MATHEMATICS

K. DOCHEV

ON SOME EXTREMAL PROPERTIES OF POLYNOMIALS

(Presented by Academician S. N. Bernstein on 12 VII 1963)

In this note an elementary proof is given of one inequality concerning trigonometric polynomials, from which, as a consequence, a number of extremal properties of polynomials are obtained.

Theorem. Let \(S(\theta)\) be an arbitrary trigonometric polynomial of degree \(n\). For arbitrary real \(\theta, \varphi, \tau\) the inequality
\[ \left| e^{i\varphi}S(\theta+i\tau)+e^{-i\varphi}S(\theta-i\tau)\right| \leq \]
\[ \leq 2\sqrt{\cos^2\varphi+\operatorname{sh}^2 n\tau}\,\max |S(\theta)|,\qquad 0\leq \theta<2\pi . \tag{1} \]

For the proof of this theorem we shall use the following lemma.

Lemma. If \(P(z)\) is an algebraic polynomial of degree \(m\), all of whose zeros are situated in the circular domain \(|z|>1\), then the zeros of every equation of the form
\[ P(\alpha z)+\eta \overline{\alpha}^{\,m}P(\overline{\alpha}^{-1}z)=0,\qquad |\eta|=1,\ |\alpha|\ne 1, \tag{2} \]
are situated in the same domain.

To prove this lemma, note that from \(|z|\leq 1\), \(|\alpha|>1\), and \(|\omega|>1\) it follows that
\[ |\alpha z-\omega|<|z-\overline{\alpha}\omega|, \]
and, consequently, if \(P(z)\) has the form
\[ P(z)=\prod_{\nu=1}^{m}(z-\omega_\nu),\qquad |\omega_\nu|>1, \]
and if \(|\alpha|>1\), then for \(|z|\leq 1\) we shall have
\[ |P(\alpha z)|=\prod_{\nu=1}^{m}|\alpha z-\omega_\nu| < \prod_{\nu=1}^{m}|z-\overline{\alpha}\omega_\nu| = |\overline{\alpha}^{\,m}P(\overline{\alpha}^{-1}z)|. \]

From this inequality it is clear that in the case \(|\alpha|>1\) equation (2) has no roots in the unit disk \(|z|\leq 1\). The case \(|\alpha|<1\) can be considered in the same way or reduced to the preceding one.

We return to the proof of the theorem formulated at the beginning. For this purpose we represent \(S(\theta)\) in the form
\[ S(\theta)=e^{-ni\theta}Q(e^{i\theta}), \]
where \(Q=Q(z)\) is an algebraic polynomial of degree \(m\leq 2n\). Without loss of generality one may suppose that \(m=2n\), since this can always be achieved by small changes in the coefficients of \(S(\theta)\). In addition, we shall assume that \(\tau\ne 0\). Put
\[ \max_\theta |S(\theta)|=L\qquad (0\leq \theta<2\pi); \]
then
\[ |Q(z)|\leq L\qquad \text{for } |z|=1. \tag{3} \]

Obviously, we shall have
\[ 2\sqrt{\cos^2\varphi+\operatorname{sh}^2 n\tau} = |e^{i\varphi}\alpha^{-n}+e^{-i\varphi}\alpha^{n}|, \]
where \(\alpha=e^{-\tau}\), and inequality (1) can be written in the form
\[ \left|e^{i\varphi}\alpha^{-n}Q(\alpha z)+e^{-i\varphi}\alpha^{n}Q(\alpha^{-1}z)\right| \leq L\left|e^{i\varphi}\alpha^{-n}+e^{-i\varphi}\alpha^{n}\right|, \qquad |z|=1. \tag{4} \]

Suppose that for some real \(\theta=\theta_0\) inequality (1) is false. Then, for \(z=e^{i\theta_0}\), inequality (4) will also be violated. Conse-

Consequently, we shall have an equality of the form

\[ e^{i\varphi}\alpha^{-n}P(\alpha\xi)+e^{-i\varphi}\alpha^nP(\alpha^{-1}\xi)=0, \qquad \xi=e^{i\theta_0}, \tag{5} \]

where \(P(z)=Q(z)-\lambda\), and \(\lambda\) is some complex number with modulus \(>L\). From (3) and from the maximum-modulus principle it follows that all the zeros of the polynomial \(P(z)=Q(z)-\lambda\) \((|\lambda|>L)\) lie outside the unit circle \(|z|\leqslant1\). On the basis of the lemma already proved, the zeros of the equation
\[ P(\alpha z)+e^{-2i\varphi}\alpha^{2n}P(\alpha^{-1}z)=0 \]
must also be situated in the region \(|z|>1\), which, however, contradicts equality (5). This proves the theorem. We note that an analogous method of proof was used in [^6].

Let us note the following special cases of inequality (1):

\[ |S(\theta+i\tau)-S(\theta-i\tau)|\leqslant 2L\,\operatorname{sh} n|\tau|, \tag{6} \]

\[ |S(\theta+i\tau)+S(\theta-i\tau)|\leqslant 2L\,\operatorname{ch} n\tau. \tag{7} \]

From (6) and (7) follows the well-known inequality

\[ |S(\theta+i\tau)|\leqslant Le^{n|\tau|}. \tag{8} \]

In the case when \(S(\theta)\) is a trigonometric polynomial with real coefficients, from (1) one obtains the inequality

\[ |S(\theta+i\tau)|\leqslant L\sqrt{\cos^2\psi+\operatorname{sh}^2 n\tau}, \qquad \psi=\arg S(\theta+i\tau). \tag{9} \]

In particular, we have:

\[ |S(\theta+i\tau)|\leqslant L\,\operatorname{ch} n\tau \quad (S(\theta)\ \text{with real coefficients}). \tag{10} \]

Inequality (6) is analogous to the following inequality:

\[ |S(\theta+h)-S(\theta-h)| \leqslant 2\sin nh\cdot \max_{\theta}|S(\theta)|, \qquad 0<h\leqslant \frac{\pi}{2n}, \]

which was proved by S. N. Bernstein in general form for integral functions of finite degree in [^4]. This inequality was generalized by B. Ya. Levin [^5].

We should indicate several assertions which are consequences of the inequalities proved above.

A. From (6), by passing to the limit as \(\tau\to0\), we obtain S. N. Bernstein’s inequality
\[ |S'(\theta)|\leqslant nL. \]

B. From (6) (and also from (7)), by passing to the limit as \(\tau\to\infty\), we easily obtain the inequality (see [^1], p. 28)

\[ \max_{0\leqslant\theta\leqslant2\pi} \left| \sum_{\nu=0}^{n}(a_\nu\cos\nu\theta+b_\nu\sin\nu\theta) \right| \geqslant \max_{0\leqslant\theta\leqslant2\pi} |a_n\cos n\theta+b_n\sin n\theta|. \tag{11} \]

C. Let \(P(x)\) be an arbitrary algebraic polynomial of degree \(n\). Then \(S(\theta)=P(\cos\theta)\) is an even trigonometric polynomial of degree \(n\), and inequality (7) for \(\theta=0\) takes the form

\[ |P(\operatorname{ch}\tau)|\leqslant L\,\operatorname{ch} n\tau, \tag{12} \]

where
\[ L=\max_{0\leqslant\theta\leqslant2\pi}|S(\theta)| =\max_{-1\leqslant x\leqslant1}|P(x)|. \]
Inequality (12) was first proved by S. N. Bernstein ([^2], p. 21).

Let \(z\) be an arbitrary point of the complex plane not belonging to the interval \([-1,1]\), and let \(R\) be the semisum of the axes of the ellipse passing-

passing through \(z\) and having foci at the points \(-1\) and \(+1\). Inequality (8), in the case where \(S(\theta)\) is the polynomial \(P(\cos\theta)\), can be written in the form

\[ |P(z)|\leq LR^n . \tag{13} \]

This inequality was first given by S. N. Bernstein \((^{2,3})\).

We note that in the case where the polynomial \(P(x)\) has real coefficients, instead of (13), as is evident from (10), one can write the more precise inequality (see \((^1)\), note on p. 75):

\[ |P(z)|\leq {L\over 2}(R^n+R^{-n}). \tag{14} \]

D. Let \(S(\theta)\) be a real trigonometric polynomial, \(\tau\) a fixed real number, and \(\xi=\operatorname{Re}S(\theta+i\tau)\), \(\eta=\operatorname{Im}S(\theta+i\tau)\). Then inequality (1) can be transformed into the form

\[ |\xi\cos\varphi-\eta\sin\varphi|\leq L\sqrt{\cos^2\varphi+\operatorname{sh}^2 n\tau},\qquad 0\leq \varphi<2\pi . \tag{15} \]

It is easily verified that the intersection of all the strips (15) coincides with the ellipse

\[ {\xi^2\over \operatorname{ch}^2 n\tau} + {\eta^2\over \operatorname{sh}^2 n\tau} \leq L^2 . \tag{16} \]

Thus, the point \(S(\theta+i\tau)\) must lie in the ellipse (16).

E. If item D is applied to the polynomial \(S(\theta)=P(\cos\theta)\), we obtain the following assertion, more general than inequality (14):

Under the mapping \(\xi+i\eta=P(x+iy)\), where \(P(z)\) is a real algebraic polynomial of degree \(n\), the image of every ellipse

\[ {x^2\over \operatorname{ch}^2\tau}+{y^2\over \operatorname{sh}^2\tau}\leq 1 \]

(\(\tau>0\)) belongs to the ellipse

\[ {\xi^2\over \operatorname{ch}^2 n\tau} + {\eta^2\over \operatorname{sh}^2 n\tau} \leq L^2,\qquad L=\max_{-1\leq x\leq 1}|P(x)|. \]

Here the extremal case is the one in which \(P(z)\) coincides with the Chebyshev polynomial.

Mathematical Institute
Bulgarian Academy of Sciences
Sofia, Bulgaria

Received
19 VI 1963

REFERENCES

\(^{1}\) S. N. Bernstein, Extremal properties of polynomials, 1937.
\(^{2}\) S. N. Bernstein, On the best approximation of continuous functions by polynomials of a given degree, Collected Works, 1, 1952, p. 8.
\(^{3}\) S. N. Bernstein, On one property of polynomials, Collected Works, 1, 1952, p. 146.
\(^{4}\) S. N. Bernstein, DAN, 60, No. 9, 1487 (1948).
\(^{5}\) B. Ya. Levin, Izv. AN SSSR, ser. matem., 14, 45 (1950).
\(^{6}\) K. Dochev, DAN, 146, No. 1, 17 (1962).