Convolutions of planar harmonic strip mappings. Issue 11 (2nd November 2021)

• Record Type: Journal Article
• Authors: Dorff, Michael; Hamidi, Samaneh G.; Jahangiri, Jay M.; Yaar, Elif
• Is Part Of: Complex variables and elliptic equations. Volume 66:Issue 11(2021)
• Page Start: 1904
• Page End: 1921
• Publication Date: 2021-11-02
• Subjects: Convolutions -- planar harmonic mappings -- directional convexity; 30C45 -- 30C50; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2020.1789864
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : We consider classes of harmonic univalent functions f k = h k + g k ¯, ( k = 1, 2 ) that are shears of the analytic map h k − g k = 1 / 2 log ⁡ [ 1 + z / 1 − z ] with dilatation ω k = e i θ k z k . We prove that if the convolution f 1 ∗ f 2 is locally one-to-one and sense-preserving, then f 1 ∗ f 2 is convex in the direction of the real axis.