MATHEMATICS

R. P. BOAS, K. I. RAHMAN

SOME INEQUALITIES FOR POLYNOMIALS AND ENTIRE FUNCTIONS

(Presented by Academician M. A. Lavrentiev on 11 VIII 1962)

I. I. Ibragimov and R. G. Mamedov \((^{1})\) proved an inequality for the norm in the space \(L^p\) of a polynomial that does not vanish inside the unit circle. The following theorem gives, possibly, the best inequality of this kind.

Theorem 1. If \(P_n(z)\) is a polynomial of degree \(\leq n\), \(P_n(z)\ne 0\) for \(|z|<1\), then for \(R>1\) and \(1\leq p\leq \infty\) the inequality
\[ \|P_n(Re^{i\theta})\|_p \leq \frac{\|1+R^n e^{in\theta}\|_p}{\|1+e^{in\theta}\|_p} \|P_n(e^{i\theta})\|_p \tag{1} \]
holds.

Inequality (1) can also be written in the form
\[ \|P_n(Re^{i\theta})\|_p \leq \frac{p+1}{2^{p+1}} \mathrm{B}\left(\frac12,\frac12 p+1\right)^{1/p} (R^n-1)\times \]
\[ \times \left\{ \frac{1}{2\pi} \int_0^{2\pi} \left[1+4\cos^2\omega\,(R^{n/2}-R^{-n/2})^{-2}\right]^{p/2} \,d\omega \right\}^{1/p} \|P_n(e^{i\theta})\|_p^{*}, \]
where \(\mathrm{B}\) is the beta function. In the case \(p=\infty\) this inequality was proved by Ankeny and Rivlin \((^{2})\). The proof is based on Boas’s interpolation formula \((^{3})\) for entire functions and on the theorem of B. Ya. Levin \((^{4})\), p. 412.

The following theorem is also valid for entire functions of finite degree.

Theorem 2. If \(f(z)\) is an entire function of degree not exceeding \(\tau\), and
\[ |f(x+iy)|\leq |f(x-iy)|,\qquad y>0, \]
then for \(y>0\) the estimate
\[ \int_{-\infty}^{\infty} |f(x+iy)|^p\,dx \leq \frac{\displaystyle \int_0^{2\pi/\tau} |\cos \tau(x+iy)|^p\,dx} {\displaystyle \int_0^{2\pi/\tau} |\cos \tau x|^p\,dx} \int_{-\infty}^{\infty} |f(x)|^p\,dx. \tag{2} \]
holds.

Inequality (2) can also be written in the form
\[ \int_{-\infty}^{\infty} |f(x+iy)|^p\,dx \leq \int_{-\infty}^{\infty} |f(x)|^p\,dx\, \frac{\displaystyle \int_0^{2\pi} (1-\sin^2\omega\,\operatorname{sech}^2 s)^{p/2}\,d\omega} {\displaystyle 2\mathrm{B}\left(\frac{1}{2p}+\frac12,\frac12\right)} \operatorname{ch}\tau y. \tag{3} \]

* As is shown by the example of the function \(P_n(z)=1+z^n\).

In the case \(p=\infty\), inequality (3) takes the form

\[ |f(x+iy)| \leq (\operatorname{ch}\tau y)\sup_{-\infty<x<\infty}|f(x)| \]

\[ (({}^{5},{}^{3})), \]

and in the case \(p=2\) (3)

\[ \int_{-\infty}^{\infty}|f(x+iy)|^{2}\,dx \leq (\operatorname{ch}2\tau y)\int_{-\infty}^{\infty}|f(x)|^{2}\,dx . \]

Northwestern University
Evanston, Illinois, USA

Regional Engineering College
Srinagar, Kashmir, India

Received
7 VIII 1962

REFERENCES

\(^{1}\) I. I. Ibragimov, R. G. Mamedov, Some inequalities for polynomials of a complex variable, DAN, 138, 526 (1961).
\(^{2}\) N. C. Ankeny, T. J. Rivlin, Pacific J. Math., 5, 849 (1955).
\(^{3}\) R. P. Boas, Math. Scand., 4, 29 (1956).
\(^{4}\) B. Ya. Levin, Distribution of zeros of entire functions, Moscow, 1956.
\(^{5}\) R. J. Duffin, A. C. Schaeffer, Bull. Am. Math. Soc., 44, 236 (1938).