Some Properties Of The Integral Means Of Analytic Functions

In this paper,we mainly investigate the ordinary convexity and logarithmic convexity of Gaussian integral means Mp,?(f,r),area integral means Mp,?(f,r)and the weighted integral means for analytic function.The conditions for the integral means to have the convexity or the logarithmic convexity are obtained.The main contents are as follows.In chapter one,the research background and present situation of the analytic function integral means are given,and provide some basic concepts and related marks to be used in this paper.In chapter two,the Gaussian integral means Mp,?(f,r)of analytic functions is dis-cussed for the convexity of r,and it is proved that when ??<0,r ?(0,?),the integral means is a convex function of r.In chapter three,in the case of a<0,the weighted integral means A?,?(f,r)of the mixed area of f(rD)and the weighted integral means L?,?(f,r)of the mixed length of(?)f(rD)with respect to the convexity of r are explored.Furthermore,it is proved that?? 0 is the optimal range of the convex function by example.In chapter four,we investigate the logarithmic convexity of the area integral means Mp,?(f,r)of some different analytic functions in the case of p = 2,? = 1.