Bounded p-valent Robertson functions defined by using a differential operator

Let F M ( λ , α , p , q ) ( M > 1 2 , | λ | < π / 2 , 0 ≤ α < p − q , p ∈ N , q ∈ N 0 , p > q ) denote the class of functions f ( z ) = z p + ∑ k = p + 1 ∞ a k z k analytic in U = { z : | z | < 1 } which satisfy for fixed M, z = re i θ ∈ U and e i λ zf ( 1 + q ) ( z ) f ( q ) ( z ) − α cos λ − i ( p − q ) sin λ ( p − q − α ) cos λ − M < M . Also let G M ( λ , α , p , q ) ( M > 1 2 , | λ | < π / 2 , 0 ≤ α < p − q , p ∈ N , q ∈ N 0 , p > q ) denote the class of bounded p-valent λ -Robertson functions of order α , 0 ≤ α < p − q , that is, g ( q ) ( z ) ∈ G M ( λ , α , p , q ) if and only if ( zg ( 1 + q ) ( z ) / ( p − q ) ) ∈ F M ( λ , α , p , q ) . In this paper we investigate certain properties of the above-mentioned classes.