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Abstract. 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 of obtained orthogonal polynomials.

We introduce the class L(β, γ ) of holomorphic, locally univalent functions in the unit disk D = {z: |z| < 1}, which we call the class of doubly close-to-convex functions. This notion unifies the earlier known extensions. The class L(β, γ ) appears to be linear invariant. First of all we determine the region of variability {w: w = log f (r), f ∈ L(β, γ )} for fixed z, |z| = r < 1, which give us the exact rotation theorem. The rotation theorem and linear invariance allows us to find the sharp value for the radius of close-to-convexity and bound for the radius of univalence. Moreover, it was helpful as well in finding the sharp region for α ∈ R, for which the integral z 0 (f (t)) α dt, f ∈ L(β, γ ), is univalent. Because L(β, γ ) reduces to β-close-to-convex functions (γ = 0) and to convex functions (β = 0 and γ = 0), the obtained results generalize several corresponding ones for these classes. We improve as well the value of the radius of univalence for the class considered by Hengartner and Schober (Proc. Amer. Math. Soc. 28 (1971) 519-524) from 0.345 to 0.577. 

Abstract. 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 P λ n (x; θ, ψ) of a real variable x as coefficients ofwhere the parameters λ, θ, ψ satisfy the conditions: λ > 0, θ ∈ (0, π), ψ ∈ R. In the case ψ = −θ we have the well-known (MP) polynomials. The cases ψ = π − θ and ψ = π + θ leads to new sets of polynomials which we call quasi-Meixner-Pollaczek polynomials and strongly symmetric MeixnerPollaczek polynomials. If x = 0, then we have an obvious generalization of the Gegenbauer polynomials. The properties of (GMP) polynomials as well as of some families of holomorphic functions |z| < 1 defined by the Stieltjes-integral formula, where the function zG λ (x; θ, ψ; z) is a kernel, will be discussed. Linearly-invariant families of holomorphic functions(1.1) f (z) = z + a 2 z 2 + . . . , z ∈ D 2010 Mathematics Subject Classification. 30C45, 30C70, 42C05, 33C45.