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 doi: 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: Accepted/Published Online: • Final Version: 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 , () 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@ 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 < .
