ H ( U × U ¯ ) : f ( z , ζ ) = z + ∑ j = 2 ∞ a j ( ζ ) z j , z ∈ U , ζ ∈ U ¯ } . er of interesting consequences of some of these strong subordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided.