On a polynomial inequality

Let p ( z ) = a 0 + ⋯ + a n z n and q ( z ) = b 0 + ⋯ 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 | a 0 | + | a n | ⩽ | b 0 | . In this paper, we slightly improve the above inequality and discuss several sharpness aspects, including all cases of equality.