arXiv subjects
Publications and source records attributed to Milutin Obradović.
In this paper we consider some properties of the initial logarithmic coefficients for inverse functions of functions univalent in the unit disc. The case of convex functions is treated separately. We give estimate, in some cases sharp, of the modulus of the initial coefficients, as well as the difference of the modulus of two consecutive coefficients.
In this paper we use a method based on the Grunsky coefficients to find upper bounds of the modulus of the initial coefficients, difference of the moduli of two consecutive initial coefficients, of the modulus of the initial logarithmic coefficient, and of the second Hankel determinant for the class of normalized bi-univalent functions.
In this paper we investigate Toeplitz and symmetric Toeplitz determinants of inverse functions for some classes of univalent functions and improve some previous results.
In this paper, we consider estimates of symmetric Toeplitz determinants $T_{q,n}(f)$ for the class ${\mathcal U}$ and for the general class ${\mathcal S}$ for certain values of $q$ and $n$ ($q,n=1,2,3\ldots$).
Let function $f$ be normalized, analytic and univalent in the unit disk ${\mathbb D}=\{z:|z|<1\}$ and $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. Using a method based on Grusky coefficients we study several problems over that class of univalent functions: upper bounds of the special case of the generalised Zalcman conjecture $|a_2a_3-a_4|$, of the third logarithmic coefficient, and of the second Hankel determinant for the logarithmic coefficients.
In this paper we give simple proofs for the main results concerning generalized Fekete-Szegő functional of type $\left|a_{3}(f)-λa_{2}(f)^{2}\right|-μ|a_{2}(f)|$, where $λ\in\mathbb{C}$, $μ>0$ and $a_{n}(f)$ is $n$-th coefficient of the power series expansion of $f\in\mathcal{S}$. In addition, we studied this functional separately for the class $\mathcal{K}$ of convex functions and we emphasize that all the results of the paper are sharp (i.e. the best possible). The advantages of the present study are that the techniques used in the proofs are more easier and use known results regarding the univalent functions, and those that it give the best possible results not only for the entire class of univalent normalized functions $\mathcal{S}$ but also for its subclass of convex functions $\mathcal{K}$.
For $f\in \mathcal{S}$, the class univalent functions 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 improve previous bounds for the second and third Hankel determinants in case when either $a_2=0,$ or $a_3=0$. We also improve an upper bound for the coefficient difference $|a_4|-|a_3|$ when $f\in \mathcal{S}$.
Using some properties of the Grunsky coefficients we improve earlier results for upper bounds of the Hankel determinants of the second and third order for the class $\mathcal{S}$ of univalent functions.
It is well-known that the condition ${\operatorname{Re}} \left[1+\frac{zf''(z)}{f'(z)}\right]>0$, $z\in{\mathbb D}$, implies that $f$ is starlike function (i.e. convexity implies starlikeness). If the previous condition is not satisfied for every $z\in {\mathbb D}$, then it is possible to get new criteria for starlikeness by using $\left|\arg\left[α+\frac{zf''(z)}{f'(z)}\right]\right|$, $z\in{\mathbb D}$, where $α>1.$
In this paper, we give sharp bounds of the difference of the moduli of the second and the first logarithmic coefficient for the functions on the class $\mathcal U$, for the $α$-convex functions, and for the class $\mathcal{G}(α)$ introduced by Ozaki.
Let $\mathcal{U(α, λ)}$, $0<α<1$, $0 < λ<1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+α}f'(z)-1\right|<λ$$ in the unit disc ${\mathbb D}$. For $f\in \mathcal{U(α, λ)}$ we give sharp bounds of its initial logarithmic coefficients $γ_{1},\,γ_{2},\,γ_{3}.$
In this paper we determine the upper bounds of the Hankel determinants of special type $H_{2}(3)(f)$ and $H_{2}(4)(f)$ for the class of univalent functions and for the class $\mathcal{U}$ defined by \[ \mathcal{U}=\left\{ f\in\mathcal{A} : \left|\left[\frac{z}{f(z)}\right]^2 f'(z)-1 \right|<1,\, z\in{\mathbb D} \right\}, \] where $\mathcal{A}$ is the class of functions analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+\cdots$.
In this paper we determine the upper bounds of $|H_{2}(3)|$ for the inverse functions of functions of some classes of univalent functions, where $H_{2}(3)(f)=a_{3}a_{5}-a_{4}^{2}$ is the Hankel determinant of a special type.
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 ${\mathcal U}^+$.
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 ${\mathcal U}(λ)$.
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 $0.400436\ldots \leλ\le1$.