On Certain Classes of Biharmonic Mappings Defined by Convolution

and Applied Analysis 3 and satisfy the condition ∂ ∂θ ( arg f ( re )) Re { zh′ − zg ′ h g } > α 1.8 in D, where 0 ≤ α < 1. For two analytic functions f1 and f2, if f1 z ∞ ∑ j 1 ajz j , f2 z ∞ ∑ j 1 Ajz j , 1.9 then the convolution of f1 and f2 is defined by ( f1 ∗ f2 ) z f1 z ∗ f2 z ∞ ∑ j 1 ajAjz j . 1.10 By using the convolution, in 16 , Ali et al. introduced the class SH φ, σ, α of harmonic mappings in the form of 1.6 such that Re ⎧ ⎨ ⎩ z ( h ∗ φ′ z − σzg ∗ φ′ z ( h ∗ φ z σg ∗ φ z ⎫ ⎬ ⎭ > α 1.11 and the class SP 0 H φ, σ, α such that Re ⎧ ⎨ ⎩ ( 1 e )z ( h ∗ φ′ z − σzg ∗ φ′ z ( h ∗ φ z σg ∗ φ z − e ⎫ ⎬ ⎭ > α, 1.12 where σ ∈ R and α ∈ 0, 1 are constants, γ ∈ R and φ z z ∞n 2 φnz is analytic in D. Now we consider a class of biharmonic mappings, denoted by BH0 φk;σ, a, b , as follows: F ∈ BH0 D with the form 1.4 is said to be in BH0 φk;σ, a, b if and only if Re { a Φ z Ψ z − b } > 0, 1.13