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