# The norm of pre-Schwarzian derivatives on bi-univalent functions of order $\alpha$

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O‎. ‎Box 19395-3697 Tehran‎, ‎Iran.

Abstract

‎In the present investigation‎, ‎we give the best estimates for the norm of the pre-Schwarzian derivative $T_{f}(z)=\dfrac{f^{''}(z)}{f^{'}(z)}$ for bi-starlike functions and a new subclass of bi-univalent functions of order $\alpha$‎, ‎where‎ ‎$\Vert T_{f} \Vert= \sup_{|z|<1} (1-|z|^{2})|\dfrac{f^{''}(z)}{f^{'}(z)}|$.

Keywords

Main Subjects

### References

J. Becker, Lownersche differentialgleichung and quasikonform fortsetzbare schlichte funktionen, J. Reine Angew. Math. 255 (1972) 23--43.

J. Becker and Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math 354 (1984) 74--94.

P.L. Duren, Univalen Functions, Springer, New York 1978.

Y.C Kim and T. Sugawa, Norm estimates of the pre-schwarzian derivatives for certain classes of univalent functions, Proc. Edinb. Math. Soc. (2) 49 (2006), no. 1, 131--143.

Y. Okuyama, The norm estimates of pre-schwarzian derivatives of spiral-like functions, Complex Var. Theory Appl. 42 (2000), no. 3, 225--239.

S. Porwal and M. Darus, On a new subclass of bi-univalent functions, J. Egyptian Math. Soc. 21 (2013), no. 3, 190--193.

H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), no. 10, 1188--1192.

S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J. 28 (1999) 217--230.

### History

• Receive Date: 13 June 2015
• Revise Date: 27 March 2016
• Accept Date: 31 March 2016