Certain extremal properties of polynomials and typically real functions (Q810664)

Let \(f(z)=zp(z)/q(z)\), where p and q are non-vanishing polynomials of degree \(\mu\) and \(\gamma\) in the unit disc \(D=\{z:| z| <1\}\), respectively. The author first proves that if \(| f(z)| \leq 1\) for \(x\in (-1,1)\), then \[ | f'(0)| \leq \min_{0<x\leq 1}\{(1+x^ 2)^{\gamma /2}/[x(1-x^ 2)^{\mu /2}]\}. \] For a typically real function g, \(g(z)=a_ 1z+a_ 2z^ 2+..\). in D, the author shows that if \(| g(xe^{i\alpha})| \leq 1\) for \(x\in (0,1)\) and some \(\alpha\in [0,\pi)\), then \[ | g'(0)| \leq \lambda_{\alpha}=\inf_{0<x<1}\{\frac{(1+x)}{x(1-x)}\sqrt{1+x^ 4- 2x^ 2 \cos 2\alpha}\}. \] Similarly if \(| g(xe^{i\alpha})| \leq 1\) for \(x\in (-1,1)\) and some \(\alpha\in [0,\pi)\), then one has \[ | g'(0)| \leq \mu_{\alpha}=\inf_{0<x<1}\{\frac{(1+x^ 2)}{x(1-x^ 2)}\sqrt{1+x^ 4-2x^ 2 \cos 2\alpha}\}. \] These estimates ae sharp for each \(\alpha\). In addition to these, some interesting remarks and special cases are also given. For some of the recent results of this type see [\textit{Q. I. Rahman} and \textit{St. Ruscheweyh}, J. Math. Anal. Appl. 146, No.2, 374-388 (1990; Zbl 0698.30005)].