On a polynomial inequality

Let p(z)=a0+⋯+anznp(z)=a0+⋯+anzn and q(z)=b0+⋯q(z)=b0+⋯ be polynomials of degree respectively n and less than n such that|p(z)|<|q(z)|,|z|<1. A result due to Q.I. Rahman states that|a0|+|an|⩽|b0|.|a0|+|an|⩽|b0|. In this paper, we slightly improve the above inequality and discuss several sharpness aspects, including all cases of equality.