A Class of Analytic Functions with Missing Coefficients. (3rd July 2011) - Yang, Ding-Gong; Liu, Jin-Lin
Abstract : LetT n ( A, B, γ, α )   ( − 1 ≤ B < 1, B < A, 0 < γ ≤ 1 andα > 0 ) denote the class of functions of the formf ( z ) = z + ∑ k = n + 1 ∞ a k z k   ( n ∈ N = { 1, 2, 3, … } ), which are analytic in the open unit diskU and satisfy the following subordination conditionf ′ ( z ) + α z f ′′ ( z ) ≺ ( ( 1 + A z ) / ( 1 + B z ) ) γ, for( z ∈ U ; A ≤ 1 ; 0 < γ < 1 ), ( 1 + A z ) / ( 1 + B z ), for( z ∈ U ; γ = 1 ) . We obtain sharp bounds on R e f ' ( z ), R e f ( z ) / z, | f ( z ) |, and coefficient estimates for functionsf ( z ) belonging to the classT n ( A, B, γ, α ) . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.