New Applications of the Bernardi Integral Operator - Shigeyoshi Owa; H. Özlem Güney (Mathematics, 2020)
Let <semantics> A ( p , n ) </semantics> be the class of <semantics> f ( z ) </semantics> which are analytic p-valent functions in the closed unit disk <semantics> <mover> U ¯ </mover> = <mfenced separators="" open="{" close="}"> z ∈ C : <mfenced open="|" close="|"> z </mfenced> ≤ 1 </mfenced> </semantics> . The expression <semantics> B − m − λ f ( z ) </semantics> is defined by using fractional integrals of order <semantics> λ </semantics> for <semantics> f ( z ) ∈ A ( p , n ) . </semantics> When <semantics> m = 1 </semantics> and <semantics> λ = 0 , </semantics> <semantics> B − 1 f ( z ) </semantics> becomes Bernardi integral operator. Using the fractional integral <semantics> B − m − λ f ( z ) , </semantics> the subclass <semantics> T p , n <mfenced separators="" open="(" close=")"> α s , β , ρ ; m , λ </mfenced> </semantics> of <semantics> A ( p , n ) </semantics> is introduced. In the present paper, we discuss some interesting properties for <semantics> f ( z ) </semantics> concerning with the class <semantics> T p , n <mfenced separators="" open="(" close=")"> α s , β , ρ ; m , λ </mfenced> . </semantics> Also, some interesting examples for our results will be considered.