Inclusion properties for a class of analytic functions defined by a second-order differential inequality

For β < 1 , and α ≥ γ ≥ 0 , let W β ( α , γ ) consist of normalized analytic functions f in the unit disk satisfying Re e i ϕ ( 1 - α + 2 γ ) f ( z ) / z + ( α - 2 γ ) f ′ ( z ) + γ z f ′ ′ ( z ) - β > 0 , z ∈ D , for some ϕ with | ϕ | < π / 2 . The extreme points and sharp coefficient bounds for this class are determined. Estimates on β are also found that would ensure functions in W β ( α , γ ) are starlike. When β = 0 , a sharp radius of univalence is obtained for the class W 0 ( α , γ ) . Additionally sufficient conditions in terms of the Schwarzian derivative and the second Taylor coefficient are obtained for functions f to belong to the class W 0 ( α , γ ) .