# Convolution theorems for starlike and convex functions in the unit disc

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

Annales Polonici Mathematici (2004)

- Volume: 84, Issue: 1, page 27-39
- ISSN: 0066-2216

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topM. Anbudurai, et al. "Convolution theorems for starlike and convex functions in the unit disc." Annales Polonici Mathematici 84.1 (2004): 27-39. <http://eudml.org/doc/281088>.

@article{M2004,

abstract = {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
$P⁰_\{β\} = \{f ∈ A: Re f^\{\prime \}(z) > β, z ∈ Δ\}$.
For λ > 0, suppose that denotes any one of the following classes of functions:
$M^\{(1)\}_\{1,λ\} = \{f ∈ : Re\{z(zf^\{\prime \}(z))^\{\prime \prime \}\} > -λ, z ∈ Δ\}$,
$M^\{(2)\}_\{1,λ\} = \{f ∈ : Re\{z(z²f^\{\prime \prime \}(z))^\{\prime \prime \}\} > -λ, z ∈ Δ\}$,
$M^\{(3)\}_\{1,λ\} = \{f ∈ : Re\{1/2 (z(z²f^\{\prime \}(z))^\{\prime \prime \})^\{\prime \} - 1\} > -λ, z ∈ Δ\}$.
The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in $_\{γ\}$ or $_\{γ\}$, γ ∈ [0,1/2]. Here $_\{γ\}$ and $_\{γ\}$ 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 $M_\{1,α\} ∗ ⊂ _\{γ\}$ and $M_\{1,α\} ∗ ⊂ _\{γ\}$, where is either $⁰_\{β\}$ or $M_\{1,β\}$. Here $M_\{1,λ\}$ denotes the class of all functions f in such that Re(zf”(z)) > -λ for z ∈ Δ.},

author = {M. Anbudurai, R. Parvatham, S. Ponnusamy, V. Singh},

journal = {Annales Polonici Mathematici},

keywords = {univalent; starlike and convex functions; Hadamard product and subordination},

language = {eng},

number = {1},

pages = {27-39},

title = {Convolution theorems for starlike and convex functions in the unit disc},

url = {http://eudml.org/doc/281088},

volume = {84},

year = {2004},

}

TY - JOUR

AU - M. Anbudurai

AU - R. Parvatham

AU - S. Ponnusamy

AU - V. Singh

TI - Convolution theorems for starlike and convex functions in the unit disc

JO - Annales Polonici Mathematici

PY - 2004

VL - 84

IS - 1

SP - 27

EP - 39

AB - 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
$P⁰_{β} = {f ∈ A: Re f^{\prime }(z) > β, z ∈ Δ}$.
For λ > 0, suppose that denotes any one of the following classes of functions:
$M^{(1)}_{1,λ} = {f ∈ : Re{z(zf^{\prime }(z))^{\prime \prime }} > -λ, z ∈ Δ}$,
$M^{(2)}_{1,λ} = {f ∈ : Re{z(z²f^{\prime \prime }(z))^{\prime \prime }} > -λ, z ∈ Δ}$,
$M^{(3)}_{1,λ} = {f ∈ : Re{1/2 (z(z²f^{\prime }(z))^{\prime \prime })^{\prime } - 1} > -λ, z ∈ Δ}$.
The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in $_{γ}$ or $_{γ}$, γ ∈ [0,1/2]. Here $_{γ}$ and $_{γ}$ 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 $M_{1,α} ∗ ⊂ _{γ}$ and $M_{1,α} ∗ ⊂ _{γ}$, where is either $⁰_{β}$ or $M_{1,β}$. Here $M_{1,λ}$ denotes the class of all functions f in such that Re(zf”(z)) > -λ for z ∈ Δ.

LA - eng

KW - univalent; starlike and convex functions; Hadamard product and subordination

UR - http://eudml.org/doc/281088

ER -

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