## Criteria for univalence, integral means and Dirichlet integral for meromorphic functions.(English)Zbl 1377.30008

Summary: Let $$\mathcal{A}(p)$$ be the class consisting of functions $$f$$ that are holomorphic in $$\mathbb D\setminus \{p\}$$, $$p\in (0,1)$$ possessing a simple pole at the point $$z=p$$ with nonzero residue and normalized by the condition $$f(0)=0=f'(0)-1$$. In this article, we first prove a sufficient condition for univalency for functions in $$\mathcal{A}(p)$$. Thereafter, we consider the class denoted by $$\Sigma(p)$$ that consists of functions $$f \in \mathcal{A}(p)$$ that are univalent in $$\mathbb D$$. We obtain the exact value for $$\displaystyle\max_ {f\in \Sigma(p)}\Delta(r,z/f)$$, where the Dirichlet integral $$\Delta(r,z/f)$$ is given by $\Delta(r,z/f)=\int\int_{|z|<r}|(z/f(z))'|^2dx dy,\quad (z=x+iy),0<r\leq 1.$ We also obtain a sharp estimate for $$\Delta(r,z/f)$$ whenever $$f$$ belongs to certain subclasses of $$\Sigma(p)$$. Furthermore, we obtain sharp estimates of the integral means for the aforementioned classes of functions.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods

### Keywords:

concave function; star-like function; Dirichlet integral
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