K-th Order Symmetric Set And Integral Mean Value

Determining the extreme values and extremal functions of the analytic functions on D={Z$C: Z <1} is very important in theprinciples of univalent functions. Baernstein[2] gave the conclusion by using Koebe function as the extremal function , Glenn Schober[6] studied the classes such as S, P, K, S* of H(D) and represented these functions with integral formulations. Wang Jian[3] and others investigated the integral mean values. Our work is continuation of their work.In this paper, we considered the properties of TG* h functions. The definition of symmetric set in [31 is extended to k-th order symmetric set. We have discussed the properties of symmetric set and gained some interesting results .Some estimations for integral mean of P S*, K, CoK, St(a), B type functions in the k-th order symmetric set also be given. By the properties of k-th order symmetric function , we obtained some estimations of integral mean value of the solutions of the boundary value problems for some kind of special partial differential equations in Dk which can be changed into Laplace equations in the unit circle by way of suitable transform . Some specific results are give for the case of k=l and k=2 .