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On starlikeness of certain integral transforms

100%

Ponnusamy S.

nr 3

227-232

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 $F(z) = ∫_0^z (f(t)/t)dt$ and $G(z) = (2/z) ∫_0^z g(t)dt$ are starlike (univalent) in U, whenever f ∈ A and g ∈ A satisfy Ref'(z) > -δ and Re g'(z) > -ϱ respectively in U.

Coefficient inequalities for concave and meromorphically starlike univalent functions

63%

Bhowmik B. , Ponnusamy S.

nr 2

177-186

Let 𝔻 denote the open unit disk and f:𝔻 → ℂ̅ be meromorphic and univalent in 𝔻 with a simple pole at p ∈ (0,1) and satisfying the standard normalization f(0) = f'(0)-1 = 0. Also, assume that f has the expansion $f(z) = ∑_{n=-1}^{∞} aₙ(z-p)ⁿ$, |z-p| < 1-p, and maps 𝔻 onto a domain whose complement with respect to ℂ̅ is a convex set (starlike set with respect to a point w₀ ∈ ℂ, w₀ ≠ 0 resp.). We call such functions concave (meromorphically starlike resp.) univalent functions and denote this class by $Co(p)(Σ^{s}(p,w₀)$ resp.). We prove some coefficient estimates for functions in these classes; the sharpness of these estimates is also established.

Convolution theorems for starlike and convex functions in the unit disc

51%

Anbudurai M. , Parvatham R. , Ponnusamy S. , Singh V.

nr 1

27-39

Let A denote the space of all analytic functions in the unit disc Δ with the normalization f(0) = f'(0) − 1 = 0. For β < 1, let $P⁰_{β} = {f ∈ A: Re f'(z) > β, z ∈ Δ}$. For λ > 0, suppose that 𝓕 denotes any one of the following classes of functions: $M^{(1)}_{1,λ} = {f ∈ 𝓐 : Re{z(zf'(z))''} > -λ, z ∈ Δ}$, $M^{(2)}_{1,λ} = {f ∈ 𝓐 : Re{z(z²f''(z))''} > -λ, z ∈ Δ}$, $M^{(3)}_{1,λ} = {f ∈ 𝓐 : Re{1/2 (z(z²f'(z))'')' - 1} > -λ, z ∈ Δ}$. The main purpose of this paper is to find conditions on λ and γ so that each f ∈ 𝓕 is in $𝓢_{γ}$ or $𝒦_{γ}$, γ ∈ [0,1/2]. Here $𝓢_{γ}$ and $𝒦_{γ}$ respectively denote the class of all starlike functions of order γ and the class of all convex functions of order γ. As a consequence, we obtain a number of convolution theorems, namely the inclusions $M_{1,α} ∗ 𝓖 ⊂ 𝓢_{γ}$ and $M_{1,α} ∗ 𝓖 ⊂ 𝒦_{γ}$, where 𝓖 is either $𝓟⁰_{β}$ or $M_{1,β}$. Here $M_{1,λ}$ denotes the class of all functions f in 𝓐 such that Re(zf''(z)) > -λ for z ∈ Δ.

Region of variability for spiral-like functions with respect to a boundary point

51%

Ponnusamy S. , Vasudevarao A. , Vuorinen M.

Colloquium Mathematicum

2009

tom 116

nr 1

31-46

For μ ∈ ℂ such that Re μ > 0 let $ℱ_{μ}$ denote the class of all non-vanishing analytic functions f in the unit disk 𝔻 with f(0) = 1 and $Re(2π/μ zf'(z)/f(z) + (1+z)/(1-z)) > 0$ in 𝔻. For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ 𝔻̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_{μ}(λ) = {f ∈ ℱ_{μ}: f'(0) = (μ/π)(λ - 1) and f''(0) = (μ/π)(a(1-|λ|²) + (μ/π)(λ-1)² - (1-λ²))}$. In the final section we graphically illustrate the region of variability for several sets of parameters.