A note on sharpening of a theorem of Ankeny and Rivlin

Let p(z) = ?n?=0 a?z? be a polynomial of degree n, M(p,R) := max|z|=R?0 |p(z)|, and M(p,1) ||p||. Then according to well-known result Ankeny Rivlin, we have for R ? 1, (Rn+1/2) This inequality has been sharpened among others by Govil, who proved that ||p||-n/2 (||p||2-4|an|2/||p||) {(R-1)||p||/||p||+2|an|- ln (1+ (R-1)||p||/||p||+2|an|)}. In this paper, sharpen the above which in turn sharpens Rivlin. We present our terms LerchPhi function ?(z,s,a), implemented Wolfram's MATHEMATICA as [z,s,a], can evaluated arbitrary numerical precision, is suitable both symbolic manipulations. Also, an example using MATLAB show some polynomials improvement bound considerably significant.