Starlikeness associated with lemniscate of Bernoulli

Abstract

For an analytic function f on the unit disk D = {z : |z| < 1} satisfying f (0) = 0 = f ′ (0) − 1, we obtain sufficient conditions so that f satisfies |(zf ′ (z)/f (z)) 2 − 1| < 1. The technique of differential subordination of first or second order is used. The admissibility conditions for lemniscate of Bernoulli are derived and employed in order to prove the main results.2010 Mathematics Subject Classification. 30C45; 30C80.

Third‐order differential subordinations for multivalent functions in the theory of source‐sink dynamics

“…In contrast, in Hästö et al, 22 the technique of differential subordinations was used to study the order of starlikeness and the boundedness of Gaussian hypergeometric functions. On the other hand, Madaan et al 23 studied a family of admissible functions associated with the lemniscate of Bernoulli—they succeeded in obtaining various first‐ and second‐order differential subordination results. Recently, Naz et al 24 have established some generalized first‐order differential subordination implications by defining the admissibility conditions for the exponential function.…”

Sufficiency for Nephroid Starlikeness using Hypergeometric Functions

“…Although similar type of differential subordination implication problems have been studied for several other function families (for instance see [2,4,6,8,11,14,21,22,24,29,36,37]), the approach of utilizing the properties of hypergeometric functions to arrive at the desired implication is totally new. In addition, this paper verifies analytically certain crucial facts which some of the above cited authors have concluded geometrically without providing any analytic clarification.…”

Lemniscate and Exponential Starlikeness of Regular Coulomb Wave Functions

“…In [24], the authors discussed the class of admissible functions Ψ[Ω, q] when the function q maps D onto a disk or a half-plane. Very recently, taking the subordinate function q(z) = √ 1 + z Madaan et al [21] discussed the admissible function class Ψ[Ω, √ 1 + z] and gave a particular case of the Theorem(1) as follow:…”

“…In the last few decades, many mathematicians started to investigate geometric properties (like univalence, starlikeness, convexity and close-to convexity) of some special functions including Bessel, Struve, Lommel, Wright and thier some generalizations. For these investigations the readers are referred to the papers [2][3][4][5][6]8,[11][12][13][14][15]19,[21][22][23]26,27,29,32] and the references therein. Also, the authors studied some geometric properties of regular Coulomb wave function in [9,10], while the author investigated the zeros of regular Coulomb wave functions and their derivatives in [16].…”