Certain Extremal Problems on a Classical Family of Univalent Functions

Lateef Ahmad Wani and Saiful R. Mondal ()
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Lateef Ahmad Wani: Department of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147001, Punjab, India
Saiful R. Mondal: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 8, 1-12

Abstract: 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.

Keywords: univalent functions; starlike and convex functions; extremal problems; subordination; arc length; partial sums (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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