The real numbers are governed by a collection of rules that have to do with addition, multiplication, and inequalities. In the rules below, x, y, z 2 R. (In other words, x, y, and z are real numbers.) Rules of addition. • (x + y) + z = x + (y + z) (Law of associativity) • x + y = y + x (Law of commutativity) • x + 0 = x (Law of identity) • x + x = 0 (Law of inverses) Rules of multiplication. • ...

As usual, R denotes the real numbers, Z denotes the integers, and C denotes the complex numbers. Thus each α ∈ C can be expressed as a + b i, where a, b are real numbers and i = −1. We call a, b the real and imaginary parts of α, and denote them Reα, Imα, respectively. If z is a complex number, with z = x + y i where x, y are the real and imaginary parts of z, then we write z for the complex co...

Let p and q be analytic functions in the unit disc E = {z : |z| < 1}, with p(0) = q(0) = 1. Assume that α and δ are real numbers such that 0 < δ ≤ 1, α+ δ ≥ 0. Let β and γ be complex numbers with β 6= 0. In the present paper, we investigate the differential subordination (p(z)) [ p (z) + zp′(z) βp(z) + γ ]δ ≺ (q(z)) [ q(z) + zq′(z) βq(z) + γ ]δ , z ∈ E, and as applications, find several suffici...

The article describes Euler’s group Γ (z), formed by the invertible elements of the ring of the residues of the integer complex numbers modulo numbers divisible by z. The answers are proved for the primary cases z = (1+ i)m and z = (p+ iq)m, p2 +q2 = 4k+3 being prime.