On differential subordinations for a class of analytic functions defined by a linear operator.
We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc.
Let A denote the class of normalized analytic functions in the unit disc U = z: |z| < 1. The author obtains fixed values of δ and ϱ (δ ≈ 0.308390864..., ϱ ≈ 0.0903572...) such that the integral transforms F and G defined by and are starlike (univalent) in U, whenever f ∈ A and g ∈ A satisfy Ref’(z) > -δ and Re g’(z) > -ϱ respectively in U.
We investigate extreme points of some classes of analytic functions defined by subordination and classes of functions with varying argument of coefficients. By using extreme point theory we obtain coefficient estimates and distortion theorems in these classes of functions. Some integral mean inequalities are also pointed out.
For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let denote the class of all analytic functions P in the unit disk with P(0) = 1 and in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability for when P ranges over the class As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.