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.
@article{bwmeta1.element.doi-10_2478_v10062-011-0017-2,
author = {Iwona Naraniecka and Jan Szynal and Anna Tatarczak},
title = {An extension of typically-real functions and associated orthogonal polynomials},
journal = {Annales UMCS, Mathematica},
volume = {65},
year = {2011},
pages = {99-112},
zbl = {1255.30021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0017-2}
}
Iwona Naraniecka; Jan Szynal; Anna Tatarczak. An extension of typically-real functions and associated orthogonal polynomials. Annales UMCS, Mathematica, Tome 65 (2011) pp. 99-112. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0017-2/
Chihara, T. S., An Introduction to Orthogonal Polynomials, Mathematics and its Applications. Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. | Zbl 0389.33008
Goluzin, G. M., On typically real functions, Mat. Sbornik N.S. 27(69) (1950), 201-218 (Russian).
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. | Zbl 0679.33001
Gasper, G., Rahman, M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35. Cambridge University Press, Cambridge, 1990. | Zbl 0695.33001
Kiepiela, K., Klimek, D., An extension of the Chebyshev polynomials, J. Comput. Appl. Math. 178 (2005), no. 1-2, 305-312. | Zbl 1068.33030
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). | Zbl 0377.30014
Koczan, L., Zaprawa, P., Domains of univalence for typically-real odd functions, Complex Var. Theory Appl. 48 (2003), no. 1, 1-17. | Zbl 1040.30004
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. | Zbl 0012.21201
Robertson, M. S., The sum of univalent functions, Duke Math. J. 37 (1970), 411-419. | Zbl 0201.40702
Rogosinski, W., Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), no. 1, 93-121. | Zbl 0003.39303