Differential Subordination Properties of Sokół-Stankiewicz Starlike Functions

Abstract

Let piz) be an analytic function defined on the open unit disk D and p(0) = 1. Condition ß in terms of complex numbers D and real E with -1<E<1 and IDI < 1 is determined such that 1 -I-ßzp'iz) -< j± §f implies piz) -< VI + z. Furthermore, the expression 1 -|-J'^^Ç and 1 4-'^^f^y are considered in obtaining similar results.

Starlikeness of Functions Defined by Third-Order Differential Inequalities and Integral Operators

“…The problem of determining sufficient conditions to ensure starlikeness of functions has been widely investigated. These include conditions in terms of differential inequalities; see, for example, [2][3][4][5][6][7][8][9][10][11]. Miller and Mocanu [12], Kuroki and Owa [13], and, more recently, Ali et al [14] determined conditions for starlikeness of functions defined by an integral operator of the form…”

Differential Subordination Properties of Certain Analytic Functions

“…Furthermore in [1], the class of Sokó l-Stankiewicz starlike functions was considered in obtaining conditions on β so that the above implications holds. In [12], the authors determined values of β so that the subordination of 1 + βzp (z), 1 + βzp (z) p(z)…”

Applications of first order differential subordination for functions with positive real part

“…In 2007, Ali et al [2] extended this result and determined the estimates on β for which the subordination 1 + βzp ′ (z)/p j (z) ≺ (1 + Dz)/(1 + Ez) (j = 0, 1, 2) implies the subordination p(z) ≺ (1 + Az)/(1 + Bz), where A, B, D, E ∈ [−1, 1]. In 2013, Omar and Halim [15] determined the condition on β in terms of complex number D and real E with −1 < E < 1 and |D| ≤ 1 such that 1+βzp ′ (z)/p j (z) ≺…”