An extension of typically-real functions and associated orthogonal polynomials

Iwona Naraniecka, Jan Szynal, Anna Tatarczak


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.


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

Full Text:



Chihara, T. S., An Introduction to Orthogonal Polynomials, Mathematics and its Applications. Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978.

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.

Gasper, G., Rahman, M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35. Cambridge University Press, Cambridge, 1990.

Kiepiela, K., Klimek, D., An extension of the Chebyshev polynomials, J. Comput. Appl. Math. 178 (2005), no. 1-2, 305-312.

Koczan, L., Szapiel, W., Sur certaines classes de fonctions holomorphes definies par une integrale de Stieltjes, Ann. Univ. Mariae Curie-Skłodowska Sect. A 28 (1974),

-51 (1976).

Koczan, L., Zaprawa, P., Domains of univalence for typically-real odd functions, Complex Var. Theory Appl. 48 (2003), no. 1, 1-17.

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.

Robertson, M. S., The sum of univalent functions, Duke Math. J. 37 (1970), 411-419.

Rogosinski, W., Uber positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), no. 1, 93-121.

Date of publication: 2016-07-27 21:54:10
Date of submission: 2016-07-26 21:50:50


Total abstract view - 805
Downloads (from 2020-06-17) - PDF - 446



  • There are currently no refbacks.

Copyright (c) 2011 Iwona Naraniecka, Jan Szynal, Anna Tatarczak