2019/05/05 by Ali, Rosihan M., Obradović, Milutin, Ponnusamy, Saminathan
#30C45 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.01694
Let \mathcal M be the class of analytic functions in the unit disk \ID with the normalization f(0)=f'(0)-1=0, and satisfying the condition |z2 ((z)/(f(z)) )''+ f'(z)((z)/(f(z)) )2-1 |≤ 1, z∈ \ID. Functions in M are known to be univalent in \ID. In this paper, it is shown that the harmonic mean of two functions in \mathcal M are closed, that is, it belongs again to \mathcal M. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in M are shown to be starlike in \ID. However we conjecture that functions in M are not necessarily starlike, as apparently supported by other examples.