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|.

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The purpose of this article is to derive some subordination and superordination results for certain families of normalized analytic functions in the open unit disk, which are acted upon by a class of multiplier transformations. Relevant connections of the results, which are presented in this article, with various known results are also considered.

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.

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Let R denote the family of functions f(z)=z+∑n=2∞anzn of bounded boundary rotation so that Ref′(z)>0 in the open unit disk U={z:z<1}. We obtain sharp bounds for Toeplitz determinants whose elements are the coefficients of functions f∈R.

The purpose of this present paper is to derive some subordination and superordination results for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in the paper, with various known results are also considered.

Abstract. The purpose of this present paper is to derive some subordination and superordination results for certain normalized analytic functions involving complex order in the open unit disk, acted upon by Carlson-Shaffer operator. Relations of the results, which are obtained in this paper, with various known results are also presented. (2000): 30C80, 30C45. Mathematics subject classification