A note on Briot-Bouquet-Bernoulli differential subordination. (English) Zbl 1121.34004

Kanas, Stanisława; Kowalczyk, Joanna

Commentat. Math. Univ. Carol. 46, No. 2, 339-347 (2005).

Summary: Let \(p, q\) be analytic functions in the unit disk \(\mathcal U\). For \(\alpha \in [0,1)\) the authors consider the differential subordination and the differential equation of the Briot-Bouquet type: \[ p^{1-\alpha }(z)+\frac {zp'(z)}{\delta p^{\alpha }(z) + \lambda p(z)}\prec h(z), \quad z\in \mathcal U, \]
\[ q^{1-\alpha }(z)+\frac {nzq'(z)}{\delta q^{\alpha }(z)+\lambda q(z)} = h(z), \quad z\in \mathcal U, \] with \(p(0) =q(0) =h(0)=1\). The aim of the paper is to find the dominant and the best dominant of the above subordination. In addition, the authors give some particular cases of the main result obtained for appropriate choices of functions \(h\).

MSC:

34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
30C35	General theory of conformal mappings
30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

Briot-Bouquet-Bernoulli differential subordination