Argument inequalities for certain analytic functions

Let q be analytic in the open unit disk U with q ( 0 ) = 1 and q ( z ) ≠ 0 ( z ∈ U ) . By using the method of differential subordinations, we derive certain conditions involving q and z q ′ under which the functions q satisfy the following two-sided inequality: − α 2 π 2 < arg q ( z ) < α 1 π 2 ( z ∈ U ) for some α 1 and α 2 ( 0 < α 1 , α 2 ≤ 1 ) . Several interesting consequences of the main results are also given. All these results presented here are sharp.