Convolution theorems for starlike and convex functions in the unit disc

• Autorzy: M. Anbudurai , R. Parvatham , S. Ponnusamy , V. Singh
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
• 30C55: General theory of univalent and multivalent functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2004
• Tom: 84
• Numer: 1
• Strony: 27-39

Daty

wydano

2004

Twórcy

autor

M. Anbudurai

• Department of Mathematics, D.G. Vaishnav College, Chennai 600 106, India

autor

R. Parvatham

• The Ramanujan Institute, University of Madras, Chennai 600 005, India

autor

S. Ponnusamy

• Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai 600 036, India

autor

V. Singh

• 3A/95, Azad Nagar, Kanpur 208 002, India

Identyfikatory

DOI

10.4064/ap84-1-2