On harmonic combination of univalent functions

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (λ)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$|f'(z)(\frac{z}{f(z)})^{2}-1| <λ~for $z\in \ID$, $$ for some $λ\in (0,1]$. In this paper, among other things, we study a "harmonic mean" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $\frac{z}{F(z)}=1/2(\frac{z}{f(z)}+\frac{z}{g(z)}),$ where $f,g\in \mathcal{S}$ or $f,g\in \mathcal{U}(1)$. In particular, we determine the radius of univalency of $F$, and propose two conjectures concerning the univalency of $F$.

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