DOI: 10.1515/ms-2017-0354 Math. Slovaca 70 (2020), No. 2, 319–328 INVESTIGATION OF THE FIFTH HANKEL DETERMINANT FOR A FAMILY OF FUNCTIONS WITH BOUNDED TURNINGS Muhammad Arif* — Inayat Ullah* — Mohsan Raza** — Pawe l Zaprawa*** (Communicated by Stanislawa Kanas ) ABSTRACT. The main aim of this paper is to study the fifth Hankel determinant for the class of functions with bounded turnings. The results are also investigated for 2-fold symmetric and 4-fold symmetric functions. c 2020 Mathematical Institute Slovak Academy of Sciences 1. Introduction and definitions Let the notation A represent the set all functions f which are analytic or holomorphic in the region D = {z ∈ C : |z| < 1} and normalized by the conditions f (0) = f ′ (0) − 1=0. Alternatively, if f ∈A, then it has the Taylor-Maclaurin series f (z)= z + ∞ n=2 a n z n (z ∈ D) . (1.1) Let S denote the class of all functions in A which are univalent in D. Also, let S ∗ , C and K g denote the classes of starlike, convex, close-to-convex respectively and are analytically defined as S ∗ = f ∈S : zf ′ (z) f (z) ∈P , (z ∈ D) , C = f ∈S : (zf ′ (z)) ′ f ′ (z) ∈P , (z ∈ D) , K g = f ∈S : zf ′ (z) g (z) ∈P , for g (z) ∈S ∗ , (z ∈ D) , where P denotes the class of analytic functions p such that Re p(z) > 0 in D and having the series form: p (z)=1+ ∞ n=1 c n z n ,z ∈ D. (1.2) For g (z)= z in the class K g , we have the class R of bounded turning defined as R = f ∈S : f ′ (z) ∈P , (z ∈ D) . 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 30C45, 30C50. K e y w o r d s: Bounded turning functions, Hankel determinant. 319 Brought to you by | Uppsala University Library Authenticated Download Date | 3/12/20 3:55 PM