Abstract
We solve problems concerning the coefficients of functions in the class \(\mathcal {T}(\lambda )\) of typically real functions associated with Gegenbauer polynomials. The main aim is to determine the estimates of two expressions: \(|a_4-a_2 a_3|\) and \(|a_2 a_4 -a_3{}^2|\). The second one is known as the second Hankel determinant. In order to obtain these bounds, we consider the regions of variability of selected pairs of coefficients for functions in \(\mathcal {T}(\lambda )\). Furthermore, we find the upper and the lower bounds of functionals of Fekete–Szegö type. Finally, we present some conclusions for the classes \(\mathcal {T}\) and \(\mathcal {T}(1/2)\).
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Zaprawa, P., Figiel, M. & Futa, A. On Coefficients Problems for Typically Real Functions Related to Gegenbauer Polynomials. Mediterr. J. Math. 14, 99 (2017). https://doi.org/10.1007/s00009-017-0863-4
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DOI: https://doi.org/10.1007/s00009-017-0863-4