Convolution theorems for starlike and convex functions in the unit disc

M. 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 -