Certain Extremal Problems on a Classical Family of Univalent Functions

Consider the collection A of analytic functions f defined within the open unit disk D , subject to the conditions f ( 0 ) = 0 and f ′ ( 0 ) = 1 . For the parameter λ ∈ [ 0 , 1 ) , define the subclass R ( λ ) as follows: R ( λ ) : = f ∈ A : Re f ′ ( z ) > λ , z ∈ D . In this paper, we derive sharp bounds on z f ′ ( z ) / f ( z ) for f in the class R ( λ ) and compute the boundary length of f ( D ) . Additionally, we investigate the inclusion properties of the sequences of partial sums f n ( z ) = z + ∑ k = 2 n a k z k for functions f ( z ) = z + ∑ n = 2 ∞ a n z n ∈ R ( λ ) . Our results extend and refine several classical results in the theory of univalent functions.