In a Banach algebra A it is well known that the usual spectrum has the following property: σ(ab)\\{0}=σ(ba)\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sigma (ab) \\setminus \\{0\\} = \\sigma (ba) \\setminus \\{0\\} \\end{aligned}$$\\end{document}for elements a, b in A. In this note we are interested in subsets of A that have the Jacobson Property, i.e. X subset A such that for a, b in A: 1-ab∈X⇒1-ba∈X.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} 1 - ab \\in X \\implies 1 - ba \\in X. \\end{aligned}$$\\end{document}We are interested in sets with this property in the more general setting of a ring. We also look at the consequences of ideals having this property. We show that there are rings for which the Jacobson radical has this property.

Let $$A, B, D, E \in [-1,1]$$ and let p(z) be an analytic function with fixed initial coefficient defined in the open unit disk. Conditions on A, B, D, and E are determined so that $$1+ \alpha z p'(z)$$ being subordinated to $$(1+Dz)/(1+Ez)$$ implies that p(z) is subordinated to $$(1+Az)/(1+Bz)$$ and other similar implications involving $$1+ \alpha z p'(z)/p(z)$$ , $$\alpha p^2(z) + \lambda z p'(z) $$ , $$ \alpha p(z) + (1-\alpha ) p^2(z) + \lambda z p'(z)$$ , and $$ (1-\alpha ) p(z) + \alpha \left( 1+ z p'(z)/p(z) \right) $$ . Also, sufficient conditions for Janowski starlikeness with fixed second coefficient are obtained.

In this article, we study the problem of the existence and nonexistence of warping function associated with constant scalar curvature on pseudo-Riemannian Poisson warped product space under the assumption that fiber space has constant scalar curvature. We characterize the warping function on Einstein Poisson warped space by taking the various dimensions of base space B (i.e; (1). \(dim B=1,\) (2). \(dimB\ge 2\)).