A Gap Condition for the Zeros and Singularities of a Certain Class of Products

Abstract

We carry out complete membership to Kaplan classes of functions given by formula {ζ∈C:|ζ|<1}∋z↦∏k=1n(1-ze-itk)pk,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\{\\zeta \\in {\\mathbb {C}}:|\\zeta |<1\\}\ i z\\mapsto \\prod \\limits _{k=1}^n (1-z\ extrm{e}^{-\ extrm{i}t_k})^{p_k}, \\end{aligned}$$\\end{document}where n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\in \\mathbb N$$\\end{document}, tk∈[0;2π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_k\\in [0;2\\pi )$$\\end{document} and pk∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_k\\in \\mathbb R$$\\end{document} for k∈N∩[1;n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in \\mathbb N\\cap [1;n]$$\\end{document}. In this way we extend Sheil-Small’s, Jahangiri’s and our previous results. Moreover, physical and geometric applications of the obtained gap condition are given. The first one is an interpretation in terms of mass and density. The second one is a visualization in terms of angular inequalities between vectors in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}.