An extension of typically-real functions and associated orthogonal polynomials (Q2890385)

Let \(\mathbb D\) be the complex unit disc and \(\mathcal{H}(\mathbb D)\) the class of all analytic functions \(f\) in \(\mathbb D\). The classic set of the so-called ``typically-real'' functions was defined and studied by Rogosinski and is denoted by \(T_{\mathbb R}\). Hence, it is known that the integral representation of such functions is given by NEWLINE\[NEWLINET_{\mathbb R}=\left\{f: f(z)=\int_0^{\pi}\frac{z}{(1-ze^{\text{i}\theta})(1-ze^{-\text{i}\theta})}d\mu(\theta),\;\mu\in\mathcal P_{[0,\pi]}\right\},NEWLINE\]NEWLINE where \(P_{[0,\pi]}\) is the set of probability measures on \([0,\pi]\). Observing that the class \(T_{\mathbb R}\) is closely related to the generating function \(\psi\) for the Chebyshev polynomials of the second kind and pointing out that the Koebe-function plays an essential role in the study of univalent functions, the authors study first a ``more symmetric'' \((p,q)\)-Koebe function and a new class defined in a similar way like \(T_{\mathbb R}\), but with \(\mu\in\mathcal P_{[-\pi,\pi]}\). For particular \(p\) and \(q\) (namely \(p=q=1\)), the class \(T_{\mathbb R}\) is obtained. The new class, denoted by \(T^{(p,q)}\), is studied carefully in connection to extremal problems, with the aid of the Stieltjes integral formula. Generalizations of Chebyshev polynomials of the second kind occur. The obtained results include the exact region of univalence for the class, bounds for the radius of univalence, the coefficient problem and also some properties of the generalized Chebyshev polynomials.