A function analytic in the open unit disk is said to be bi-univalent in if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions.

A function is said to be bi-univalent in the open unit disk D if both the function and its inverse map are univalent in D. By the same token, a function is said to be bi-subordinate in D if both the function and its inverse map are subordinate to certain given function in D. The behavior of the coefficients of such functions are unpredictable and unknown. In this paper, we use the Faber polynomial expansions to find upper bounds for the n-th (n≥3) coefficients of classes of bi-subordinate functions subject to a gap series condition as well as determining bounds for the first two coefficients of such functions.

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.

Complex-valued harmonic functions that are univalent and sense preserving in the open unit disk can be written in the form f = h + g ‾ $f=h+\overline{g}$ , where h and g are analytic. In this paper we investigate some classes of univalent harmonic functions with varying coefficients related to Janowski functions. By using the extreme points theory we obtain necessary and sufficient convolution conditions, coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. The radii of starlikeness and convexity for these classes are also determined.

We consider meromorphic starlike univalent functions that are also bi-starlike and find Faber polynomial coefficient estimates for these types of functions. A function is said to be bi-starlike if both the function and its inverse are starlike univalent.

Data envelopment analysis (DEA) is used to evaluate the performance of decision making units (DMUs) with multiple inputs and outputs in a homogeneous group. In this way, the acquired relative efficiency score for each decision making unit lies between zero and one where a number of them may have an equal efficiency score of one. DEA successfully divides them into two categories of efficient DMUs and inefficient DMUs. A ranking for inefficient DMUs is given but DEA does not provide further information about the efficient DMUs. One of the popular methods for evaluating and ranking DMUs is the common set of weights (CSW) method. We generate a CSW model with considering nondiscretionary inputs that are beyond the control of DMUs and using ideal point method. The main idea of this approach is to minimize the distance between the evaluated decision making unit and the ideal decision making unit (ideal point). Using an empirical example we put our proposed model to test by applying it to the data of some 20 bank branches and rank their efficient units.