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(x)],}^{(k)}(x)+(n^2-2n+k)T_{n-1}^{(k-1)
]

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