Inequalities for polynomials satisfying \(p(z)\equiv z^np(1/z)\)

Finding the sharp estimate of \(\max_{|z|=1} |p'(z)|\) in terms of \(\max_{|z|=1} |p(z)|\) for the class of polynomials p(z) satisfying \(p(z) \equiv z^n p(1/z)\) has been a well-known open problem for a long time and many papers in this direction have appeared. The earliest result is due to Govil, Jain and Labelle [9] who proved that for polynomials p(z) satisfying \(p(z) \equiv z^n p(1/z)\) and having all the zeros either in left half or right half-plane, the inequality \(\max_{|z|=1} |p'(z)| \le \frac{n}{\sqrt{2}} \max_{|z|=1} |p(z)|\) holds. A question was posed whether this inequality is sharp. In this paper, we answer this question in the negative by obtaining a bound sharper than \(\frac{n}{\sqrt{2}}\). We also conjecture that for such polynomials

$$\max_{|z|=1} |p'(z)| \le \Big(\frac{n}{\sqrt{2}} - \frac{\sqrt{2}-1}{4}(n-2)\Big) \max_{|z|=1} |p(z)|$$

and provide evidence in support of this conjecture.