New Applications of the Bernardi Integral Operator

Abstract

Let A ( p , n ) be the class of f ( z ) which are analytic p-valent functions in the closed unit disk U ¯ = z ∈ C : z ≤ 1 . The expression B − m − λ f ( z ) is defined by using fractional integrals of order λ for f ( z ) ∈ A ( p , n ) . When m = 1 and λ = 0 , B − 1 f ( z ) becomes Bernardi integral operator. Using the fractional integral B − m − λ f ( z ) , the subclass T p , n α s , β , ρ ; m , λ of A ( p , n ) is introduced. In the present paper, we discuss some interesting properties for f ( z ) concerning with the class T p , n α s , β , ρ ; m , λ . Also, some interesting examples for our results will be considered.

Keywords

• analytic <i>p</i>-valent function
• Bernardi integral operator
• Libera integral operator
• fractional integral
• gamma function
• Miller–Mocanu lemma