Reports of the Academy of Sciences of the USSR
1963. Volume 151, No. 6

MATHEMATICS

D. I. MAMEDKHANOV

SOME EXTREMAL PROBLEMS IN THE CLASS OF POLYNOMIALS AND RATIONAL FUNCTIONS

(Presented by Academician I. N. Vekua on March 6, 1963)

Let

\[ M(f,r)=\max_{|z|=r}|f(z)|. \]

It is known \(\left(^{1}\right.\), p. 167) that for any polynomial of degree \(n\) the inequality

\[ M(P_n,R)\leq R^n M(P_n,1), \tag{1} \]

holds, where \(R\geq 1\), and for \(r\leq 1\)

\[ M(P_n,r)\geq r^n M(P_n,1). \tag{2} \]

Inequalities (1) and (2) are sharp in the class of polynomials of degree \(n\).

Denote by \(K[n,d]\) the class of polynomials of degree \(n\) having no zeros in the disk \(|z|<d\). For this class of polynomials, inequalities (1) and (2) have been sharpened by various authors.

In the paper \(\left(^{2}\right)\) it was proved that if \(P_n(z)\in K[n,1]\) and \(R\geq 1\), then

\[ M(P_n,R)\leq \frac{1+R^n}{2}M(P_n,1). \tag{3} \]

This inequality was generalized in the metric of the space \(\mathscr L_p\) by I. I. Ibragimov and R. G. Mamedov \(\left(^{3}\right)\).

We \(\left(^{4}\right)\) sharpened this result for the class \(K[n,\sqrt R]\), and it was also proved that if \(P(z)\in K[n,1]\) and \(r\leq 1\), then

\[ |P_n(re^{i\alpha})|\geq \left(\frac{1+r}{2}\right)^n |P_n(e^{i\alpha})|. \]

For a rational function of the form

\[ R(z)=\frac{P_n(z)}{Q_m(z)} =\frac{a_1z^n+\cdots+a_n}{b_1z^m+\cdots+b_m} \]

it was proved in the same paper \(\left(^{4}\right)\) that if \(P_n(z)\in K[n,1]\) and \(Q_m(z)\in K[m,1+\rho]\) \((\rho>0)\), then

\[ |R(re^{i\alpha})|\geq \left(\frac{1+r}{2}\right)^n \left(\frac{\rho}{1-r+\rho}\right)^m |R(e^{i\alpha})|, \tag{4} \]

where \(r\leq 1\).

In the present article all the results listed above are sharpened in a corresponding way, and also, for the given class of polynomials, an inequality is obtained which is an analogue of the classical inequality of A. A. Markov \(\left(^{5}\right)\)

\[ \max_{-1\leq x\leq 1}|P'_n(x)|\leq n^2M. \tag{5} \]

and S. N. Bernstein’s inequality (7)

\[ |P_n'(x)| \leq \frac{nM}{\sqrt{1-x^2}} . \tag{5′} \]

Theorem 1. Let \(P_n(z)\) be of the class \(K[n,\sqrt R+\gamma]\), \(R \geq 1\), \(\gamma>0\); let \(z_j=r_j e^{i\varphi_j}\) be the zeros of this polynomial. Then for \(r \leq 1\) the inequality

\[ \left|P_n\left(Re^{i\omega}\right)\right| \leq \left(\frac{R+d}{r+d}\right)^n \left|P_n\left(re^{i\psi}\right)\right| \tag{6} \]

holds under the condition
\[ \cos(\psi-\varphi_j)\leq \cos(\omega-\varphi_j), \qquad d=\min r_j \quad (j=1,2,\ldots,n). \]

Corollary. Let the polynomial \(P_n(z)\) be of the class \(K[n,1]\). Then for the trigonometric polynomial \(P_n(e^{i\theta})=T_n(\theta)\) of degree \(n\) the inequality

\[ |T_n(\omega)|\leq |T_n(\psi)| \]

holds under the condition
\[ \cos(\psi-\varphi_j)\leq \cos(\omega-\varphi_j). \]

Theorem 2. Let \(P_n(z)\) be of the class \(K[n,R+\rho]\), where \(R \geq 1\), \(\rho>0\), and let \(z_j=r_j e^{i\varphi_j}\) be the zeros of this polynomial. Then for \(r \leq 1\) the inequality

\[ \left|P_n\left(Re^{i\omega}\right)\right| \geq \left(\frac{\sigma-R}{\sigma-r}\right)^n \left|P_n\left(re^{i\psi}\right)\right| \tag{7} \]

holds under the condition
\[ \cos(\psi-\varphi_j)\leq \cos(\omega-\varphi_j), \qquad \sigma=\min r_j \quad (j=1,2,\ldots,n). \]

Let \(P_n(z)\in K[n,\sqrt R]\); then inequality (7) is true if in it the moduli of the polynomials are replaced by the maximum of their moduli.

Theorem 3. Let \(P_n(z)\) be of the class \(K[n,\sqrt{r'}+\gamma]\), where \(r'\geq 1\), \(\gamma>0\); let \(z_j=r_j e^{i\varphi_j}\) \((j=1,2,\ldots,n)\) be the zeros of this polynomial, and let \(Q_m(z)\in K[m,r'+\rho]\), where \(r'\geq 1\), \(\rho>0\), and \(\tau_k=\rho_k e^{i\alpha_k}\) \((k=1,2,\ldots,m)\) be the zeros of \(Q_m(z)\). Then for \(r\leq 1\) the inequality

\[ \left|R\left(r'e^{i\omega}\right)\right| \leq \left(\frac{r'+d}{r+d}\right)^n \left(\frac{\sigma-r'}{\sigma-r}\right)^m \left|R\left(re^{i\psi}\right)\right| \tag{8} \]

holds under the conditions that
\[ \cos(\psi-\varphi_j)\leq \cos(\omega-\varphi_j), \qquad \cos(\psi-\alpha_k)\leq \cos(\omega-\alpha_k), \]
\[ d=\min r_j, \qquad \sigma=\min \rho_k. \]

Remark. Inequalities (6) and (8) are sharp. Indeed, in paper (5) polynomials and rational functions for which equality is attained are given for a special case.

Theorem 4*. Let \(P_n(z)\) be an arbitrary polynomial of degree \(n\); let \(Q_m(z)\) be a polynomial of the class \(K[m,\rho+\gamma]\), \(\rho\geq 1\), \(\gamma>0\); and let \(z_j=r_j e^{i\varphi_j}\) \((j=1,2,\ldots,m)\) be the zeros of this polynomial.

If

\[ |R(z)|=\left|\frac{P_n(z)}{Q_m(z)}\right|\leq L \quad \text{for } |z|\leq 1, \]

then

\[ |R(\rho e^{i\alpha})| \leq \rho^n\left(\frac{\sigma-1}{\sigma-\rho}\right)^m L, \tag{9} \]

where
\[ \sigma=\min r_j \quad (j=1,2,\ldots,m). \]

Remark. Under the conditions of Theorem 4, inequality (9) can also be written in the form

\[ |R(\rho e^{i\alpha})| \leq \left(\frac{\rho}{r}\right)^n \left(\frac{\sigma-r}{\sigma-\rho}\right)^m \max_{|z|=r}|R(re^{i\alpha})|, \tag{9′} \]

where \(r\leq 1\).

\[ \underline{\hspace{2.5cm}} \]

* This theorem is a refinement of Walsh’s inequality ((8), p. 282).

We shall give two more theorems analogous to Theorem 1.

Theorem 5. Let \(P_n(z)\in K[n,1]\); \(z_j=r_j e^{i\varphi_j}\) \((j=1,2,\ldots,n)\) be the zeros of this polynomial. Then for \(x\leqslant 1\) the inequality

\[ \left|\rho_n(e^{i\alpha})\right|\leqslant \left(\frac{1+d}{|x|+d}\right)^n |P_n(x)| \tag{10} \]

holds, provided that \(\cos\varphi_j\leqslant \cos(\alpha-\varphi_j)\), \(d=\min r_j\) \((j=1,2,\ldots,n)\).

If in the right-hand side of inequality (10) the modulus of the polynomial is replaced by the maximum of the modulus, then \(P_n(z)\) may be taken from the class \(K[n,\sqrt{|x|}]\).

Theorem 6. Let \(P_n(z)\) be any polynomial of degree \(n\); \(z_j=r_j e^{i\varphi_j}\), \((j=1,2,\ldots,n)\), be the zeros of this polynomial. If \(|P_n(x_0e^{i\alpha_0})|=\max_{|z|\leqslant x_0}|P_n(z)|\), then for \(x_0\leqslant 1\) the inequality

\[ |P_n(x_0e^{i\alpha_0})|\leqslant |P_n(x_0)| \tag{11} \]

holds, provided that \(\cos\varphi_j\leqslant \cos(\alpha_0-\varphi_j)\) \((j=1,2,\ldots,n)\).

We now give an inequality which is an analogue of the inequalities of A. A. Markov (5) and S. N. Bernstein (5′) for the given class.

Theorem 7*. Under the assumptions of Theorem 6, for any polynomial \(P_n(z)\) the inequality

\[ |P'_n(x)|\leqslant n \max_{-1\leqslant x\leqslant 1}|P_n(x)|. \tag{12} \]

holds.

If \(P_n(z)\in K[n,1]\) and the assumptions of Theorem 6 are satisfied, then

\[ |P'_n(x)|\leqslant n/2 \tag{13} \]

provided that \(|P_n(x)|\leqslant 1\) on \([-1,1]\).

It is not difficult to observe that analogous assertions hold in the metric of the space \(\mathscr{L}_p\).

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

Received
28 II 1963

References

1. G. Pólya, G. Szegő, Problems and Theorems in Analysis, 1, Moscow, 1956.
2. N. C. Ankeny, T. J. Rivlin, Pacific J. Math., 5, 849 (1955).
3. I. I. Ibragimov, R. G. Mamedov, DAN, 139, 28 (1961).
4. D. I. Mamedkhanov, Izv. AN Azerbaijan SSR, ser. matem., No. 5 (1962).
5. D. I. Mamedkhanov, Izv. AN Azerbaijan SSR, ser. matem., No. 1 (1963).
6. A. A. Markov, Selected Works, Moscow–Leningrad, 1948, p. 51.
7. S. N. Bernstein, Collected Works, 1, Moscow, 1952.
8. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Moscow, 1961.
9. P. D. Lax, Am. Math. Soc., 50, 509 (1944).

* Erdős conjectured and Lax proved in (9) that if \(P(z)\in K[n,1]\) and \(|P_n(z)|\leqslant 1\) for \(|z|\leqslant 1\), then \(|P'_n(z)|\leqslant n/2\).