Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

We consider the inverse function z ¼ gðwÞ of a (normalized) starlike function w ¼ f ðzÞ of order on the unit disk of the complex plane with 0 < < 1. Krzyż, Libera and Złotkiewicz obtained sharp estimates of the second and the third coefficients of gðwÞ in their paper (1979). Prokhorov and Szynal gave sharp estimates of the fourth coefficient of gðwÞ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of gðwÞ together with explicit forms of the extremal functions.

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“…We remark that the above result was extended to the so-called Janowski starlike functions by Ali and Vasudevarao [1].…”

Section: Introductionmentioning

confidence: 57%

On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

Sugawa

Wang

2021

IIS

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“…The mapping f (z) → g(z) := 1/f (1/z) establishes a one-to-one correspondence between functions in the classes S and Σ and also between functions in the classes ST (ϕ) and (ST ) (ϕ) because (see for more details [5])…”

Section: Ma and Mindamentioning

confidence: 99%

Coeficient estimates for a subclass of meromorphic bi-univalent functions defined by subordination

Motamednezhad¹,

Bulut²

2020

Stud. Univ. Babes-Bolyai Math.

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“…Ali [1] estimated sharp bounds for the first four coefficients and the Fekete-Szegő coefficient functional of the inverse functions which belong to the class of strongly starlike functions denoted by SS * (α) defined as |arg (zf…”

Section: Introductionmentioning

confidence: 99%

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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Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

Cited by 5 publications

References 30 publications

On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

Coeficient estimates for a subclass of meromorphic bi-univalent functions defined by subordination

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

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