A. Vasudevarao

Region of variability for close-to-convex functions

For a complex number α with Re α > 0 let 𝒦 φ (α) be the class of analytic functions f in the unit disk 𝔻 with f (0) = 0 satisfying Re(f ′(z)/φ′(z)) > 0 in 𝔻, f ′(0)/φ′(0) = α, for some convex univalent function φ in 𝔻. For any fixed z 0 ∈ 𝔻 we shall determine the region of variability for f (z 0) when f ranges over the class 𝒦 φ (α). As corollaries, we present a number of its consequences.

Region of variability for certain classes of univalent functions satisfying differential inequalities

For complex numbers α, β and M ∈ ℝ with 0 < M ≤ |α| and |β| ≤ 1, let ℬ(α, β, M) be the class of analytic and univalent functions f in the unit disk 𝔻 with f(0) = 0, f ′(0) = α and f ″(0) = Mβ satisfying |zf ″(z)| ≤ M, z ∈ 𝔻. Let 𝒫(α, M) be the another class of analytic and univalent functions in 𝔻 with f(0) = 0, f ′(0) = α satisfying Re(zf ″(z)) > −M, z ∈ 𝔻, where α ∈ ℂ∖{0}, 0 < M ≤ 1/log 4. For any fixed z 0 ∈ 𝔻, and we shall determine the region of variability V j (j = 1, 2) for f ′(z 0) when f ranges over the class 𝒮 j (j = 1, 2), where and In the final section we graphically illustrate the region of variability for several sets of parameters. † In memoriam: Keijo Vuorinen, 15 February 1914–14 May 2009.

FEKETE-SZEGÖ PROBLEM FOR CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS

For , let denote the class of locally univalent normalized analytic functions in the unit disk satisfying the condition >. In the present paper, we shall obtain the sharp upper bound for Fekete- functional for the complex parameter .

Some basic properties of certain subclass of harmonic univalent functions

Abstract For , let denote the class of sense preserving harmonic mappings in the unit disk satisfying . The main aim of this paper is to study some basic properties such as coefficient bounds, growth estimates and convolution for functions in the class . We end the paper with an application, and construct harmonic univalent polynomials belonging to .