MATHEMATICS

I. I. Ibragimov and R. G. Mamedov

SOME INEQUALITIES FOR POLYNOMIALS OF A COMPLEX VARIABLE

(Presented by Academician V. I. Smirnov on 9 January 1961)

Let \(Q_n(z)=a_0z^n+a_1z^{n-1}+\cdots+a_n\) be a polynomial of degree \(n\). S. N. Bernstein \((^1)\) proved the following assertion:

If \(|Q_n(z)|\leq 1\) for \(|z|\leq 1\), then the inequality \(|Q'_n(z)|\leq n\) holds for \(|z|=1\).

Let

\[ \|Q_n(re^{i\varphi})\|_p= \left\{\int_0^{2\pi}|Q_n(re^{i\varphi})|^p\,d\varphi\right\}^{1/p}. \]

It is not difficult to show (for example, see \((^5)\)) that for any \(p\geq 1\) the inequalities

\[ \|Q_n(Re^{i\varphi})\|_p\leq R^n\|Q_n(e^{i\varphi})\|_p \quad (R>1); \tag{1} \]

\[ \|Q_n(\rho e^{i\varphi})\|_p\geq \rho^n\|Q_n(e^{i\varphi})\|_p \quad (\rho<1). \tag{2} \]

hold.

For polynomials \(Q_n(z)\) having no zeros inside the unit disk \(|z|<1\), de Bruijn \((^4)\) proved the following assertion:

Theorem A. If the polynomial \(Q_n(z)\) of degree \(n\) has no zeros in \(|z|<1\), then for any \(p\geq 1\) the inequality

\[ \|Q'_n(e^{i\varphi})\|_p \leq \frac{n}{2} \left[ \frac{\sqrt{\pi}\,\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} \|Q_n(e^{i\varphi})\|_p . \tag{3} \]

In relation (3), the equality sign is attained only for polynomials of the form \(Q_n(z)=\lambda+\mu z^n\), where \(|\lambda|=|\mu|\).

In this note we refine inequalities (1) and (2) for polynomials \(Q_n(z)\) of degree \(n\) that have no zeros in \(|z|<1\).

Theorem 1. If \(Q_n(z)\) has no zeros in \(|z|<1\), then for \(p\geq 1\) the inequality

\[ \|Q_n(Re^{i\varphi})\|_p < \left\{ \frac12 \left[ \frac{\sqrt{\pi}\,\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} (R^n-1)+1 \right\} \|Q_n(e^{i\varphi})\|_p \tag{4} \]

holds for any \(R>1\).

Proof. Suppose first that \(Q_n(z)\ne \lambda+\mu z^n\), where \(|\lambda|=|\mu|\). From the obvious inequality

\[ |Q_n(Re^{i\varphi})| \leq \int_1^R |Q'_n(re^{i\varphi})|\,dr + |Q_n(e^{i\varphi})|, \]

where \(\varphi\) (\(0 \leqslant \varphi \leqslant 2\pi\)) is any number and \(R>1\), for any \(\rho \geqslant 1\), the inequality follows:
\[ \|Q_n(Re^{i\varphi})\|_p \leqslant \int_1^R \|Q_n'(re^{i\varphi})\|_p\,dr + \|Q_n(e^{i\varphi})\|_p . \]

Taking (1) into account, the last inequality may be written in the form
\[ \|Q_n(Re^{i\varphi})\|_p \leqslant \|Q_n'(e^{i\varphi})\|_p \int_1^R r^{\,n-1}\,dr + \|Q_n(e^{i\varphi})\|_p . \]

Hence, by Theorem A it follows that
\[ \|Q_n(Re^{i\varphi})\|_p < \left\{ \frac{n}{2} \left[ \frac{\sqrt{\pi}\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} \frac{R^n-1}{n} +1 \right\} \|Q_n(e^{i\varphi})\|_p, \]
i.e. (4) is valid.

It remains to verify the validity of (4) for polynomials of the form \(Q_n(z)=\lambda+\mu z^n\), where \(|\lambda|=|\mu|\). This follows from the inequality
\[ \|\lambda+\mu e^{i\varphi}R^n\|_p < (R^n-1)|\mu|(2\pi)^{1/p} + \|\lambda+\mu e^{i\varphi n}\|_p = \]
\[ = \left\{ \frac12 \left[ \frac{\sqrt{\pi}\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} (R^n-1) +1 \right\} \|\lambda+\mu z^n\|_p . \]

In the case \(p=\infty\), Theorem 1 was proved by Ankeny and Rivlin (3), and in the case \(p=1\), by Rahman (5).

Theorem 2. If \(Q_n(z)\) has no zeros in \(|z|<1\), then there exists a positive number \(\delta\) such that, for \((1-\delta)<\rho<1\), the inequality
\[ \|Q_n'(\rho e^{i\varphi})\|_p > \left\{ 1-\frac12 \left[ \frac{\sqrt{\pi}\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} (1-\rho^n) \right\} \|Q_n(e^{i\varphi})\|_p, \tag{5} \]
holds, where \(\rho \geqslant 1\) is any number.

Proof. The validity of inequality (5) for polynomials \(Q_n(z)=\lambda+\mu z^n\), where \(|\lambda|=|\mu|\), having no zeros in \(|z|<1\), is verified directly. Indeed,
\[ \|\lambda+\mu\rho^n e^{i\varphi n}\|_p > \|\lambda+\mu e^{i\varphi n}\|_p - (1-\rho^n)|\mu|(2\pi)^{1/p} = \]
\[ = \left\{ 1-\frac12 \left[ \frac{\sqrt{\pi}\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} (1-\rho^n) \right\} \|\lambda+\mu e^{i\varphi n}\|_p, \]
i.e. (5) is valid.

Now we prove the theorems for polynomials \(Q_n(z)\ne \lambda+\mu z^n\) of degree \(n\), where \(|\lambda|=|\mu|\), having no zeros in \(|z|<1\). For the proof suppose the contrary, i.e. that (5) does not hold. This means that there exists a polynomial \(Q_n(z)\ne \lambda+\mu z^n\) (\(|\lambda|=|\mu|\)) of degree \(n\), having no zeros in \(|z|<1\), and a sequence of values \(1-\delta<\rho_m<1\) \((m=1,2,\ldots)\) with
\(\lim\limits_{m\to\infty}\rho_m=1\), such that
\[ \|Q_n(\rho_m e^{i\varphi})\|_p \leqslant \left\{ 1-\frac12 \left[ \frac{\sqrt{\pi}\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} (1-\rho_m^n) \right\} \|Q_n(e^{i\varphi})\|_p . \tag{6} \]

Then
\[ \|Q_n'(e^{i\varphi})\|_p = \left\| \lim_{m\to\infty} \frac{Q_n(e^{i\varphi})-Q_n(\rho_m e^{i\varphi})} {e^{i\varphi}-\rho_m e^{i\varphi}} \right\|_p \geqslant \]
\[ \geqslant \lim_{m\to\infty} \frac{1}{1-\rho_m} \left( \|Q_n(e^{i\varphi})\|_p-\|Q_n(\rho_m e^{i\varphi})\|_p \right). \]

Hence, by virtue of (6), we have

\[ \begin{aligned} \left\|Q_n'\left(e^{i\varphi}\right)\right\|_p &\geq \frac{1}{2} \left[ \frac{\sqrt{\pi}\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} \left\|Q_n\left(e^{i\varphi}\right)\right\|_p \lim_{m\to\infty}\frac{1-\rho_m^n}{1-\rho_m} \\ &= \frac{n}{2} \left[ \frac{\sqrt{\pi}\Gamma\left(\frac12 p+1\right)} {\Gamma\left(\frac12(p+1)\right)} \right]^{1/p} \left\|Q_n\left(e^{i\varphi}\right)\right\|_p, \end{aligned} \]

which contradicts Theorem A. Thus, inequality (5) is proved also for all \(Q_n(z)\ne \lambda+\mu z^n\), where \(|\lambda|=|\mu|\). For \(p=\infty\), it follows from Theorem 2 that if \(Q_n(z)\) has no zeros in \(|z|<1\) and \(\max_{|z|=1}|Q_n(z)|=1\), then there exists a positive number \(\delta>0\) such that

\[ \max_{|z|=\rho}|Q_n(z)|\geq \frac{1+\rho^n}{2} \]

for \(1-\delta<\rho<1\).

For \(p=1\), one theorem of Rahman follows from Theorem 2 \((^5)\).

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
5 I 1961

References

\(^1\) S. N. Bernstein, Extremal Properties of Polynomials, Moscow, 1947.
\(^2\) P. D. Lax, Bull. Am. Math. Soc., 50, 509 (1944).
\(^3\) N. C. Ankeny, T. J. Rivlin, Pacific J. Math., 5, 849 (1955).
\(^4\) N. G. de Bruijn, Nederl. Akad. Proc., 50, 1265 (1947).
\(^5\) Q. J. Rahman, Proc. Am. Math. Soc., 10, 800 (1959).