Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli

Abstract

In this paper, our aim is to define a new subclass of close-to-convex functions in the open unit disk U that are related with the right half of the lemniscate of Bernoulli. For this function class, we obtain the upper bound of the third Hankel determinant. Various other related results are also considered.

Third Hankel Determinant Problem for Certain Subclasses of Analytic Functions Associated with Nephroid Domain

“…Then he along with coauthors in [28] investigated the class of close to convex functions associated with lemniscate of Bernouli and evaluated its Hankel determinant. Continuing the same trend he in [29] incorporated the research on Toeplitz forms and Hankel determinant for some q-starlike functions associated with a generalized domain. Many other domains were also investigated for its Hankel determinant like a class of starlike functions associated with k-Fibonacci numbers.…”

Third Hankel determinant and Zalcman functional for a class of starlike functions with respect to symmetric points related with sine function

“…Note that H 2,1 (f) = a 3 − a 2 2 , is classical Fekete-Szegö functional. In year 1933, the maximum value of |H 2,1 (f)| was obtained for a class S. For various subclasses of class A, the maximum value of |H 2,1 (f)| was investigated by different authors, for details see [7,[16][17][18][19][29][30][31][32][33][34]. Furthermore, second Hankel determinant when j = 2 and k = 2 is…”

“…is known as third order Hankel determinant. Babalola [2] was the first person to study the upper bound of H 3,1 (f) for subclass of S. For more details on this topic readers are advised to see the work of several researchers like Zaprawa [35], Raza et al [27], Khan et al [11], Cho et al [5], Lecko et al [13], Srivastava et al [32], and Khan et al [12]. Very recently, Arif et al [1], investigated upper bounds for third hankel determinant for the class of functions S * (Ψ) associated with trigonometric sine function.…”

Sharp estimation of Hermitian-Toeplitz determinants for Janowski type starlike and convex functions

“…The Hankel determinants have been among the most studied topics in Geometric Function Theory (GFT) in recent years; such studies can be back to the 1960s (see [10,18]). In many of the recently-published works dealing extensively with the Hankel and Toeplitz determinants, use is made also of the basic (or q-) calculus (see, for example, [9,17,20,22,23]; see also a survey-cum-expository review article by Srivastava [21]).…”