Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• # Artykuł - szczegóły

## Annales Polonici Mathematici

2004 | 84 | 1 | 27-39

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

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.

27-39

wydano
2004

### Twórcy

autor
• Department of Mathematics, D.G. Vaishnav College, Chennai 600 106, India
autor
• The Ramanujan Institute, University of Madras, Chennai 600 005, India
autor
• Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai 600 036, India
autor
• 3A/95, Azad Nagar, Kanpur 208 002, India