ON A SPECIAL CLASS OF LOGHARMONIC MAPPINGS

Abstract

This paper considers a special class SL(h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{L}(h)$$\\end{document} of logharmonic mappings of the form f(z)=h(z)h′(z)¯,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(z)=h(z)\\overline{h^{\\prime }(z)},$$\\end{document} where h is analytic in the unit disk U, normalized by h(0)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h(0)=0$$\\end{document}, h′(0)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{\\prime }(0)=1$$\\end{document}, and h(U) is starlike. For this class of functions, a distortion theorem is proved, and Bohr’s inequality along with some improvements and refinements is investigated. In addition, the radius of starlikeness and an estimate for arclength are obtained.