A certain subclass of analytic and close-to-convex functions

Coefficient, distortion and growth inequalities for certain close-to-convex functions

“…Remark 2 We note that Xu et al [7] also obtained a similar estimates and our results differ from their in the hypothesis. Also we have shown that the results are sharp.…”

“…for an analytic function w with w(0) = 0 and |w(z)| < 1 which is sharp for the functions w(z) = z 2 or w(z) = z, the desired result follows upon using the estimate that Though Xu et al [7] have given an estimate of |a n | for all n, their result is not sharp in general. For n = 2, 3, our results provide sharp bounds.…”

“…The idea here is to replace the average of f(z) and -f (-z) by the corresponding product -g(z) g(-z), and the factor z is included to normalize the expression, so that -z 2 f'(z)/(g(z) g(-z)) takes the value 1 at z = 0. To make the functions univalent, it is further assumed that g is starlike of order 1/2 so that the function -g(z) g(-z)/z is starlike, which in turn implies the close-to-convexity of f. For some recent works on the problem, see [4][5][6][7]. Instead of requiring the quantity -z 2 f'(z)/(g(z)…”

Coefficient Estimates for Starlike and Convex Classes of -fold Symmetric Bi-univalent Functions

“…In recent times, the study of bi-univalent functions gained momentum mainly due to the work of Srivastava et al [11]. Motivated by this, many researchers (see [4,11,12,13,14,15,17]) recently investigated several interesting subclasses of the class and found non-sharp estimates on the first two Taylor-Maclaurin coefficients. For each function in , the function ℎ( ) = � ( ) is univalent and maps the unit disk into a region withfold symmetry.…”

“…Let ∈ , ( , ) . Then there are analytic functions , : → , with (0) = (0) = 0 satisfying (12), (13), and (36), we get From (37) and (39), we get…”