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On a new subclass of bi-univalent functions defined by using Salagean operator

In this manuscript, by using the Salagean operator, new subclasses of bi-univalent functions in the open unit disk are defined. Moreover, for functions belonging to these new subclasses, upper bounds for the second and third coefficients are found. | Turk J Math (2018) 42: 2891 – 2896 © TÜBİTAK doi:10.3906/mat-1507-100 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article On a new subclass of bi-univalent functions defined by using Salagean operator Bilal ŞEKER∗, Department of Mathematics, Faculty of Science, Dicle University, Diyarbakır, Turkey Received: 28.07.2015 • Accepted/Published Online: 23.03.2017 • Final Version: 27.11.2018 Abstract: In this manuscript, by using the Salagean operator, new subclasses of bi-univalent functions in the open unit disk are defined. Moreover, for functions belonging to these new subclasses, upper bounds for the second and third coefficients are found. Key words: Univalent functions, bi-univalent functions, coefficient bounds and coefficient estimates, Salagean operator 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 ∈ C : |z| 0, p(z) = 1 + c1 z + c2 z 2 + . for z ∈ U. 2. Coefficient bounds for the function class HΣm,n (α) By introducing the function class HΣm,n (α) , we start by means of the following definition. Definition 2.1 A function f (z) given by (1.1) is said to be in the class HΣm,n (α) (0 n) if the following conditions are satisfied: ( m ) D f (z) απ f ∈ Σ and arg n) . Then 2α |a2 | ≤ √ m n m 2α(3 − 3 ) + (2 − 2n )2 − α(22m − 22n ) (2.3) and |a3 | ≤ 2892 4α2 2α + . 3m − 3n (2m − 2n )2 (2.4) ŞEKER/Turk J Math Proof It can be written that the inequalities (2.1) and (2.2) are equivalent to Dm f (z) α = [p(z)] Dn f (z) (2.5) Dm g(w) α = [q(w)] Dn g(w) (2.6) and where p(z) and q(w) are in P and have the forms p(z) = 1 + p1 z + p2 z 2 + p3 z 3 + · · · (2.7) q(w) = 1 + q1 w + q2 w2 + q3 w3 + · · · . (2.8) and Now, equating the coefficients in (2.5) and (2.6), we obtain (2m − 2n )a2 = αp1 (2.9) (3m − 3n )a3 − 2n (2m − 2n )a22 = αp2 + α(α − 1) 2 p1 2 (2.10) −(2m − 2n )a2 = αq1 (2.11) and (3m −