Let P b ( α ) {\mathcal {P}_b}(\alpha ) denote the class of functions P ( z ) = 1 + b ( 1 − α ) z + ⋯ P(z) = 1 + b(1 - \alpha )z + \cdots which are analytic and satisfy Re ⁡ { P ( z ) } > α \operatorname {Re} \{ P(z)\} > \alpha for | z | > 1 |z| > 1 where α ∈ [ 0 , 1 ) \alpha \in [0,1) and b ∈ [ 0 , 2 ] b \in [0,2] . We demonstrate some inequalities involving | P ′ ( z ) | |P’(z)| and | P ′ ( z ) / P ( z ) | |P’(z)/P(z)| dependent on b and α \alpha which are subsequently applied to the class of functions whose derivative lies in P b ( α ) {\mathcal {P}_b}(\alpha ) to obtain distortion, covering, and radius of convexity properties.