On Certain Families of Analytic Functions in the Hornich Space

Let ( H , ⊕ , ⊙ ) denote the Hornich space consisting of all locally univalent and analytic functions f on the unit disk D : = { z ∈ C : | z | < 1 } with f ( 0 ) = 0 = f ′ ( 0 ) - 1 for which arg f ′ is bounded in D . For f , g ∈ H and r , s ∈ R , we consider the integral operator I r , s ( z ) : = ∫ 0 z ( f ′ ( ξ ) ) r ( g ′ ( ξ ) ) s d ξ and determine all values of r and s for which the operator ( f , g ) ↦ I r , s maps a specified subclass of H into another specified subclass of H . We also determine the set of extreme points for different subclasses of H with respect to the Hornich space structure. Using the extreme points, we develop a new approach to obtain the pre-Schwarzian norm estimate for different subclasses of H . We also consider a larger space H ~ , whose linear structure is same as that of H and study the same problems as stated above for some subclasses of H ~ .