Sharp Coefficients Bounds for Starlike Functions Associated with Generalized Telephone Numbers
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Sharp Coefficients Bounds for Starlike Functions Associated with Generalized Telephone Numbers Erhan Deniz1 Received: 11 May 2020 / Revised: 17 August 2020 / Accepted: 1 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we introduced the class ST∗ (λ) of analytic functions which is related to starlike functions and generating function of generalized telephone numbers. By using bounds on some coefficient functionals for the family of functions with positive real part, we obtain for functions in the class ST∗ (λ) several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Keywords Univalent functions · Starlike function · Generalized telephone numbers · Hankel determinant Mathematics Subject Classification Primary 30C45 · Secondary 30C50 · 30C80
1 Introduction Let A be the class of functions f which are analytic in the open unit disc U = {z : |z| < 1} and normalized by the conditions f (0) = f (0)−1 = 0. Let us denote by S the subclass of A containing functions which are univalent in U. An analytic function f is subordinate to an analytic function g (written as f ≺ g) if there exists an analytic function w with w (0) = 0 and |w (z)| < 1 for z ∈ U such that f (z) = g (w (z)) . In particular, if g is univalent in U, then f (0) = g (0) and f (U) ⊂ g (U) . For arbitrary fixed numbers A and B satisfying −1 ≤ B < A ≤ 1, denote by P [A, B] the class of analytic functions p such that p (0) = 1 and satisfy the subordination p (z) ≺ (1 + Az) (1 + Bz) (z ∈ U) . Note that for 0 ≤ β < 1, P [1 − 2β, −1] is the class of analytic functions p with p (0) = 1 satisfying p (z) > β in U. We call the
Communicated by V. Ravichandran.
B 1
Erhan Deniz [email protected] Department of Mathematics, Faculty of Science and Letters, Kafkas University, Kars, Turkey
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E. Deniz
functions in P = P [1, −1] as Carathéodory functions. The class S ∗ [A, B] consists of functions f ∈ A such that z f (z) f (z) ∈ P [A, B] for z ∈ U. The functions in the class S ∗ [A, B] are called the Janowski starlike functions, introduced by Janowski [11]. For 0 ≤ β < 1, S ∗ [1 − 2β, −1] := S ∗ (β) is the usual class of starlike functions of order β. Note that S ∗ = S ∗ 0) is the classical class of starlike functions. Moreover, the classes S ∗ [1 − β, 0] := Sβ∗ = { f ∈ A : |z f (z) f (z) − 1| < 1 − β} and ∗ (β) = { f ∈ A : |z f (z) f (z) − 1| < β|z f (z) f (z) + 1|} have S ∗ [β, −β] := S been studied in [1,2]. In terms of subordination, the class of starlike functions is given by z f (z) f (z) ≺ (1 + z)(1 − z). Ma and Minda [19] gave a unified presentation of various subclasses of starlike and convex functions by replacing the subordinate function (1 + z)(1 − z) by a more general analytic function ϕ with positive real part and normalized by the conditions ϕ(0) = 1, ϕ (0) > 0, and ϕ maps U onto univalently a region starlike with respect to 1 and symmetric with respect to the real axis.
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