# An extension of typically-real functions and associated orthogonal polynomials

Iwona Naraniecka; Jan Szynal; Anna Tatarczak

Annales UMCS, Mathematica (2011)

- Volume: 65, Issue: 2, page 99-112
- ISSN: 2083-7402

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topIwona Naraniecka, Jan Szynal, and Anna Tatarczak. "An extension of typically-real functions and associated orthogonal polynomials." Annales UMCS, Mathematica 65.2 (2011): 99-112. <http://eudml.org/doc/268336>.

@article{IwonaNaraniecka2011,

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.},

author = {Iwona Naraniecka, Jan Szynal, Anna Tatarczak},

journal = {Annales UMCS, Mathematica},

keywords = {Typically-real functions; univalent functions; local univalence; univalence; starlikeness; Chebyshev polynomials; orthogonal polynomials; typically-real functions},

language = {eng},

number = {2},

pages = {99-112},

title = {An extension of typically-real functions and associated orthogonal polynomials},

url = {http://eudml.org/doc/268336},

volume = {65},

year = {2011},

}

TY - JOUR

AU - Iwona Naraniecka

AU - Jan Szynal

AU - Anna Tatarczak

TI - An extension of typically-real functions and associated orthogonal polynomials

JO - Annales UMCS, Mathematica

PY - 2011

VL - 65

IS - 2

SP - 99

EP - 112

AB - 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.

LA - eng

KW - Typically-real functions; univalent functions; local univalence; univalence; starlikeness; Chebyshev polynomials; orthogonal polynomials; typically-real functions

UR - http://eudml.org/doc/268336

ER -

## References

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- Mason, J. C., Handscomb, D. C., Chebyshev Polynomials, Chapman and Hall/ CRC, Boca Raton, FL, 2003.
- Robertson, M. S., On the coefficients of typically-real function, Bull. Amer. Math. Soc. 41 (1935), no. 8, 565-572. Zbl0012.21201
- Robertson, M. S., The sum of univalent functions, Duke Math. J. 37 (1970), 411-419. Zbl0201.40702
- Rogosinski, W., Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), no. 1, 93-121. Zbl0003.39303

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