Induction principle on proving an inequality

Question

If $P(z)$ having no zeros in $|z|<1,$ then $$\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}\leq \frac{n}{2}.$$ Can we prove this by induction on $n$? or is there any alternative way?

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Attempt at a proof: let us try to show that the inequality holds by induction on the degree $n$ of the polynomial $P(z)$.

If $n=1$ then $P(z)=z-w$ with $|w|\geq 1$, and we have
$$\displaystyle\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}=\frac{1}{1+|w|}\leq \frac{1}{1+1},$$ which is nothing but the given inequality when $n=1$.

Let $Q(z)=(z-w)P(z)$ with $|w|\geq 1$, where $P(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\geq 1$. $$\displaystyle\frac{\max_{|z|=1}|Q'(z)|}{\max_{|z|=1}|Q(z)|}=\frac{\max_{|z|=1}|(z-w)P'(z)+P(z)|}{\max_{|z|=1}|(z-w)P(z)|}$$ \begin{equation}\label{p1}\displaystyle\leq\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}+\frac{1}{\max_{|z|=1}|z-w|} (?).\end{equation}

Thus the proof by induction will be complete once we establish (?) part of the above.

Answer 1

The statement is known in the literature as the Theorem of Erdős and Lax.

It was conjectured by Erdős and first proved by Lax. Later additional proofs were given by de Bruijn, Aziz--Mohammad, Rahman, and Boas. The proofs of these authors do not use induction.

1. A. Aziz and Q. G. Mohammad: Simple Proof of a Theorem of Erdős and Lax, Proc. Amer. Math. Soc. 80, no. 1 (1980), 119--122
2. R. P. Boas: Inequalities for asymmetric entire functions, Illinois J. Math. 1, no. 1 (1957), 94--97
3. N. G. de Bruijn: Inequalities concerning polynomials in the complex domain, Indag. Math. 9 (1947), 591--598
4. P. D. Lax: Proof of a conjecture of P. Erdős on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509--513
5. Q. I. Rahman: Inequalities for polynomials, Proc. Amer. Math. Soc. 10 (1959), 800--806