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.

In this paper, we introduce and investigate two new subclasses H σ μ ( λ , φ ) and M σ γ ( λ , μ , φ ) of Ma-Minda bi-univalent functions defined by using subordination in the open unit disk D = { z ∈ C : | z | < 1 } . For functions belonging to these new subclasses, we obtain estimates for the initial coefficients | a 2 | and | a 3 | . The results presented in this paper would generalize those in related works of several earlier authors.

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.

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 article, we study differential subordnations in q-analogue. Some properties of analytic functions in q-analogue associated with cardioid domain and limacon domain are considered. In particular, we determine conditions on $\alpha$ such that $1+\alpha \frac{z{\partial }_{q}h\left(z\right)}{{\left(h\left(z\right)\right)}^n}(n=0,1,2,3)$ are subordinated by Janowski functions and $h\left(z\right)\prec 1+\frac{4}{3}z+\frac{2}{3}{z}^2.$ We also consider the same implications such that $h\left(z\right)\prec 1+\sqrt{2}z+\frac{1}{2}{z}^2$ . We apply these results on analytic functions to find sufficient conditions for q-starlikeness related with cardioid and limacon.