The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛^*

Abstract

In this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

Sharp bounds of third Hankel determinant for a class of starlike functions and a subclass of $q$-starlike functions

“…This bound apart from being sharp is an improvement over the bound obtained in [10]. Moreover, for q → 1 − , this bound reduces to earlier known sharp bound for SL * [4]. We also give extremal functions to justify our claims.…”

“…Let q → 1 − in the above Theorem, then it reduces to the following result obtained by Banga and Kumar [4].…”

“…It is achieved by Kowalczyk et al [12], for functions in A satisfying Re(f (z)/z) > α, α ∈ [0, 1) and in [11] for convex functions. Following which, Banga and Kumar [4] recently derived sharp bound of third Hankel determinant as |H 3 (1)| ≤ 1/36 for functions in SL * , which earlier was calculated to be 43/576 in [23]. Lecko et al [16] found that 1/9 is the sharp bound of the third Hankel determinant for starlike functions of order 1/2.…”

Third Hankel Determinant For A Class Of Starlike Functions Associated With Exponential Function

“…Some interesting domains on which the Hankel determinant for starlike class was estimated by including k-Fibonacci numbers, Shafiq et al [21] estimated the bounds of the third Hankel determinant for the same. To read more on this work, one can refer [22,6,7,3,11].…”

Hankel Determinants and Coefficient Estimates for Starlike Functions Related to Symmetric Booth Lemniscate

“…These Hankel determinants for different subclasses of analytic and univalent functions have been investigated by many authors. Recently, sharp bounds for |H 3 (1)( f )| were obtained using a result from [31]; see [32][33][34][35][36][37] for some detailed work on Hankel determinants. A new form for the fourth Hankel determinant is given in [38], which is studied for a new subclass of analytic functions introduced, and the upper bound of the fourth Hankel determinant for this class is obtained.…”