Hardy's inequalities for the Jacobi weight

4 Jul 2021, 15:25


Ramil Nasibullin (Kazan Federal University)


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 $\mathbb{D}$.

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Ramil Nasibullin (Kazan Federal University)

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