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1
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On starlikeness of certain integral transforms

100%
Ponnusamy S.
Annales Polonici Mathematici
|
1991-1992
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tom 56
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nr 3
227-232
EN
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.
2
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Criteria for univalence, starlikeness and convexity

64%
Ponnusamy S., Vasundhra P.
Annales Polonici Mathematici
|
2005
|
tom 85
|
nr 2
121-133
EN
Let 𝓐 denote the class of all normalized analytic functions f (f(0) = 0 = f'(0)-1) in the open unit disc Δ. For 0 < λ ≤ 1, define 𝓤(λ) = {f ∈ 𝓐 : |(z/f(z))²f'(z) - 1| < λ, z ∈ Δ} and 𝓟(2λ) = f ∈ 𝓐 : |(z/f(z))''| < 2λ, z ∈ Δ.cr Recently, the problem of finding the starlikeness of these classes has been considered by Obradović and Ponnusamy, and later by Obradović et al. In this paper, the authors consider the problem of finding the order of starlikeness and of convexity of 𝓤(λ) and 𝓟(2λ), respectively. In particular, for fi ∈ 𝓐 with f''(0) = 0, we find conditions on λ, β*(λ) and β(λ) so that 𝓤(λ) ⊊ 𝓢*(β*(λ)) and 𝓟(2λ) ⊊ 𝒦(β(λ)). Here, 𝓢*(β) and 𝒦(β) (β < 1) denote the classes of functions in 𝓐 that are starlike of order β and convex of order β, respectively. In addition to these results, we also provide a coefficient condition for functions to be in 𝒦(β). Finally, we propose a conjecture that each function f ∈ 𝓤(λ) with f''(0) = 0 is convex at least when 0 < λ ≤ 3 - 2√2.
3
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Coefficient inequalities for concave and meromorphically starlike univalent functions

64%
Bhowmik B., Ponnusamy S.
Annales Polonici Mathematici
|
2008
|
tom 93
|
nr 2
177-186
EN
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.
4
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Sufficient conditions for starlike and convex functions

64%
Ponnusamy S., Vasundhra P.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 3
277-288
EN
For n ≥ 1, let 𝓐 denote the class of all analytic functions f in the unit disk Δ of the form $f(z) = z + ∑_{k=2}^∞ a_kz^k$. For Re α < 2 and γ > 0 given, let 𝓟(γ,α) denote the class of all functions f ∈ 𝓐 satisfying the condition |f'(z) - α f(z)/z + α - 1| ≤ γ, z ∈ Δ. We find sufficient conditions for functions in 𝓟(γ,α) to be starlike of order β. A generalization of this result along with some convolution results is also obtained.
5
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Starlikeness of functions satisfying a differential inequality

51%
Ali R., Ponnusamy S., Singh V.
Annales Polonici Mathematici
|
1995
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tom 61
|
nr 2
135-140
EN
In a recent paper Fournier and Ruscheweyh established a theorem related to a certain functional. We extend their result differently, and then use it to obtain a precise upper bound on α so that for f analytic in |z| < 1, f(0) = f'(0) - 1 = 0 and satisfying Re{zf''(z)} > -λ, the function f is starlike.
6
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Convolution theorems for starlike and convex functions in the unit disc

51%
Anbudurai M., Parvatham R., Ponnusamy S., Singh V.
Annales Polonici Mathematici
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2004
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tom 84
|
nr 1
27-39
EN
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.
7
Content available remote

Coefficient bounds for certain classes of analytic functions

51%
Juneja O., Ponnusamy S., Rajasekaran S.
Annales Polonici Mathematici
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1995
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tom 62
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nr 3
231-244
EN
We determine coefficient estimates for α-spiral functions of order ϱ with respect to N-symmetric points (|α| = π/2, 0 ≤ ϱ = 1$ and N is a positive integer). Sharp coefficient bounds are also obtained for functions of the form $f(z)^{-t}$, where t is a positive integer and f(z) is an α-spiral function of order ϱ. Using this we deduce coefficient estimates for inverses of univalent α-spiral and meromorphic univalent α-spiral functions with vanishing early coefficients.
8
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Landau's theorem for p-harmonic mappings in several variables

51%
Chen S., Ponnusamy S., Wang X.
Annales Polonici Mathematici
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2012
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tom 103
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nr 1
67-87
EN
A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^pf=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:Ω → ℂ^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ {1,...,m}) of $f = (f₁,..., f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch's theorem for a class of p-harmonic mappings f from the unit ball 𝔹ⁿ into ℂⁿ with the form $f(z) = ∑_{(k₁,..., kₙ) = (1,...,1)}^{(p,...,p)} |z₁|^{2(k₁-1)} ⋯ |zₙ|^{2(kₙ-1)}G_{p-k₁+1,...,p-kₙ+1}(z)$, where each $G_{p-k₁+1,..., p-kₙ+1}$ is harmonic in 𝔹ⁿ for $k_{i} ∈ {1,...,p}$ and i ∈ {1,. .., n}.
9
Content available remote

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

51%
Ponnusamy S., Vasudevarao A., Vuorinen M.
Colloquium Mathematicum
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2009
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tom 116
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nr 1
31-46
EN
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.
10
Content available remote

Univalence, strong starlikeness and integral transforms

51%
Obradović M., Ponnusamy S., Vasundhra P.
Annales Polonici Mathematici
|
2005
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tom 86
|
nr 1
1-13
EN
Let 𝓐 represent the class of all normalized analytic functions f in the unit disc Δ. In the present work, we first obtain a necessary condition for convex functions in Δ. Conditions are established for a certain combination of functions to be starlike or convex in Δ. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on f and μ so that the non-linear integral transform $∫_0^z (ζ/f(ζ))^{μ} dζ$ is univalent in Δ. Finally, we give sufficient conditions for functions to be strongly starlike of order α.
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