We consider functions f ( z ) = z + … that are analytic in the unit disk and satisfy there the inequality \alpha $"> Re ( f ′ ( z ) + z f ″ ( z ) ) > α , α < 1 . We find extreme points and then determine sharp lower bounds on Re f ′ ( z ) and Re ( f ( z ) / z ) . Sharp results for the sequence of partial sums are also found.