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

Abstract

In 1955, Waadeland considered the class of m-fold symmetric starlike functions of the formmn+1 ; m ≥ 1; |z| < 1 and obtained the sharp coefficient bounds. Pommerenke in 1962, proved the same coefficient bounds for m-fold symmetric close-to-convex functions. Nine years later, Keogh and Miller confirmed the same bounds for the class of m-fold symmetric Bazilevic functions. Here we will show that these bounds can be improved even further for the m-fold symmetric bi-close-toconvex functions. Moreover, our results improve those corresponding coefficient bounds given by Srivastava et al that appeared in 7(2) (2014) issue of this journal. 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.

Initial coefficient estimates for subclasses of bi-univalent functions

“…where g is the extension of f −1 in U. Recently, several researchers such as (see [2,11,13,15,16,19]) obtained coefficients The main aim of the present investigation is to introduce and study two new subclasses of the function class Σ and find estimates on the initial coefficients |a 2 | and |a 3 | for functions in these new subclasses of the function class Σ using q-differential operator.…”