Some convolution properties of a certain class of -valent analytic functions

Let A ( p ) denote the class of functions f ( z ) = z p + ∑ n = p + 1 ∞ a n z n ( p ∈ N = { 1 , 2 , … } ) which are analytic and p -valent in the unit disc U = { z : | z | < 1 } . The objective of the present work is to obtain some convolution properties for the class P ( p , α ) = { f ( z ) ∈ A ( p ) : Re { f ′ ( z ) z p − 1 } > α , 0 ≤ α < p , p ∈ N , z ∈ U } . Also we prove that the integral operator J p , c ( f ) ∈ P ( p , β ) ( f ( z ) ∈ A ( p ) ) , where the value β is sharp.