Abstract
A planar harmonic mapping is a complex-valued function f : U ? C of the form f (x + iy) = u(x,y) + iv(x,y), where u and v are both real harmonic. Such a function can be written as f = h + g?, where h and g are both analytic; the function ? = g'=h' is called the dilatation of f. We consider the linear combinations of planar harmonic mappings that are the vertical shears of the asymmetrical vertical strip mappings j(z) = 1/2isin?j log (1+zei?j/ 1+ze-i?j) with various dilatations, where ?j ? [?/2,?), j=1,2. We prove sufficient conditions for the linear combination of this class of harmonic univalent mappings to be univalent and convex in the direction of the imaginary axis.
Submitted Version
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
