Dual Smale's mean value conjecture for $n=7$
We give an analytic proof of the dual Smale's mean value conjecture in the case $n=7$.
math.CV↗
arXiv subjects
Ilgiz R. Kayumov
Publications and source records attributed to Ilgiz R. Kayumov.
Weissler and Bernoulli type inequalities in Bergman spaces
We consider Weissler type inequalities for Bergman spaces with general radial weights and give conditions on the weight $w$ in terms of its moments ensuring that $\|f_r\|_{A^{2n}(w)}\leq \|f\|_{A^2(w)}$ whenever $n\in \mathbb{N}$ and $0< r\le 1/\sqrt{n}$. For noninteger exponents a special case of this inequality is proved which can be considered as a certain analog of the Bernoulli inequality. An example of a monotonic weight is constructed for which these inequalities are no longer true.
math.CV↗
On Bloch seminorm of finite Blaschke products in the unit disk
We prove that, for any finite Blaschke product $w=B(z)$ in the unit disk, the corresponding Riemann surface over the $w$--plane contains a one-sheeted disk of the radius $0.5$. Moreover, it contains a unit one-sheeted disk with a radial slit. We apply this result to obtain a universal sharp lower estimate of the Bloch seminorm for finite Blaschke products.
math.CV↗
On a conjecture about univalent polynomials
In this paper we prove a recent conjecture formulated by Dmitrishin, Smorodin and Stokolos about that certain polynomials are univalent in the unit disk. As a consequence we get an upper estimate for the Koebe radius of univalent polynomials.
math.CV↗
Bohr--Rogosinski inequalities for bounded analytic functions
In this paper we first consider another version of the Rogosinski inequality for analytic functions $f(z)=\sum_{n=0}^\infty a_nz^n$ in the unit disk $|z| < 1$, in which we replace the coefficients $a_n$ $(n= 0,1,\ldots ,N)$ of the power series by the derivatives $f^{(n)}(z)/n!$ $(n= 0,1,\ldots ,N)$. Secondly, we obtain improved versions of the classical Bohr inequality and Bohr's inequality for the harmonic mappings of the form $f = h + \overline{g}$, where the analytic part $h$ is bounded by $1$ and that $|g'(z)| \le k|h'(z)|$ in $|z| < 1$ and for some $k \in [0,1]$.
math.CV↗