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Register a new userOn maximal area integral problem for analytic functions in the starlike family
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For an analytic function $f$ defined on the unit disk $|z|<1$, let $Delta(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<rle 1$. In 1990, Yamashita conjectured that $Delta(r,z/f)le pi r^2$ for convex functions $f$ and it was finally settled in 2013 by Obradovi{c} and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation $zf(z)/f(z)prec (1+(1-2beta)alpha z)/(1-alpha z)$ for $0le beta<1$ and $0<alphale 1$. We prove Yamashitas conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.
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Harmonic functions are natural generalizations of conformal mappings. In recent years, a lot of work have been done by some researchers who focus on harmonic starlike functions. In this paper, we aim to introduce two classes of harmonic univalent functions of the unit disk, called hereditarily $lambda$-spirallike functions and hereditarily strongly starlike functions, which are the generalizations of $lambda$-spirallike functions and strongly starlike functions, respectively. We note that a relation can be obtained between this two classes. We also investigate analytic characterization of hereditarily spirallike functions and uniform boundedness of hereditarily strongly starlike functions. Some coefficient conditions are given for hereditary strong starlikeness and hereditary spirallikeness. As a simple application, we consider a special form of harmonic functions.
In this note, we study the boundedness of integral operators $I_{g}$ and $T_{g}$ on analytic Morrey spaces. Furthermore, the norm and essential norm of those operators are given.
In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass $mathcal{P}_{tau,mu}(k,delta,gamma)$ of analytic and univalent functions in the open unit disk $mathbb{U}$. In particular, for functions in the class $mathcal{P}_{tau,mu}(k,delta,gamma)$, we derive sufficient coefficient inequalities, distortion theorems involving the above-mentioned fractional derivative operators, and the radii of starlikeness and convexity. In addition, some applications of functions in the class $mathcal{P}_{tau,mu}(k,delta,gamma)$ are also pointed out.
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}
We introduce the class of analytic functions $$mathcal{F}(psi):= left{fin mathcal{A}: left(frac{zf(z)}{f(z)}-1right) prec psi(z),; psi(0)=0 right},$$ where $psi$ is univalent and establish the growth theorem with some geometric conditions on $psi$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $alpha$ and $beta$ for the functions in the classes $mathcal{BS}(alpha):= {fin mathcal{A} : ({zf(z)}/{f(z)})-1 prec {z}/{(1-alpha z^2)},; alphain [0,1) }$ and $mathcal{S}_{cs}(beta):= {fin mathcal{A} : ({zf(z)}/{f(z)})-1 prec {z}/({(1-z)(1+beta z)}),; betain [0,1) }$, respectively which improves the earlier known bounds. The sharp Bohr-radii for the classes $S(mathcal{BS}(alpha))$ and $mathcal{BS}(alpha)$ are also obtained. A few examples as well as certain newly defined classes on the basis of geometry are also discussed.
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