Differential Inequalities and Univalent Functions

Let \(\mathcal{M}\) be the class of analytic functions in the unit disk \(\mathbb{D}\) with the normalization f(0) = f′(0) − 1 = 0, and satisfying the condition $$\left|{{z^2}{{\left({{z\over{f(z)}}}\right)}^{\prime\prime}}\;+\;f'(z){{\left({{z\over{f(z)}}} \right)}^2}\;-\;1}\right|\le 1,\;\;\;z\;\in\;\mathbb{D}.$$ Functions in \(\mathcal{M}\) are known to be univalent in \(\mathbb{D}\). 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 \(\mathcal{M}\) are shown to be starlike in \(\mathbb{D}\). However we conjecture that functions in \(\mathcal{M}\) are not necessarily starlike, as apparently supported by other examples.