The residual Julia set of a transcendental meromorphic function is the subset of the Julia set consisting of all points (equivalently, all components) that do not lie on the boundary of any Fatou component. In this paper, we establish several conditions ensuring that the residual Julia set of a transcendental meromorphic function is non-empty.
In this work, we study idempotent preradicals with the properties of being hereditary, radical, and stable. We prove that if every idempotent preradical is radical, then every semiartinian left R-module is semisimple. Also, if every idempotent preradical is hereditary, then R is a left self-injective and a left MAX ring. And, if every idempotent preradical is stable, then R is a left V-ring and a left Noetherian ring. Finally, we characterize the rings where every idempotent preradical is a hereditary radical; in the commutative case R must be semisimple, and in the noncommutative case R must be semisimple and must also satisfy that the left annihilator of every element in R is a two sided ideal of R.
We determine sharp radii of Ma–Minda starlikeness for the class 𝒲_n(β ) consisting of functions f∈𝒜_n satisfying Re (f(z)/z)>β , where 0≤β <1 . A sharp estimate for the logarithmic derivative zf'(z)/f(z) is first obtained, which leads to a unified radius theorem. By combining this estimate with known disk inclusion results for several Ma–Minda domains, we obtain sharp radius constants for the classes associated with the nephroid, sine, lune, sigmoid, and hyperbolic starlike functions. The sharpness of each radius is established by exhibiting an extremal function.
Product formulas for γ , e^γ and rational approximations for γ are given. This approximation is based on the construction of a linear form in logarithms.
