Convolution theorems for starlike and convex functions in the unit disc

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

Annales Polonici Mathematici, Tome 83 (2004), p. 27-39 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2004-01-01

Zbl 1098.30013

EUDML-ID : urn:eudml:doc:281088