The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

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An exact estimate of the third Hankel determinants for functions inverse to convex functions

Rath,

Kumar,

Krishna

2023

Mat. Stud.

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Invesigation of bounds for Hankel determinat of analytic univalent functions is prominent intrest of many researcher from early twenth century to study geometric properties. Many authors obtained non sharp upper bound of third Hankel determinat for different subclasses of analytic univalent functions until Kwon et al. obtained exact estimation of the fourth coefficeient of Caratheodory class. Recently authors made use of an exact estimation of the fourth coefficient, well known second and third coefficient of Caratheodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions. Let $w=f(z)=z+a_{2}z^{2}+\cdots$ be analytic in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$, and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0)=0$, and $f'(0)=1$. Let $z=f^{-1}$ be the inverse function of $f$, given by $f^{-1}(w)=w+t_2w^2+\cdots$ for some $|w|<r_o(f)$. Let $\mathcal{S}^c\subset\mathcal{S}$ be the subset of convex functions in $\mathbb{D}$. In this paper, we estimate the best possible upper bound for the third Hankel determinant for the inverse function $z=f^{-1}$ when $f\in \mathcal{S}^c$.Let $\mathcal{S}^c$ be the class of convex functions. We prove the following statements (Theorem):If $f\in$ $\mathcal{S}^c$, then\begin{equation*}\big|H_{3,1}(f^{-1})\big| \leq \frac{1}{36}\end{equation*} and the inequality is attained for $p_0(z)=(1+z^3)/(1-z^3).$

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The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

Cited by 6 publications

References 12 publications

Fourth Hankel Determinant for the Set of Star-Like Functions

Fourth Hankel Determinant for the Set of Star-Like Functions

Third Hankel determinants for two classes of analytic functions with real coefficients

An exact estimate of the third Hankel determinants for functions inverse to convex functions

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