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Integral mean estimates for polynomials whose zeros are within a circle. (English) Zbl 0874.30001
The authors prove four theorems concerning inequalities of \(L_p\) norms on the unit circle of a polynomial with restricted zeros and its derivative. In all theorems the numbers \(p\) and \(q\) satisfy \(p>1\), \(q> 1\), \(p^{-1} +q^{-1} =1\) and \(P(z)\) is a polynomial with complex coefficients of degree \(n\). Theorem 1. If all the zeros of \(P(z)\) lie in the closed disk \(|z|\leq K\), \(K\leq 1\), then for each \(r>0\) the inequality \[ n\left \{\int_0^{2\pi} \bigl|P (e^{i\theta}) \bigr|^r d\theta \right\}^{1 \over r} \leq\left \{\int^{2\pi}_0 |1+Ke^{i\theta} |^{ qr} d\theta \right\}^{1 \over qr} \left\{\int_0^{2\pi} \bigl|P' (e^{i\theta}) \bigr|^{pr} d \theta \right\}^{1\over pr} \] holds. Theorem 2. If all the zeros of \(P(z)\) lie in the closed unit disk, then for each \(r>0\) and every \(\alpha\) such that \(|\alpha |\leq 1\) the inequality \[ n \left\{\int_0^{2 \pi} \bigl|P(e^{i\theta}) +\alpha m\bigr|^r d\theta\right\}^{1 \over r} \leq\left \{\int_0^{2\pi} |1+e^{i\theta} |^{qr} d\theta \right\}^{1 \over qr} \left\{\int_0^{2\pi} \bigl|P'(e^{i \theta}) \bigr|^{pr} d\theta \right\}^{1 \over pr} \] holds, where \(m=\min_{|z |=1} |P(z) |\). Theorem 3. If all the zeros of \(P(z)\) lie in the closed disk \(|z|\leq K\) where \(K\geq 1\) then for each \(r\geq 1\) the inequality \[ n\left \{\int_0^{2\pi} \bigl|P(e^{i\theta}) \bigr|^r d\theta \right\}^{1\over r} \leq C_r \left\{\int_0^{2\pi} |1+e^{i\theta} |^{qr} d\theta \right\}^{1 \over qr} \left\{\int_0^{2\pi} \bigl|P'(e^{i\theta}) \bigr|^{pr} d\theta \right\}^{1\over pr} \] holds, where the constant \(C_r\) is given by \[ C_r= {\{\int_0^{2\pi} |1+ K^ne^{i\theta} |^r d\theta\}^{1\over r} \over\{\int_0^{2\pi} |1+ e^{i\theta} |^rd \theta\}^{1\over r}}. \] Theorem 4. If all the zeros of \(P(z)\) lie in the closed disk \(|z|\leq K\), \(K\geq 1\), then for each \(r\geq 1\) and every \(\alpha\) such that \(|\alpha |\leq 1\) the inequality \[ n\left \{\int_0^{2\pi} \bigl |P(e^{i\theta}) +\alpha m\bigr|^r d\theta \right\}^{1\over r} \leq\left\{\int_0^{2\pi} |1+K^n e^{i\theta}|^r d\theta\right\}^{1 \over r} \max_{|z|=1} \bigl|P'(z) \bigr| \] holds. The result is best possible for \(P(z)= z^n+ K^n\). Here \(m= \min_{|z|=K} |P(z)|\). Theorems 1 and 3 for \(C\)-polynomials and \(K=1\) were proven by the reviewer (Proc. Am. Math. Soc. 80, 78-82 (1980; Zbl 0441.30010) using an inequality of Rogosinski [E. Hille, Analytic function theory. II. (1962; Zbl 0102.29401), Theorem 18.5.3]. In the present paper the authors use an inequality due to N. G. De Bruijn [Proc. Akad. Wet. Amsterdam 50, 1265-1272 (1947; Zbl 0029.19802), Theorem 1].

MSC:
30A10 Inequalities in the complex plane
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