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Register a new userOn a generalization of close-to-convex functions
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A motivation comes from {em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to-convex functions by means of a $q$-analog of a difference operator acting on analytic functions in the unit disk $mathbb{D}={zin mathbb{C}:,|z|<1}$. We use the terminology {em $q$-close-to-convex functions} for the $q$-analog of close-to-convex functions. The $q$-theory has wide applications in special functions and quantum physics which makes the study interesting and pertinent in this field. In this paper, we obtain some interesting results concerning conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in the $q$-close-to-convex family. As a result we find certain dilogarithm functions that are contained in this family. Secondly, we also study the famous Bieberbach conjecture problem on coefficients of analytic $q$-close-to-convex functions. This produces several power series of analytic functions convergent to basic hypergeometric functions.
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Let $es$ be the family of analytic and univalent functions $f$ in the unit disk $D$ with the normalization $f(0)=f(0)-1=0$, and let $gamma_n(f)=gamma_n$ denote the logarithmic coefficients of $fin {es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families $F(c)$ and $G(delta)$ of functions $fin {es}$ defined by $$ {rm Re} left ( 1+frac{zf(z)}{f(z)}right )>1-frac{c}{2}, mbox{ and } , {rm Re} left ( 1+frac{zf(z)}{f(z)}right )<1+frac{delta}{2},quad zin D $$ for some $cin(0,3]$ and $deltain (0,1]$, respectively. We obtain the sharp upper bound for $|gamma_n|$ when $n=1,2,3$ and $f$ belongs to the classes $F(c)$ and $G(delta)$, respectively. The paper concludes with the following two conjectures: begin{itemize} item If $finF (-1/2)$, then $ displaystyle |gamma_n|le frac{1}{n}left(1-frac{1}{2^{n+1}}right)$ for $nge 1$, and $$ sum_{n=1}^{infty}|gamma_{n}|^{2} leq frac{pi^2}{6}+frac{1}{4} ~{rm Li,}_{2}left(frac{1}{4}right) -{rm Li,}_{2}left(frac{1}{2}right), $$ where ${rm Li}_2(x)$ denotes the dilogarithm function. item If $fin G(delta)$, then $ displaystyle |gamma_n|,leq ,frac{delta}{2n(n+1)}$ for $nge 1$. end{itemize}
This paper mainly uses the nonnegative continuous function ${zeta_n(r)}_{n=0}^{infty}$ to redefine the Bohr radius for the class of analytic functions satisfying $real f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of the alternating series $A_f(r)$ with analytic functions $f$ of the form $f(z)=sum_{n=0}^{infty}a_{pn+m}z^{pn+m}$ in $|z|<1$. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series $M_f(r)$ and the odd and even bits of $f(z)$ are also established. We will prove that most of results are sharp.
We study the smoothness of the Siciak-Zaharjuta extremal function associated to a convex body in $mathbb{R}^2$. We also prove a formula relating the complex equilibrium measure of a convex body in $mathbb{R}^n$ to that of its Robin indicatrix. The main tool we use are extremal ellipses.
The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forellis theorem on the complex analyticity of the functions that are: (1) $mathcal{C}^infty$ smooth at a point, and (2) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.
In this paper, we determine the radii of $beta -$uniformly convex of order $alpha $ for three kinds of normalized Lommel and Struve functions of the first kind. In the cases considered the normalized Lommel and Struve functions are $beta -$uniformly convex functions of order $alpha $ on the determined disks. The basic tool of this study is Lommel and Struve functions in series.
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