univalent functions

One of the most important and interesting topics in the geometric function theory is the classes of starlike and convex functions of Ma-Minda type on the unit disk D = {z ∈ C: |z| < 1} which are defined by subordination. Let A be the class of functions f, analytic in the unit disc D = {z : |z| < 1} on the complex plane C, with the normalization f(0) = f 0 (0) − 1 = 0, and ST N (s) and CVN (s) be a family of starlike and convex functions f ∈ A so that for each z ∈ D, such that the quantity zf0 (z)/f(z) or 1 + zf00(z)/f0 (z) respectively are lying in the region bounded by the Nephroid (u − 1)2 + v 2 − 4s 2 3 = 108s 4 v 2 , 0 < s ≤ √ 2 4 . In this paper, some properties and characteristics of classes ST N (s) and CVN (s) defined by the Ma-Minda type, like the structure of functions in these classes, we study extreme functions, growth theorem, distoration, and rotation theorem.

Let  be the open unit disc in the complex plane  and  be the class of all functions of  which are analytic and normalized in  The subclass of consisting of all univalent functions  in  is denoted by  We say that a function  is said to be starlike function if and only if for all We denote by  the class of all satrlike functions in If  and  are two of the functions in  then we say that  is subordinate to  written  or  if there exists a Schwartz function  such that  for all  Furthermore, if the function  is univalent in  then we have the following equivalence: Also for and   their Hadamard product (or convolution) is defined by The logarithmic coefficients   of , denoted by , are defined by These coefficients play an important role for various estimates in the theory of univalent functions. For example, consider the Koebe function where  It is easy to see that the above function  has logarithmic coefficients where  and  Also for the function  we have and the sharp estimates and hold. We remark that the Fekete-Szego theorem is used. For  , the problem seems much harder and no significant upper bounds for  when  appear to be known. Moreover, the problem of finding the sharp upper bound for for  is still open for . The sharp upper bounds for modulus of logarithmic coefficients are known for functions in very few subclasses of . For functions in the class  it is easy to prove that  for  and the equality holds for the Koebe function. The celebrated de Brangeschr('39') inequalities (the former Milin conjecture) for univalent functions  state that where   with the equality if and only if De Branges used this inequality to prove the celebrated Bieberbach conjecture. Moreover, the de Brangeschr('39') inequalities have also been the source of many other interesting inequalities involving logarithmic coefficients of  such as    Let  denote the class of functions  and satisfying the following subordination relation where .

In this paper, first we obtain a subordination relation for the class  and by making use of this relation we give two sharp estimates for the logarithmic coefficients of the function 

We obtain two sharp estimates for the logarithmic coefficients of the function 

The following conclusions were drawn from this research. Logarithmic coefficients  of the function  are estimated.