arXiv subjects
Publications and source records attributed to Nikola Tuneski.
Let $\mathcal{S}$ denote the class of functions $f$ which are analytic and univalent in the unit disk ${\mathbb D}=\{z:|z|<1\}$ and normalized with $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. Using a method based on Grusky coefficients we study two problems over the class $\mathcal{S}$: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference $|a_5|-|a_4|$.
Let ${\mathcal U}^+$ be the class of analytic functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its inverse. In this paper we give sharp estimates of the initial coefficients and initial logarithmic coefficients for $f$, as well as, sharp estimates of the second and the third Hankel determinant for $f$ and $f^{-1}$. We also show that the Zalcman conjecture holds for functions $f$ from ${\mathcal U}^+$.
For $f\in \mathcal{S}$, the class of normalized functions, analytic and univalent in the unit disk $\mathbb{D}$ and given by $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$ for $z\in \mathbb{D}$, we give an upper bound for the coefficient difference $|a_4|-|a_3|$ when $f\in \mathcal{S}$. This provides an improved bound in the case $n=3$ of Grispan's 1976 general bound $||a_{n+1}|-|a_n||\le 3.61\dots .$ Other coefficients bounds, and bounds for the second and third Hankel determinants when $f\in \mathcal{S}$ are found when either $a_2=0,$ or $a_3=0$.
Let ${\mathcal A}$ be the class of functions that are analytic in the unit disc ${\mathbb D}$, normalized such that $f(z)=z+\sum_{n=2}^\infty a_nz^n$, and let class ${\mathcal U}(λ)$, $0<λ\le1$, consists of functions $f\in{\mathcal A}$, such that \[ \left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < λ\quad (z\in {\mathbb D}). \] In this paper we determine the sharp upper bounds for the Hankel determinants of second and third order for the inverse functions of functions from the class ${\mathcal U}(λ)$.
In this paper we study an interpolation problem involving the Fekete--Szegö functional. It turns out that this problem links to the so-called filtration of infinitesimal generators. We introduce new filtration classes using the non-linear differential operator \[α\frac{f(z)}{z}+β\frac{zf'(z)}{f(z)}+(1-α-β)\left(1+\frac{zf''(z)}{f'(z)}\right)\] and establish certain properties of these classes. Sharp upper bounds of the modulus of the Fekete--Szegö functional over some filtration classes are found. We also present open problems for further study.
Introducing a new method we give sharp estimates of the Hermitian Toepliz determinants of third order for the class $\mathcal{S}$ of functions univalent in the unit disc. The new approach is also illustrated on some subclasses of the class $\mathcal{S}$.
Let $f$ be function that is analytic in the unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$, i.e., of type $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. If additionally, \[ \left| \left(\frac{z}{f(z)}\right)^2 f'(z) -1\right|<λ\quad\quad (z\in{\mathbb D}), \] then $f$ belongs to the class ${\mathcal U}(λ)$, $0<λ\le1$. In this paper we prove sharp upper bound of the modulus of the fifth coefficient of $f$ from ${\mathcal U}(λ)$ satisfying \[ \frac{f(z)}{z}\prec \frac{1}{(1+z)(1+λz)}, \] ("$\prec$" is the usual subordination) in the case when $0.400436\ldots \leλ\le1$.
Let ${\mathcal A}$ be the class of functions $f$ that are analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. Let $0<λ\le1$ and \[ {\mathcal U}(λ) = \left\{ f\in{\mathcal A}: \left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < λ,\, z\in{\mathbb D} \right\}. \] In this paper sharp upper bounds of the first three coefficients of the inverse function $f^{-1}$ are given in the case when \[\frac{f(z)}{z}\prec \frac{1}{(1-z)(1-λz)}.\]
In this paper we determine the disks $|z| 0\qquad (|z|<r).$$
In this paper we improve the bounds of the third order Hankel determinant for two classes of univalent functions with bounded turning. The bounds are not sharp, but the sharp ones are conjectured.
In this paper we improve the upper bound of the third order Hankel determinant for the class of Ozaki close-to-convex functions. The sharp bound is conjectured.
In this paper we give improved, probably not sharp, upper bounds of the Hankel determinant of third order for various classes of univalent functions and conjecture the sharp one.
Function $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$, normalized, analytic and univalent in the unit disk $\mathbb D=\{z:|z|<1\}$, belongs to the class $\mathcal{U}$. if, and only if, \[ \left| \left(\frac{z}{f(z)}\right)^2 -1\right|<1 \quad\quad (z\in \mathbb D). \] In this paper, we prove the Zalcman and the generalized Zalcman conjecture for the class $\mathcal{U}$ and some values of parameters in the conjectures.
In this paper we give an upper bound of the third logarithmic coefficient for the class $\mathcal{S}$ of univalent functions in the unit disc.
Let $f$ be analytic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order $α$, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.
In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class $\mathcal{S}$ of univalent functions in the unit disc.
Adequate sampling is essential for the well-functioning of a market surveillance system. As small as possible statistically significant sample size is the main factor that determines the costs of market surveillance actions. This paper studies various possibilities for calculation of the size of the sample with an emphasis on the method based on the binomial distribution. Examples, comparisons, and conclusions are provided.
Let $f$ be analutic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bound of Hankel determinant of the second order for the class of analytic unctions satisfying \[ \left|\arg \left[\left(\frac{z}{f(z)}\right)^{1+α}f'(z) \right] \right|<γ\fracπ{2} \quad\quad (z\in\mathbb D),\] for $0<α<1$ and $0<γ\leq1$.