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In this paper we consider the Hankel determinant \(H_2(3) = a_3a_5 - a_4{}^2\) defined for the coefficients of a function f which belongs to the class \(\mathcal {S}\) of univalent functions or to its subclasses: \(S^*\) of starlike functions, \(\mathcal {K}\) of convex functions and \(\mathcal {R}\) of functions whose derivative has a positive real part. Bounds of \(|H_2(3...

The main aim of this paper is to discuss the third Hankel determinants for three classes: \(S^*\) of starlike functions, \(\mathcal {K}\) of convex functions and \(\mathcal {R}\) of functions whose derivative has a positive real part. Moreover, the sharp results for twofold and threefold symmetric functions from these classes are obtained.

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...

November 2015; Revised 22 December 2015; Accepted 3 January 2016
Academic Editor: Marco Donatelli
Copyright © 2016 **Paweł** **Zaprawa**. This is an open access article distributed under the Creative Commons

November 2015; Revised 22 December 2015; Accepted 3 January 2016
Academic Editor: Marco Donatelli
Copyright © 2016 **Paweł** **Zaprawa**. This is an open access article distributed under the Creative Commons

Let \(D\subset \mathbb {C}\) and \(0\in D\). A set D is circularly symmetric if, for each \(\varrho \in \mathbb {R}^+\), a set \(D\cap \{\zeta \in \mathbb {C}:|\zeta |=\varrho \}\) is one of three forms: an empty set, a whole circle, a curve symmetric with respect to the real axis containing \(\varrho \). A function f analytic in the unit disk \(\Delta \equiv \{\zeta \in \mathbb...

Let \({\mathcal {F}}\) denote the class of all functions univalent in the unit disk \(\Delta \equiv \{\zeta \in {\mathbb {C}}\,:\,\left| \zeta \right| <1\}\) and convex in the direction of the real axis. The paper deals with the subclass \({\mathcal {F}}^{(n)}\) of these functions \(f\) which satisfy the property \(f(\varepsilon z)=\varepsilon f(z)\) for all \(z\in \Delta...