On some first-order differential subordination

Let AA denote the class of functions f   that are analytic in the unit disc DD and normalized by f(0) = f′(0) − 1 = 0. In this paper, we investigate the class of functions such that Re{f′(z)+zf″(z)-β}>αRe{f′(z)+zf″(z)-β}>α in DD. We determine conditions for α and β under which the function f is univalent, close-to-convex, and convex. To obtain this, we first estimate ∣Arg{f′(z)}∣ which improves the earlier results.