Speaker
Description
We prove onedimensional new Hardy type inequalities for Jacoby eights. Using this inequality, we obtain Nehari-Pokornii type univalence conditions for analytic in the unite disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ functions. The following theorem holds.
Theorem 1. Suppose that $f$ is meromorphic in $\mathbb{D}$ function. If $n\in \mathbb{N}$, $a_k$ and $\mu_k$, $k=\overline{1,n}$, are positive real numbers and
$$
|S_f(z)| \leq \sum_{k=1}^n \frac{b_k A(\mu_k)}{(1-|z|^2)^{\mu_k}}, \quad z\in \mathbb{D},
$$
where $b_k=\frac{2P_{2-\mu_k}}{A(\mu_k)} \ a_k$, $a_1+a_2+ \ldots + a_n\leq 1$, $0\leq \mu_1\leq \mu_2\leq \ldots \leq \mu_n\leq 2$ and
$$
A(\mu)=
\begin{cases}
2^{3\mu-1}\pi^{2(1-\mu)}, & 0\leq \mu \leq 1, \\
2^{3-\mu}, & 1\leq \mu \leq 2;
\end{cases}
$$
Then the function $f$ is univalent in $\mathbb{D}$.
