Special Classes of Univalent Functions with Missing Coefficients and Integral Transforms

S. Ponnusamy ; P. Sahoo

Bulletin of the Malaysian Mathematical Society, Tome 28 (2005) no. 2

Résumé

Let A n be the class of all analytic functions f of the form f ( z )= z + ∑ k = n + 1 ∞ a k z k , z ∈Δ, where n ∈ N is fixed. For λ>0 and α1, define U n (λ)={ f ∈ A n : |( z / f ( z )) n + 1 f '( z ) - 1|, z ∈Δ} and S α ∗ ={ f ∈ S ∗ (α): | z f '( z )/ f ( z ) - 1|1 - α, z ∈Δ}. In this paper, we find suitable conditions on λ and α so that U n (λ) is included in S α and S ∗ (α). Here S α and S ∗ (α) denote the usual classes of strongly starlike and starlike of order α, respectively. We determine necessary conditions so that f ∈ U n (λ) implies that | z f '( z )/ f ( z ) - 1/(2β)|1/(2β), z ∈Δ, or |1 + z f ''( z )/ f '( z ) - 1/(2β)|1/(2β), | z | r , where r = r (λ, n ) will be specified. For c + 1 - n >0, define [ I ( f )]( z )= F ( z )= z [( c + 1 - n )/ z c + 1 - n .∫ 0 z ( t / f ( t )) n t c - n d t ] 1/ n . We also find conditions on λ, α and c so that I ( U n (λ))⊂ S α ∗ .

Détail

Classification : 30C45, 30C55.