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Milutin Obradović

Publications and source records attributed to Milutin Obradović.

29 records · Page 2Linked to original sources
arXiv2021

Coefficients of the inverse of functions for the subclass of $\mathcal U (λ)$

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 \[\frac{f(z)}{z}\prec \frac{1}{(1-z)(1-λz)}.\]

math.CV↗
arXiv2020

Zalcman and generalized Zalcman conjecture for the class $\mathcal{U}$

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.

math.CV↗
arXiv2019

Differential Inequalities and Univalent Functions

Let ${\mathcal M}$ be the class of analytic functions in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition $$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq 1, \quad z\in \ID. $$ Functions in $\mathcal{M}$ are known to be univalent in $\ID$. In this paper, it is shown that the harmonic mean of two functions in ${\mathcal M}$ are closed, that is, it belongs again to ${\mathcal M}$. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in $\mathcal{M}$ are shown to be starlike in $\ID$. However we conjecture that functions in $\mathcal{M}$ are not necessarily starlike, as apparently supported by other examples.

math.CV↗
arXiv2016

On relations between the classes $\mathcal S$ and $\mathcal U$

Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the unit disk $\ID$ and satisfying the normalization $f(0)=0= f'(0)-1$. Let $\mathcal{S}$ denote the subclass of ${\mathcal A}$ consisting of univalent functions in $\ID$. We consider the subclass $\mathcal{U} $ of $\mathcal{S}$ that is defined by the condition that for its members $f$ the condition $$\left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < 1 ~\mbox{ for $z\in \ID$} $$ holds. To theses relations belong striking similarities and on the other hand big differences. We show that some results about $\mathcal{S}$ can be improved for $\mathcal{U}$, while others cannot.

math.CV↗
arXiv2015

Geometric studies on the class ${\mathcal U}(λ)$

The article deals with the family ${\mathcal U}(λ)$ of all functions $f$ normalized and analytic in the unit disk such that $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. The family ${\mathcal U}(λ)$ has been studied extensively in the recent past and functions in this family are known to be univalent in $\ID$. However, the problem of determining sharp bounds for the second coefficients of functions in this family was solved recently in \cite{VY2013} by Vasudevarao and Yanagihara but the proof was complicated. In this article, we first present a simpler proof. We obtain a number of new subordination results for this family and their consequences. In addition, we show that the family ${\mathcal U}(λ)$ is preserved under a number of elementary transformations such as rotation, conjugation, dilation and omitted value transformations, but surprisingly this family is not preserved under the $n$-th root transformation for any $n\geq 2$. This is a basic here which helps to generate a number of new theorems and in particular provides a way for constructions of functions from the family ${\mathcal U}(λ)$. Finally, we deal with a radius problem.

math.CV↗
arXiv2015

Where is $f(z)/f'(z)$ univalent?

Let ${\mathcal S}$ denote the family of all univalent functions $f$ in the unit disk $\ID$ with the normalization $f(0)=0= f'(0)-1$. There is an intimate relationship between the operator $P_f(z)=f(z)/f'(z)$ and the Danikas-Ruscheweyh operator $T_f:=\int_{0}^{z}(tf'(t)/f(t))\,dt$. In this paper we mainly consider the univalence problem of $F=P_f$, where $f$ belongs to some subclasses of ${\mathcal S}$. Among several sharp results and non-sharp results, we also show that if $f\in {\mathcal S}$, then $F \in {\mathcal U}$ in the disk $|z|<r$ with $r\leq r_6\approx 0.360794$ and conjecture that the upper bound for such $r$ is $\sqrt{2}-1$.

math.CV↗