Sains Malaysiana 44(1)(2015): 155–158

 

On Starlikeness of Libera Transform

(Jelmaan Libera Seperti Bintang)

 

 

MAMORU NUNOKAWA1 & JANUSZ SOKÓŁ2*

 

1University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba 260-0808, Japan

 

2Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12,

35-959 RzeszówPoland

 

Received: 18 January 2014/Accepted: 15 March 2014

 

ABSTRACT

Let A denote the class of functions f that are analytic in the unit disc D and normalized by f (0) = f ʹ(0) 1 = 0. In this paper conditions are determined for starlikeness of the Libera integral of functions such that arg {f ʹ} < δ in D. The results of Miller and Mocanu (2002) on Libera transform of functions with bounded turning are improved.

 

Keywords: Nunokawa’s lemma; starlike functions; strongly starlike functions

 

ABSTRAK

Biarkan  A menandakan kelas fungsi f yang analitik dalam cakera unit  D dan dinormalkan oleh f (0) = f' (0) 1 = 0. Dalam kertas ini keadaan ditentukan untuk fungsi kamiran Libera seperti bintang adalah arg {f ʹ} < δ dalam D . Keputusan Miller dan Mocanu (2002) atas transformasi fungsi Libera dengan pusingan terbatas bertambah baik.

Kata kunci: Fungsi seperti bintang; fungsi seperti bintang yang kuat; lema Nunokawa

REFERENCES

Brannan, D.A. & Kirwan, W.E. 1969. On some classes of bounded univalent functions. J. London Math. Soc. 1(2): 431-443.

Miller, S.S. & Mocanu, P.T. 2002. Libera transform of functions with bounded turning. J. Math. Anal. Appl. 276: 90-97.

Mocanu, P.T. 1995. New extensions of a theorem of R. Singh and S. Singh. Mathematica (Cluj) 37(60): 171-182.

Mocanu, P.T. 1986. On starlikness of Libera transform. Mathematica (Cluj) 28(51): 153-155.

Nunokawa, M. 1993. On the order of strongly starlikeness of strongly convex functions. Proc. Japan Acad. 69 (Ser. A): 234-237.

Nunokawa, M., Cho, N.E., Kwon, O.S. & Sokół, J. On extensions of Mocanu’s P.T. results, submitted for publication.

Robertson, M.S. 1936. On the theory of univalent functions. Ann. Math. 37: 374-408.

Sokół, J. 2008. Starlikeness of the Libera transform of functions with bounded turning. Appl. Math. Comput. 203: 273-276.

Stankiewicz, J. 1966. Quelques probl`emes extremaux dans les classes des fonctions -angulairement etoiles. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 20: 59-75.

 

 

*Corresponding author; email: jsokol@prz.edu.pl

 

 

previous