Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 721-731

ON THE PROBLEM OF GROMOVA AND VASIL'EV ON INTEGRAL MEANS, AND YAMASHITA'S CONJECTURE FOR SPIRALLIKE FUNCTIONS

Saminathan Ponnusamy and Karl-Joachim Wirths

Indian Statistical Institute (ISI), Chennai Centre, SETS, MGR Knowledge City
CIT Campus, Taramani, Chennai 600 113, India; samy 'at' isichennai.res.in, samy 'at' iitm.ac.in

Technische Universität Braunschweig, Institut für Analysis und Algebra
38106 Braunschweig, Germany; kjwirths 'at' tu-bs.de

Abstract. In this article we consider some integral means problem for certain classes of univalent analytic functions, in particular for the class of the starlike functions of order β and for the class of α-spirallike functions of order β. Our investigation settles one of the open problems of Gromova and Vasil'ev. In addition, we solve another problem concerning area maximum property of α-spirallike functions of order β in the setting of Yamashita and hence, we find the solution to Yamashita's conjecture for certain Dirichlet-finite functions in a general form.

2010 Mathematics Subject Classification: Primary 30C45, 30C70; Secondary 30H10, 33C05.

Key words: Analytic, univalent, convex, close-to-convex, starlike functions and spirallike functions, Dirichlet-finite, area integral, and Gaussian hypergeometric functions, Bessel's functions.

Reference to this article: S. Ponnusamy and K.-J. Wirths: On the problem of Gromova and Vasil'ev on integral means, and Yamashita's conjecture for spirallike functions. Ann. Acad. Sci. Fenn. Math. 39 (2014), 721-731.

Full document as PDF file

doi:10.5186/aasfm.2014.3922