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Bounds for the second Hankel determinant of certain bi-univalent functions

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We investigate the second Hankel determinant inequalities for a certain class of analytic and bi-univalent functions. Some interesting applications of the results presented here are also discussed. | Turk J Math (2016) 40: 679 – 687 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1505-3 Research Article Bounds for the second Hankel determinant of certain bi-univalent functions Halit ORHAN1,∗, Nanjundan MAGESH2 , Jagadeesan YAMINI3 Department of Mathematics, Faculty of Science, Atat¨ urk University, Erzurum, Turkey 2 Post-Graduate and Research Department of Mathematics, Government Arts College for Men Krishnagiri, Tamil Nadu, India 3 Department of Mathematics, Government First, Grade College, Vijayanagar, Bangalore, Karnataka, India 1 • Received: 01.05.2015 Accepted/Published Online: 14.10.2015 • Final Version: 08.04.2016 Abstract: We investigate the second Hankel determinant inequalities for a certain class of analytic and bi-univalent functions. Some interesting applications of the results presented here are also discussed. Key words: Bi-univalent functions, bi-starlike, bi-Bˇ azileviˇc, second Hankel determinant 1. Introduction Let A denote the class of functions of the form f (z) = z + ∞ ∑ an z n , (1.1) n=2 which are analytic in the open unit disk U = {z : z ∈ C and |z| β; z ∈ U; 0 ≤ β β; z ∈ U; 0 ≤ β 0; z ∈ U . f (z) f (z) ∗Correspondence: orhanhalit607@gmail.com 2010 AMS Mathematics Subject Classification: 30C45. 679 ORHAN et al./Turk J Math In [17], it was shown that if the above analytical criteria hold for z ∈ U, then f is in the class of starlike functions S ∗ (0) for α real and is in the class of convex functions K(0) for α ≥ 1. In general, the class of α convexity. It is well known that every function f ∈ S has an inverse f −1 , defined by f −1 (f (z)) = z and f (f −1 (w)) = w (z ∈ U) ) ( 1 |w| 0 for c ∈ (0, 2), we conclude that Fγ1 γ1 Fγ2 γ2 − (Fγ1 γ2 )2 < .