Abstract
In the present paper, we introduce and study certain subclasses of analyticfunctions in the open unit disk U which is defined by the differentialoperator . We study and investigate some inclusionproperties of these classes. Furthermore, a generalizedBernardi-Libera-Livington integral operator is shown to be preserved for theseclasses. MSC: 30C45.
Highlights
1 Introduction Let A be a class of functions f in the open unit disk U = {z ∈ C : |z| < } normalized by f ( ) = f ( ) – =
Let R be a class of all functions φ which are analytic and univalent in U and for which φ(U) is convex with φ( ) = and Re φ(z) >, z ∈ U
For two functions f and g analytic in U, we say that the function f is subordinate to g in U and write f (z) ≺ g(z), z ∈ U, if there exists a Schwarz function w(z) which is analytic in U with w( ) = and |w(z)| < such that f (z) = g(w(z)), z ∈ U
Summary
Let R be a class of all functions φ which are analytic and univalent in U and for which φ(U) is convex with φ( ) = and Re φ(z) > , z ∈ U. Making use of the principle of subordination between analytic functions, denote by S(ξ , φ) [ ] a subclass of the class A for ≤ ξ < and φ ∈ R which are defined by Let f , g ∈ A, where f and g are defined by f (z) = z +
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