For μ ≥ 0 , we consider a linear operator L μ : A → A defined by the convolution f μ ∗ f , where f μ = ( 1 − μ ) z 2 F 1 ( a , b , c ; z ) + μ z ( z 2 F 1 ( a , b , c ; z ) ) ′ . Let φ ∗ ( A , B ) denote the class of normalized functions f which are analytic in the open unit disk and satisfy the condition z f ′ / f ≺ ( 1 + A z ) / 1 + B z , − 1 ≤ A < B ≤ 1 , and let R η ( β ) denote the class of normalized analytic functions f for which there exits a number η ∈ ( − π / 2 , π / 2 ) such that 0$"> Re ( e i η ( f ′ ( z ) − β ) ) > 0 , ( β < 1 ) . The main object of this paper is to establish the connection between R η ( β ) and φ ∗ ( A , B ) involving the operator L μ ( f ) . Furthermore, we treat the convolution I = ∫ 0 z ( f μ ( t ) / t ) d t   ∗ f ( z ) for f ∈ R η ( β ) .