Close-to-Convexity of Convolutions of Classes of Harmonic Functions. (2nd May 2018) - Garg, Raj Kumar; Jahangiri, Jay M.
Abstract : Forj = 1, 2 and for positive integersm andn, we consider classes of harmonic functionsf j = h j + g j ¯, whereg 1 ( z ) = z n h 1 ( z ) andg 2 ′ ( z ) = z n h 2 ′ ( z ) org 1 ′ ( z ) = z n h 1 ′ ( z ) andg 2 ′ ( z ) = z m h 2 ′ ( z ), and we prove that their convolutionf 1 ⁎ f 2 = h 1 ⁎ h 2 + g 1 ⁎ g 2 ¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic inz < 1 .