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

Abstract

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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.

How to cite

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Iwona 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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  1. Chihara, T. S., An Introduction to Orthogonal Polynomials, Mathematics and its Applications. Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Zbl0389.33008
  2. Goluzin, G. M., On typically real functions, Mat. Sbornik N.S. 27(69) (1950), 201-218 (Russian). 
  3. Gasper, G., q-extensions of Clausen's formula and of the inequalities used by de Branges in his proof of the Bieberbach, Robertson and Milin conjectures, SIAM J. Math. Anal. 20 (1989), no. 4, 1019-1034. Zbl0679.33001
  4. Gasper, G., Rahman, M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35. Cambridge University Press, Cambridge, 1990. Zbl0695.33001
  5. Kiepiela, K., Klimek, D., An extension of the Chebyshev polynomials, J. Comput. Appl. Math. 178 (2005), no. 1-2, 305-312. Zbl1068.33030
  6. Koczan, L., Szapiel, W., Sur certaines classes de fonctions holomorphes définies par une intègrale de Stieltjes, Ann. Univ. Mariae Curie-Skłodowska Sect. A 28 (1974), 39-51 (1976). Zbl0377.30014
  7. Koczan, L., Zaprawa, P., Domains of univalence for typically-real odd functions, Complex Var. Theory Appl. 48 (2003), no. 1, 1-17. Zbl1040.30004
  8. Mason, J. C., Handscomb, D. C., Chebyshev Polynomials, Chapman and Hall/ CRC, Boca Raton, FL, 2003. 
  9. Robertson, M. S., On the coefficients of typically-real function, Bull. Amer. Math. Soc. 41 (1935), no. 8, 565-572. Zbl0012.21201
  10. Robertson, M. S., The sum of univalent functions, Duke Math. J. 37 (1970), 411-419. Zbl0201.40702
  11. Rogosinski, W., Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), no. 1, 93-121. Zbl0003.39303

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