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Register a new userSecond Hankel determinant for bi-starlike and bi-convex functions of order b{eta}
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In the present investigation the authors obtain upper bounds for the second Hankel determinant of the classes bi-starlike and bi-convex functions of order beta.
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In the present work, we propose to investigate the second Hankel determinant inequalities for certain class of analytic and bi-univalent functions. Some interesting applications of the results presented here are also discussed.
In this sequel to the recent work (see Azizi et al., 2015), we investigate a subclass of analytic and bi-univalent functions in the open unit disk. We obtain bounds for initial coefficients, the Fekete-Szego inequality and the second Hankel determinant inequality for functions belonging to this subclass. We also discuss some new and known special cases, which can be deduced from our results.
Making use of Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant of a subclass $mathcal{N}_{sigma }^{mu}left( lambda ,tright) $ of bi-univalent function class $sigma.$
In this paper, two new subclasses of bi-univalent functions related to conic domains are defined by making use of symmetric $q$-differential operator. The initial bounds for Fekete-Szego inequality for the functions $f$ in these classes are estimated.
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}
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