On Booth lemniscate and Hadamard product of analytic functions

Year, pages: 2015, 1337 - 1344

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 Polya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely $\mathcal{K}*\mathcal{K}=\mathcal{K}$. They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class $\mathcal K$. 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.

Piejko, K., Sokół, J. 2015. On Booth lemniscate and Hadamard product of analytic functions. In Mathematica Slovaca, vol. 65, no.6, pp. 1337-1344. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0093

Piejko, K., Sokół, J. (2015). On Booth lemniscate and Hadamard product of analytic functions. Mathematica Slovaca, 65(6), 1337-1344. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0093