Abstract. In this paper, we introduce several new subclasses of the class of m -fold symmetric bi-univalent functions and obtain estimates of the Taylor-Maclaurin coefficients |a m+1 | , |a 2m+1 | and Fekete-Szegö functional problems for functions in these new subclasses. The results presented in this paper improve the earlier results of Ali et al. [1], Frasin and Aouf [6], and Srivastava et al. [14] in terms of the bounds as well as the ranges of the parameter under consideration. Our results also further generalize the results of Peng et al. [19].Mathematics subject classification (2010): Primary 30C45, 33C50; Secondary 30C80.

No abstract

The area of [Formula: see text]-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of [Formula: see text]-calculus was initiated by Jackson [Jackson, On [Formula: see text]-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203; On [Formula: see text]-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.], who was the first to develop [Formula: see text]-integral and [Formula: see text]-derivative in a systematic way. In this paper, we make use of the concept of [Formula: see text]-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of [Formula: see text]-starlike and [Formula: see text]-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this [Formula: see text]-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of [Formula: see text] found in [Mishra and Gochhayat, Fekete–Szego problem for [Formula: see text]-uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563–572] for [Formula: see text], [Formula: see text].

No abstract

No abstract

In an article of Pommerenke [10] he remarked that, for an -fold symmetric functions in the class , the well known lemma stated by Caratheodary for a one fold symmetric functions in still holds good. Exploiting this concept, we introduce certain new subclasses of the bi-univalent function class in which both and −1 are -fold symmetric analytic with their derivatives in the class of analytic functions. Furthermore, for functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for | +1 | and | 2 +1 |. We remark here that the concept of -fold symmetric bi-univalent is not in the literature and the authors hope it will make the researchers interested in these type of investigations in the forseeable future. By the working procedure and the difficulty involved in these procedures, one can clearly conclude that there lies an unpredictability in finding the coefficients of a -fold symmetric bi-univalent functions.