V.K. Jain: Visser's inequality and its sharp refinement, p.171-175

Abstract:

For a polynomial $p(z) = \sum_{k=0}^{n}c_k z^k$ of degree $n$, with $\max_{\vert z\vert = 1}\vert p(z)\vert = M$, Visser had obtained

$$\vert c_0\vert + \vert c_n\vert \leq M. \nonumber$$

Using certain integral inequality for a polynomial, we have suggested a different proof of Visser's inequality (in a new but equivalent form)

$$\vert\alpha\vert\vert c_0\vert + \vert\beta\vert\vert c_n\vert \leq M\{\max(\vert\alpha\vert,\vert\beta\vert)\}.\nonumber$$

Further for polynomial $p(z)$ having all its zeros, either in $\vert z\vert \leq 1$ or in $\vert z\vert \geq 1$, a sharp refinement

$$\vert\alpha\vert\vert c_0\vert + \vert\beta\vert\vert c_n\vert \leq M\{(\vert\alpha\vert + \vert\beta\vert)/2\},\nonumber$$

of new form of Visser's inequality, has also been obtained.

Key Words: Visser's inequality, polynomial, zeros, refinement.

2000 Mathematics Subject Classification: Primary: 30C10,
Secondary 30A10.

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