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An extension of typically-real functions and associated orthogonal polynomials

Iwona NaranieckaJan SzynalAnna Tatarczak — 2011

Annales UMCS, Mathematica

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...

Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Iwona NaranieckaJan SzynalAnna Tatarczak — 2013

Annales UMCS, Mathematica

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...

Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Iwona NaranieckaJan SzynalAnna Tatarczak — 2013

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

The extremal functions  f 0 ( z )   realizing the maxima of some functionals (e.g. max | a 3 | , and  max a r g f ' ( z ) ) within the so-called universal linearly invariant family U α (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 n λ ( x ; θ , ψ ) of a real variable x as coefficients of G λ ( x ; θ , ψ ; z ) = 1 ( 1 - z e i θ ) λ - i x ( 1 - z e i ψ ) λ + i x = n = 0 P n λ ( x ; θ , ψ ) z n , | z | < 1 , where the parameters λ , θ , ψ satisfy the conditions: λ > 0 ,...

An extension of typically-real functions and associated orthogonal polynomials

Iwona NaranieckaJan SzynalAnna Tatarczak — 2011

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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...

Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian

Magdalena Sobczak-KnećViktor V. StarkovJan Szynal — 2011

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

The relation between the Jacobian and the orders of a linear invariant family of locally univalent harmonic mapping in the plane is studied. The new order (called the strong order) of a linear invariant family is defined and the relations between order and strong order are established.

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