Differential Inequalities and Univalent Functions

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