arXiv subjects

M. Obradović

Publications and source records attributed to M. Obradović.

5 recordsLinked to original sources

arXiv2018

New characterization based symmetry tests

Two new symmetry tests, of integral and Kolmogorov type, based on the characterization by squares of linear statistics are proposed. The test statistics are related to the family of degenerate U-statistics. Their asymptotic properties are explored. The maximal eigenvalue, needed for the derivation of their logarithmic tail behavior, was calculated or approximated using techniques from the theory of linear operators and the perturbation theory. The quality of the tests is assessed using the approximate Bahadur efficiency as well as the simulated powers. The tests are shown to be comparable with some recent and classical tests of symmetry.

stat.ME↗

arXiv2017

Logarithmic Coefficients and a Coefficient Conjecture for Univalent Functions

Let ${\mathcal U}(λ)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. The logarithmic coefficients $γ_n$ of $f$ are defined by the formula $\log(f(z)/z)=2\sum_{n=1}^\infty γ_nz^n$. In a recent paper, the present authors proposed a conjecture that if $f\in {\mathcal U}(λ)$ for some $0<λ\leq 1$, then $|a_n|\leq \sum_{k=0}^{n-1}λ^k$ for $n\geq 2$ and provided a new proof for the case $n=2$. One of the aims of this article is to present a proof of this conjecture for $n=3, 4$ and an elegant proof of the inequality for $n=2$, with equality for $f(z)=z/[(1+z)(1+λz)]$. In addition, the authors prove the following sharp inequality for $f\in{\mathcal U}(λ)$: $$\sum_{n=1}^{\infty}|γ_{n}|^{2} \leq \frac{1}{4}\left(\frac{π^{2}}{6}+2{\rm Li\,}_{2}(λ)+{\rm Li\,}_{2}(λ^{2})\right), $$ where ${\rm Li}_2$ denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of $\mathcal S$.

math.CV↗

arXiv2014

The radius of univalence of the reciprocal of a product of two analytic functions

Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the open unit disk $\ID$ with the normalization $f(0)=0= f'(0)-1$ and ${\mathcal S}$ be the class of univalent functions from ${\mathcal A}$. In this paper, we consider radius of univalence of $F$ defined by $F(z)=z^{3}/(f(z)g(z))$, where $f$ and $g$ belong to some subclasses of ${\mathcal A}$ (for which $f(z)/z$ and $g(z)/z$ are non-vanishing in $\ID$) and, in some cases in precise form, belonging to some subclasses of ${\mathcal S}$. All the results are proved to be sharp. Applications of our investigation through Bessel functions are also presented.

math.CV↗

arXiv2012

Univalence of the average of two analytic functions

Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\ID=\{z:\,|z|<1\}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_nz^n.$ Let $\mathcal{U}$ denote the set of all $f\in \mathcal{A}$, $f(z)/z\neq 0$ and satisfying the condition $$ | f'(z) (\frac{z}{f(z)})^{2}-1 | < 1 {for $z\in \ID$}. $$ Functions in ${\mathcal U}$ are known to be univalent in $\ID$. For $α\in [0,1]$, let $$ \mathcal{N}(α)= \{f_α:\, f_α(z)=(1-α)f(z)+α\int_0^z\frac{f(t)}{t}\,dt, {$f\in\mathcal{A}$ with $|a_n|\leq n$ for $n\geq 2$}\}. $$ In this paper, we first show that the condition $\sum_{n=2}^{\infty}n|a_n|\leq 1$ is sufficient for $f$ to be in ${\mathcal U}$ and the same condition is necessary for $f\in {\mathcal U}$ in case all $a_n$'s are negative. Next, we obtain the radius of univalence of functions in the class $\mathcal{N}(α)$. Also, for $f,g\in \mathcal{U}$ with $\frac{f(z)+g(z)}{z}\neq 0$ in $\ID$, $F(z)=(f(z)+g(z))/2$, and $G(z)=r^{-1}F(rz)$, we determine a range of $r$ such that $G\in {\mathcal U}$. As a consequence of these results, several special cases are presented.

math.CV↗

arXiv2011

On harmonic combination of univalent functions

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (λ)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$|f'(z)(\frac{z}{f(z)})^{2}-1| <λ~for $z\in \ID$, $$ for some $λ\in (0,1]$. In this paper, among other things, we study a "harmonic mean" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\frac{z}{F(z)}=1/2(\frac{z}{f(z)}+\frac{z}{g(z)}), $$ where $f,g\in \mathcal{S}$ or $f,g\in \mathcal{U}(1)$. In particular, we determine the radius of univalency of $F$, and propose two conjectures concerning the univalency of $F$.

math.CV↗