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.

There are many articles in the literature dealing with the first-order and the second-order differential subordination and superordination problems for analytic functions in the unit disk, but only a few articles are dealing with the above problems in the third-order case (see, e.g., Antonino and Miller (2011) and Ponnusamy et al. (1992)). The concept of the third-order differential subordination in the unit disk was introduced by Antonino and Miller in (2011). Let Ω be a set in the complex planeC. Also letpbe analytic in the unit diskU=z:z∈C  and  z<1and suppose thatψ:C4×U→C. In this paper, we investigate the problem of determining properties of functionsp(z)that satisfy the following third-order differential superordination:Ω⊂ψpz,zp′z,z2p′′z,z3p′′′z;z:z∈U. As applications, we derive some third-order differential subordination and superordination results for meromorphically multivalent functions, which are defined by a family of convolution operators involving the Liu-Srivastava operator. The results are obtained by considering suitable classes of admissible functions.

There are many articles in the literature dealing with the first-order and the second-order differential subordination and differential superordination problems for analytic functions in the unit disk, but there are only a few articles dealing with the third-order differential subordination problems. The concept of third-order differential subordination in the unit disk was introduced by Antonino and Miller, and studied recently by Tang and Deniz. Let be a set in the complex plane C, let p(z) be analytic in the unit disk U = {z : z ∈ C and |z| < 1}, and let ψ : C 4 ×U → C. In this paper, we investigate the problem of determining properties of functions p(z) that satisfy the following third-order differential superordination:As applications, we derive some third-order differential superordination results for analytic functions in U, which are associated with a family of generalized Bessel functions. The results are obtained by considering suitable classes of admissible functions. New third-order differential sandwich-type results are also obtained.

In this paper, we introduce and investigate two new subclasses H

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This paper introduces a new integral operator in q-analog for multivalent functions. Using as an application of this operator, we study a novel class of multivalent functions and define them. Furthermore, we present many new properties of these functions. These include distortion bounds, sufficiency criteria, extreme points, radius of both starlikness and convexity, weighted mean and partial sum for this newly defined subclass of multivalent functions are discussed. Various integral operators are obtained by putting particular values to the parameters used in the newly defined operator.

In this paper we define and consider some familiar subsets of analytic functions associated with sine functions in the region of unit disk on the complex plane. For these classes our aim is to find the Hankel determinant of order three.

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