tailieunhanh - Second Hankel determinant for certain subclasses of bi-univalent functions

In the present paper, we obtain the upper bounds for the second Hankel determinant for certain subclasses of analytic and bi-univalent functions. Moreover, several interesting applications of the results presented here are also discussed. | Turk J Math (2017) 41: 694 – 706 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Second Hankel determinant for certain subclasses of bi-univalent functions 1,∗ ˘ ˙ 1 , Hari Mohan SRIVASTAVA2,3 Murat C ¸ AGLAR , Erhan DENIZ Department of Mathematics, Faculty of Science and Letters, Kafkas University, Kars, Turkey 2 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada 3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan, Republic of China 1 Received: • Accepted/Published Online: • Final Version: Abstract: In the present paper, we obtain the upper bounds for the second Hankel determinant for certain subclasses of analytic and bi-univalent functions. Moreover, several interesting applications of the results presented here are also discussed. Key words: Analytic functions, univalent functions, bi-univalent functions, subordination between analytic functions, Hankel determinant 1. Introduction and definitions Let A denote the family of functions f analytic in the open unit disk U = {z : z ∈ C and |z| β (z ∈ U; 0 ≦ β β (w ∈ U; 0 ≦ β 0. To establish our main results, we shall require the following lemmas. Lemma 1 (see, for example, [27]) If the function p ∈ P is given by the following series: p(z) = 1 + c1 z + c2 z 2 + · · · , () then the sharp estimate given by |ck | ≦ 2 (k = 1, 2, 3, · · · ) holds true. Lemma 2 (see [13]) If the function p ∈ P is given by the series (), then 2c2 = c21 + x(4 − c21 ), () ( 4c3 = c31 + 2(4 − c21 )c1 x − c1 (4 − c21 )x2 + 2(4 − c21 ) 1 − |x| 2 ) z () for some x and z with |x| ≦ 1 and |z| ≦ 1. 2. Main results Our first main result for the class f ∈ Nσ (β) is stated as follows: Theorem 1 Let f (z) given by () be in the class Nσ (β) (0 ≦ β < 1). Then ( ) ( [ √ ]) 2 (1−β)2 11− .