Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination

Abstract

Several radii problems are considered for functions f (z) = z + a 2 z 2 + • • • with fixed second coeffcient a 2. For 0 ≤ β < 1, sharp radius of starlikeness of order β for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order β for uniformly convex functions, and sharp radius of strong-starlikeness of order γ for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases. 1. Motivation and a survey Let A denote the class of analytic functions f defined in the open unit disc D := {z ∈ C : |z| < 1} and normalized by the conditions f (0) = 0 and f ′ (0) = 1. Let S be its subclass consisting of univalent analytic functions. Thus functions in S has the form f (z) = z + a 2 z 2 + • • •. Gronwall [13] obtained lower and upper bounds for the quantities | f (z)| and | f ′ (z)| for univalent functions with preassigned second coefficient a 2. Corresponding results for convex functions were also obtained. Unaware of these results, Finkelstein [8] investigated the problem again and obtained similar results, except for an inaccurate lower bound for | f (z)|. Corresponding estimates for starlike functions of positive order with fixed second coefficient were obtained by Tepper [41], while for convex functions of positive order, such estimates were derived by Padmanabhan [24]. The problem for general classes of functions defined by subordination was investigated by Padmanabhan [27] in 2001. For close-to-star and close-to-convex functions, such estimates were investigated by Al-Amiri [4] and Silverman [33], respectively. In addition to distortion and growth estimates, Tepper [41] obtained the radius of convexity for starlike functions with fixed second coefficient. This radius result was also obtained independently by Goel [11], whom additionally obtained the radius of starlikeness for functions f with fixed second coefficient satisfying Re(f (z)/z) > 0 for z ∈ D. Following these works, several authors have investigated radii problems for functions with fixed second coefficient; we provide here a brief history of these works.

On a Subclass of Strongly Starlike Functions Associated with Exponential Function

“…There are still many open problems concerning determination of sharp coefficient bounds for various subclasses of S such as S P , S * L and S * RL (see [7,19,27,44]). The correspondence between the classes S * e and K e and [23, Theorem 3, p. 164] yield the sharp upper bound for the Fekete-Szegö functional |a 3 − μa 2 2 | in the class S * e for all real μ. If f (z) = z + a 2 z 2 + a 3 z 3 + · · · ∈ S * e , then…”

“…Similarly, S * L := S * ( √ 1 + z) is the subclass of S * introduced by Sokół and Stankiewicz [46], consisting of functions f ∈ A such that z f (z)/ f (z) lies in the domain bounded by the right-half of the lemniscate of Bernoulli given by |w 2 − 1| < 1. More results regarding these classes can be found in [2,4,7,10,13,22,29,30,32,[40][41][42][43][44][45]. Recently, the authors [27] discussed the properties of the class…”

Convex combination of analytic functions

“…Let A be the class of analytic functions f defined on the unit disc D D fz 2 C W jzj < 1g, and normalized by For recent investigation on the class SL, see [1][2][3][4][5]. Another class of our interest is the class Mˇ,ˇ> 1, consisting of f 2 A satisfying Re .zf 0 .z/=f .z// <ˇfor all z 2 D. The class Mˇwas investigated by Uralegaddi et al [6], while its subclass was investigated by Owa and Srivastava [7].…”

Sufficient Conditions for Starlikeness

“…Similarly, S * L := S * ( √ 1 + z) is the subclass of S * introduced by Sokól and Stankiewicz [18], consisting of functions f ∈ A such that zf ′ (z)/f (z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w 2 − 1| < 1. More results regarding these classes can be found in [1,3,5,11,13,16,17]. Recently, Sharma et al [14] introduced and studied the properties of the class S * (1 + (4/3)z + (2/3)z 2 ) = S * C .…”

“…Recently Ali et al [2] determined the condition on β for p(z) ≺ √ 1 + z when 1 + βzp ′ (z)/p n (z) with n = 0, 1, 2 or (1 − β)p(z) + βp 2 (z) + βzp ′ (z) is subordinated to √ 1 + z. Motivated by the works in [1,2,3,9,15,17], in Section 2, we determine the sharp conditions on β so that p(z)…”