Radius of Starlikeness for Classes of Analytic Functions
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Radius of Starlikeness for Classes of Analytic Functions See Keong Lee1 · Kanika Khatter2
· V. Ravichandran3
Received: 17 May 2020 / Revised: 28 August 2020 / Accepted: 16 September 2020 / Published online: 12 October 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We consider normalized analytic function f on the open unit disk for which either Re f (z)/g(z) > 0, | f (z)/g(z) − 1| < 1 or Re(1 − z 2 ) f (z)/z > 0 for some analytic function g with Re(1 − z 2 )g(z)/z > 0. We have obtained the radii for these functions to belong to various subclasses of starlike functions. The subclasses considered include the classes of starlike functions of order α, lemniscate starlike functions and parabolic starlike functions. Keywords Starlike functions · Exponential function · Lemniscate of Bernoulli · Radius problems · Coefficient estimate Mathematics Subject Classification 30C45 · 30C50 · 30C80
1 Introduction Let Dr := {z ∈ C : |z| < r } be an open disk of radius r centered at the origin and D = D1 be the open unit disk in C. For any two classes G and H of analytic functions defined on the unit disk D, the H-radius for the class G, denoted by RH (G) (or just
Dedicated to Prof. K. G. Subramanian on the occasion of his 72nd birthday Communicated by Rosihan M. Ali.
B
Kanika Khatter [email protected] See Keong Lee [email protected] V. Ravichandran [email protected]; [email protected]
1
School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia
2
Department of Mathematics, Hindu Girls College, Sonipat, Haryana 131001, India
3
Department of Mathematics, National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India
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RH if the class G is understood in the context), is the largest radius ρ ≤ 1 such that f ∈ G implies the function fr , defined by fr (z) = f (r z)/r , belongs to class H for all 0 < r < ρ. Among the radius problems for various subclasses of analytic functions, one direction of study focuses on obtaining the radius for classes consisting of functions characterized by ratio of the function f and another function g, where g is a function belonging to some special subclass of A of all analytic functions on D normalized by f (0) = 0 = f (0) − 1. MacGregor [11,12] obtained the radius of starlikeness for the class of functions f ∈ A satisfying either Re( f (z)/g(z)) > 0 or | f (z)/g(z) − 1| < 1 for some g ∈ K. Ali et al. [2] estimated several radii for classes of functions satisfying either (i) Re( f (z)/g(z)) > 0, where Re(g(z)/z) > 0 or Re(g(z)/z) > 1/2; (ii) | f (z)/g(z) − 1| < 1, where Re(g(z)/z) > 0 or g is convex; (iii) | f (z)/g (z) − 1| < 1, where Reg (z) > 0. The work is further investigated in [7,20]. These classes are related to the Caratheodory class P consisting of all analytic functions p with p(0) = 1 and Rep(z) > 0 for all z ∈ D. Motivated by the aforesaid studies, we consider the following three classes K1 , K2 , and K3 defined by 1 − z2 f (z) ∈ P, for some g ∈ A satis
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