Radii of Starlikeness and Convexity of Some Entire Functions
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Radii of Starlikeness and Convexity of Some Entire Functions Vibha Madaan1 · Ajay Kumar1 · V. Ravichandran2 Received: 13 June 2019 / Revised: 17 December 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract A normalized analytic function f is lemniscate starlike if the quantity z f (z)/ f (z) lies in the region bounded by the right half of the lemniscate of Bernoulli |w2 − 1| = 1. It is Janowski starlike if the quantity z f (z)/ f (z) lies in the disk whose diametric end points are (1 − A)/(1 − B) and (1 + A)/(1 + B) for −1 ≤ B < A ≤ 1. The radii of lemniscate starlikeness and Janowski starlikeness have been determined for normalizations of q-Bessel functions, Bessel functions of first kind of order ν and Lommel functions of first kind. Corresponding convexity radii are also determined. Keywords q-Bessel function · Lommel function · Lemniscate starlikeness · Janowski starlikeness · Radius problem Mathematics Subject Classification 30C10 · 30C15 · 30C45
1 Introduction Let A be the class of all analytic functions f defined on the open unit disk D and normalized by the conditions f (0) = 0 and f (0) = 1. The class S is the subclass of
Communicated by See Keong Lee. The first author is supported by Senior Research Fellowship from University Grants Commission, New Delhi, Ref. No.: 1069/(CSIR-UGC NET DEC, 2016).
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Ajay Kumar [email protected] Vibha Madaan [email protected] V. Ravichandran [email protected]; [email protected]
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Department of Mathematics, University of Delhi, New Delhi, Delhi 110 007, India
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Department of Mathematics, National Institute of Technology, Tiruchirappalli 620015, India
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V. Madaan et al.
A consisting of univalent functions. A function f ∈ A is starlike if t f (D) ⊂ f (D) for ∗ of lemniscate starlike functions introduced and studied by all t ∈ [0, 1]. The class SL Sokól and Stankiewicz [18] consists of all starlike functions f for which the quantity Q ST ( f )(z) = z f (z)/ f (z) satisfies (Q ST ( f )(z))2 − 1 < 1 for all z ∈ D and the ∗ . For class KL of the class of lemniscate convex functions consists of f with z f ∈ SL ∗ −1 ≤ B < A ≤ 1, the class S [A, B] of Janowski starlike functions consists of all starlike functions f satisfying |Q ST ( f )(z) − 1| < |A − B Q ST ( f )(z)| for all z ∈ D. The function f ∈ S ∗ [A, B] if the quantity Q ST ( f )(z) belongs to the disk whose diametric end points are (1 − A)/(1 − B) and (1 + A)/(1 + B). The class K[A, B] of Janowski convex functions (see [14]) consists of f with z f ∈ S ∗ [A, B]. The radius of ∗ ( f ) of a function f (defined on D or on a larger disk) is the lemniscate starlikeness rL largest positive real number r such that (Q ST ( f )(z))2 − 1 < 1 for all z with |z| < r . c ( f ) of Let Q C V ( f )(z) = 1 + z f (z)/ f (z). The radius of lemniscate convexity rL a function f is the largest positive real number r such that(Q C V ( f )(z))2 − 1 < 1 ∗ ( f ) or the radius of for all z with |z| < r . The radius of Janowski starlikeness r A,B c Ja
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