MATHEMATICS

V. S. VIDENSKII

A GENERALIZATION OF V. A. MARKOV’S INEQUALITIES

(Presented by Academician S. N. Bernstein on 14 I 1958)

Theorem. If on the interval \([-1,1]\) a polynomial \(P_n(x)\) of degree not exceeding \(n\) satisfies the inequality

\[ |P_n(x)| \leq |\alpha x+i\sqrt{1-x^2}| \quad (\alpha>0), \tag{1} \]

then

\[ |P_n^{(k)}(x)| \leq M_n^{(k)}(1)=\frac{\alpha+1}{2}T_n^{(k)}(1)+\frac{\alpha-1}{2}T_{n-2}^{(k)}(1) \quad (k=1,\ldots,n), \tag{2} \]

where \(T_n(x)=\cos n\arccos x\). Equality in (2) is attained only for polynomials \(P_n(x)=\gamma M_n(x)\), \(|\gamma|=1\),

\[ M_n(x)=\frac{\alpha+1}{2}T_n(x)+\frac{\alpha-1}{2}T_{n-2}(x) \tag{3} \]

at the points \(x=\pm1\).

For \(\alpha=1\) the right-hand side of (1) is equal to unity, and inequalities (2) become V. A. Markov’s inequalities. For \(k=1\) and \(2\) and any \(\alpha>0\) the theorem was proved in my note \((^1)\).

Put

\[ M_n(x)=\Re\{(\alpha x+i\sqrt{1-x^2})[T_{n-1}(x)+iS_{n-1}(x)]\}, \]

\[ L_n(x)=\sqrt{1-x^2}N_{n-1}(x)=\Im\{(\alpha x+i\sqrt{1-x^2})[T_{n-1}(x)+iS_{n-1}(x)]\}, \tag{4} \]

where \(S_n(x)=\sin n\arccos x\). The functions \(M_n(x)\) and \(N_{n-1}(x)\) are polynomials of degrees \(n\) and \(n-1\), respectively; all their zeros lie in the interval \((-1,1)\) and mutually interlace (see, for example, \((^2)\)). It is not difficult to show that from (4) there follow equality (3) and the equality

\[ L_n(x)=\frac{\alpha+1}{2}S_n(x)+\frac{\alpha-1}{2}S_{n-2}(x). \tag{5} \]

If we put

\[ H_k(x)=|M_n^{(k)}(x)+iL_n^{(k)}(x)| \quad (k=1,\ldots,n); \tag{6} \]

\[ \Phi_k(x)= \begin{cases} H_k(x), & \text{for } \xi_1^{(k)} \leq x \leq \xi_{n-k+1}^{(k)},\\ |M_n^{(k)}(x)|, & \text{for } -\infty < x \leq \xi_1^{(k)},\ \xi_{n-k+1}^{(k)} \leq x < +\infty, \end{cases} \tag{7} \]

where \(\xi_1^{(k)}\) and \(\xi_{n-k+1}^{(k)}\) are the extreme zeros of the function \(L_n^{(k)}(x)\) lying on the interval \([-1,1]\), then from my previous results \((^{3-5})\) it follows that, for polynomials satisfying (1), the following estimates of their derivatives are valid:

\[ |P_n^{(k)}(x)| \leq \Phi_k(x) \quad (k=1,\ldots,n;\ -\infty<x<\infty). \tag{8} \]

It was originally shown by S. N. Bernstein \((^6)\) that on the interval \([-1,1]\) for the first derivative the inequality \(|P'_n(x)| \leqslant H_1(x)\) holds.

Our theorem will follow directly from inequality (8), if we show that the even continuous function \(\Phi_k(x)\) increases monotonically for \(x>0\). Moreover, since all zeros of \(M_n^{(k)}(x)\) lie in the interval \((\xi_1^{(k)},\xi_{n-k+1}^{(k)})\) \((^3)\), it is necessary to establish the increase of \(\Phi_k(x)\) only on the interval \([0,\xi_{n-k+1}^{(k)}]\).

Let us prove that \(H_k^2(x)\) is expanded in the interval \((-1,1)\) in a Taylor series in even powers of \(x\) with positive coefficients. In doing so we shall rely on the fact established in the work of A. Schaeffer and R. Duffin \((^7)\) that

\[ W_{n,k}(x)=[T_n^{(k)}(x)]^2+[S_n^{(k)}(x)]^2=\sum_{p=0}^{\infty} a_{p,k}x^{2p},\qquad a_{p,k}>0 \tag{9} \]

\[ (k=1,2,\ldots,n;\ p=0,1,2,\ldots). \]

Obviously, the function \(H_k^2(x)\) can be written in the form

\[ \begin{aligned} H_k^2(x)=& \left\{\frac{\alpha}{2}[T_n^{(k)}(x)+T_{n-2}^{(k)}(x)] +\frac{1}{2}[T_n^{(k)}(x)-T_{n-2}^{(k)}(x)]\right\}^2 \\ &+\left\{\frac{\alpha}{2}[S_n^{(k)}(x)+S_{n-2}^{(k)}(x)] +\frac{1}{2}[S_n^{(k)}(x)-S_{n-2}^{(k)}(x)]\right\}^2 . \end{aligned} \tag{10} \]

On the one hand, the identities

\[ T_n(x)+T_{n-2}(x)=2xT_{n-1}(x),\qquad T_n(x)-T_{n-2}(x)=-\frac{2}{n-1}(1-x^2)T'_{n-1}(x), \]

\[ S_n(x)+S_{n-2}(x)=2xS_{n-1}(x),\qquad S_n(x)-S_{n-2}(x)=-\frac{2}{n-1}(1-x^2)S'_{n-1}(x). \tag{11} \]

hold.

On the other hand, the functions \(T_n(x)\) and \(S_n(x)\) satisfy the differential equation

\[ (1-x^2)y''-xy'+n^2y=0, \tag{12} \]

therefore \(T_n^{(k)}(x)\) and \(S_n^{(k)}(x)\) satisfy the equation

\[ (1-x^2)y^{(k+2)}-(2k+1)xy^{(k+1)}+(n^2-k^2)y^{(k)}=0. \tag{13} \]

From the identities (11) we obtain

\[ [T_n^{(k)}(x)+T_{n-2}^{(k)}(x)]^2+[S_n^{(k)}(x)+S_{n-2}^{(k)}(x)]^2= \]

\[ =4\left[x^2W_{n-1,k}(x)+kx\frac{d}{dx}W_{n-1,k-1}(x)+k^2W_{n-1,k-1}(x)\right]. \tag{14} \]

From (13) we obtain

\[ [(1-x^2)T'_{n-1}(x)]^{(k)} =-[xT_{n-1}^{(k)}(x)+(n^2-2n+k)T_{n-1}^{(k-1)}(x)], \]

\[ [(1-x^2)S'_{n-1}(x)]^{(k)} =-[xS_{n-1}^{(k)}(x)+(n^2-2n+k)S_{n-1}^{(k-1)}(x)]. \tag{15} \]

Applying (11) and (15), we may write

\[ [T_n^{(k)}(x)-T_{n-2}^{(k)}(x)]^2+[S_n^{(k)}(x)-S_{n-2}^{(k)}(x)]^2= \]

\[ =\frac{4}{(n-1)^2}\left[ x^2W_{n-1,k}(x)+(n^2-2n+k)x\frac{d}{dx}W_{n-1,k-1}(x)+ \right. \]

\[ \left. +(n^2-2n+k)^2W_{n-1,k-1}(x) \right]; \tag{16} \]

\[ \{[T_n^{(k)}(x)]^2-[T_{n-2}^{(k)}(x)]^2\}+\{[S_n^{(k)}(x)]^2-[S_{n-2}^{(k)}(x)]^2\} = \]
\[ = \frac{2}{n-1}\left[2x^2 W_{n-1,k}(x)+(n^2-2n+2k)x\frac{d}{dx}W_{n-1,k-1}(x)+\right. \]
\[ \left.{}+2k(n^2-2n+k)W_{n-1,k-1}(x)\right]. \tag{17} \]

From (14), (16), and (17), taking (9) into account, we conclude that for \(k=1,\ldots,n-2\) the function \(H_k^2(x)\) expands on \((-1,1)\) into an even power series with positive coefficients, and hence for \(1\le k\le n-2\) the theorem is proved. For \(k=n\) the theorem follows directly from (8), since at \(x=1\) we have \(|P_n^{(n)}(1)|\le M_n^{(n)}(1)\). For \(k=n-1\) the result again follows easily from (8), for the maximum of the linear function \(P_n^{(n-1)}(x)\) on the interval \([-1,1]\) is attained at one of the endpoints of the interval; but at the points \(x=\pm1\) we have \(|P_n^{(n-1)}(\pm1)|\le M_n^{(n-1)}(1)\). In the proof of the theorem it was implicitly assumed that \(n\ge2\), but it is clear that for \(n=1\) and \(2\) inequalities (2) are simple consequences of inequalities (8).

I note that, in the question of a constructive characterization of functions given on a finite interval by means of their approximations by polynomials, investigated recently by V. K. Dzyadyk \({}^{8}\) and A. F. Timan \({}^{9}\), an important role is played by estimates of the successive derivatives of polynomials satisfying the inequalities

\[ |P_n(x)|\le \left|\frac{x}{n}+i\sqrt{1-x^2}\right|^\rho,\qquad \rho>0,\qquad -1\le x\le 1. \]

In the works mentioned, neither an extremal polynomial for this problem, for any values of \(\rho\), nor exact upper bounds for the derivatives \(P_n^{(k)}(x)\) on the interval \([-1,1]\) were given.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
14 I 1958

CITED LITERATURE

\({}^{1}\) V. S. Videnskii, Zap. matem. otd. fiz.-matem. fak. i Kharkovsk. matem. obshch., 26, ser. 4 (1958).
\({}^{2}\) S. N. Bernstein, Collected Works, 1, article No. 42, 1952, pp. 452—467.
\({}^{3}\) V. S. Videnskii, DAN, 67, No. 5 (1949).
\({}^{4}\) V. S. Videnskii, DAN, 73, No. 2 (1950).
\({}^{5}\) V. S. Videnskii, Izv. AN SSSR, ser. matem., 15, 401 (1951).
\({}^{6}\) S. N. Bernstein, Collected Works, 1, article No. 46, 1952, pp. 497—499.
\({}^{7}\) C. A. Schaeffer, R. J. Duffin, Bull. Am. Math. Soc., 44, No. 4 (1938).
\({}^{8}\) V. K. Dzyadyk, Izv. AN SSSR, ser. matem., 20, 623 (1956).
\({}^{9}\) A. F. Timan, DAN, 116, No. 5 (1957).