Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties...

The extremal functions f0(z) realizing the maxima of some functionals (e.g. max |a3|, and max arg f′(z)) within the so-called universal linearly invariant family Uα (in the sense of Pommerenke [10]) have such a form that f′0(z) looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials Pλn (x; θ,ψ) of a real variable x as coefficients of [###] where the parameters...

The extremal functions ${f}_{0}\left(z\right)$ realizing the maxima of some functionals (e.g. $max|{a}_{3}|$, and $maxarg{f}^{{}^{\text{'}}}\left(z\right)$) within the so-called universal linearly invariant family ${U}_{\alpha}$ (in the sense of Pommerenke [10]) have such a form that ${f}_{0}^{{}^{\text{'}}}\left(z\right)$ looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials ${P}_{n}^{\lambda}(x;\theta ,\psi )$ of a real variable $x$ as coefficients of $${G}^{\lambda}(x;\theta ,\psi ;z)=\frac{1}{{(1-z{e}^{i\theta})}^{\lambda -ix}{(1-z{e}^{i\psi})}^{\lambda +ix}}=\sum _{n=0}^{\infty}{P}_{n}^{\lambda}(x;\theta ,\psi ){z}^{n},\phantom{\rule{4pt}{0ex}}\left|z\right|<1,$$
where the parameters $\lambda $, $\theta $, $\psi $ satisfy the conditions: $\lambda >0$,...

Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties...

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