Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions

Abstract

Let Σ denote the class of functions f (z) = z + ∞ ∑ n=2 anz n belonging to the normalized analytic function class A in the open unit disk U, which are bi-univalent in U, that is, both the function f and its inverse f −1 are univalent in U. The usual method for computation of the coefficients of the inverse function= z is too difficult to apply in the case of m-fold symmetric analytic functions in U. Here, in our present investigation, we aim at overcoming this difficulty by using a general formula to compute the coefficients of f −1 (z) in conjunction with the residue calculus. As an application, we introduce two new subclasses of the bi-univalent function class Σ in which both f (z) and f −1 (z) are m-fold symmetric analytic functions with their derivatives in the class P of analytic functions with positive real part in U. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for |am+1| and |a2m+1|.

Advances on the coefficient bounds for m-fold symmetric bi-close-to-convex functions

“…Here we will show that these bounds can be improved even further for the m-fold symmetric bi-close-to-convex functions. Moreover, the coefficient bounds presented in this paper for |a m+1 |, |a 2m+1 | and m > 1 also improve those corresponding coefficient bounds given by Srivastava et al [12]. A function is said to be bi-close-to-convex in a simply connected domain if both the function and its inverse map are close-to-convex there.…”

“…This is a subclass of m-fold symmetric bi-close-to-convex functions. For m ≥ 2 the bounds presented by our Theorem 1.1 for |a m+1 | and |a 2m+1 | are far better than those given in [12,Theorem 3].…”

“…Because the bi-univalency requirement makes the behavior of the coefficients of bi-univalent functions unpredictable, no general coefficient bounds for subclasses of bi-univalent functions was known up until the publication of article [6] by Jahangiri and Hamidi. The unpredictability of m-fold symmetric bi-starlike functions was first studied by the authors in [4] followed by the publication of the articles [12] and [13] by Srivastava et al Here we further improve the bounds given in [4] to include the larger class of m-fold symmetric bi-close-to-convex functions. We begin with the statement of the following…”

“…Srivastava et al [12,Theorem 3] considered the class of m-fold symmetric bi-univalent functions m f (z m ) for which Re{f (z)} > β; 0 ≤ β < 1. This is a subclass of m-fold symmetric bi-close-to-convex functions.…”

Certain classes of bi-univalent functions with bounded boundary variation

“…Various subclasses of the bi-univalent function class Σ were introduced and non-sharp estimates on the first two coefficients |a 2 | and |a 3 | in the Taylor-Maclaurin series expansion (1.1) were found in several recent investigations (see, for example, [1,2,4,5,6,7,8,10,11,12,13,14,15,16,17,19,21,22,23,24,25,26,27,29,30,31,32,33,34,35] and references therein). The aforecited all these papers on the subject were actually motivated by the pioneering work of Srivastava et al [28].…”