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On Dyakonov type theorems for harmonic quasiregular mappings

Miloš Arsenović, Miroslav Pavlović (2017)

Czechoslovak Mathematical Journal

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.

On starlikeness of certain integral transforms

S. Ponnusamy (1992)

Annales Polonici Mathematici

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 F ( z ) = 0 z ( f ( t ) / t ) d t and G ( z ) = ( 2 / z ) 0 z g ( t ) d t are starlike (univalent) in U, whenever f ∈ A and g ∈ A satisfy Ref’(z) > -δ and Re g’(z) > -ϱ respectively in U.

On the extreme points of subordination families

Jacek Dziok (2010)

Annales Polonici Mathematici

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.

Region of variability for functions with positive real part

Saminathan Ponnusamy, Allu Vasudevarao (2010)

Annales Polonici Mathematici

For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let γ , β denote the class of all analytic functions P in the unit disk with P(0) = 1 and R e ( e i γ P ( z ) ) > β c o s γ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability V ( z , λ ) for 0 z P ( ζ ) d ζ when P ranges over the class ( λ ) = P γ , β : P ' ( 0 ) = 2 ( 1 - β ) λ e - i γ c o s γ . 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.

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