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:Ref′(z)>λ,z∈D. In this paper, we derive sharp bounds on zf′(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 fn(z)=z+∑k=2nakzk for functions f(z)=z+∑n=2∞anzn∈R(λ). Our results extend and refine several classical results in the theory of univalent functions.