On Booth Lemniscate and Hadamard Product of Analytic Functions

Abstract

In [RUSCHEWEYH, S.-SHEIL-SMALL, T.: Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135] the authors proved the P`olya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely K ∗ K = K. They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class K. In this paper we consider similar convolution problems for some classes of functions. Especially we give a new applications of a result [SOKÓŁ, J.: Convolution and subordination in the convex hull of convex mappings, Appl. Math. Lett. 19 (2006), 303-306] on the subordinating relations in the convex hull of convex mappings under convolution. The paper deals with several ideas and techniques used in geometric function theory. Besides being an application to those results it provides interesting corollaries concerning special functions, regions and curves.

Starlike functions associated with the generalized Koebe function

“…The function F q (z) was studied in [27,28]. The function F q (z) is a starlike univalent when q 2 < 1.…”

Analytic representation of the generalized Pascal snail and its applications

“…where G is the family of function univalent, convex in one direction, and satisfying {1 + zf (z)/f (z)} > −1/2, see [12]. [8]. Summarizing, T α,β,γ is a analytic univalent function with positive real part in D, T α,β,γ (D) is symmetric with respect to the real axis, starlike with respect to T α,β,γ (0) = 1 and convex in one direction under some conditions on α and β.…”

Differential subordination and radius estimates for starlike functions associated with the Booth lemniscate

“…In 2015, Piejko and Sokó l [24] proved that the function z/(1 − αz 2 ) is convex univalent for 0 ≤ α < 3 − 2 √ 2 and also discussed some convolution properties related to the functions in the class BS * (α). Using the representation formula, we see that the function…”

“…In [11], various radius problems and subordination results were also discussed for some subclasses of analytic functions. For more details, see [24]. The Booth lemniscate is a special case of the Persian curve [29] and it was named after Booth, an Irish mathematician, who studied it in 1873.…”