Some remarks for a certain class of holomorphic functions at the boundary of the unit disc

Year 2019, Volume: 23 Issue: 3, 446 - 452, 01.06.2019

Bülent Nafi Örnek , Tuğba Akyel

https://doi.org/10.16984/saufenbilder.464294

Abstract

We consider a boundary version of the Schwarz Lemma on a certain class which is

denoted by K(alfa) . For the function  f(z)=z+c₂z²+....... which is defined in the unit disc E

such that the function f (z) belongs to the class K(alfa) , we estimate from below the modulus of the

angular derivative of the function ( )

( )

zf ' z

f z

at the boundary point b with ( ) = 1

( ) 1

bf ' b

f b 

. Moreover,

we get the Schwarz Lemma for the class K(alfa) . We also investigate some inequalities obtained in

terms of sharpness.

Keywords

Schwarz Lemma, Holomorphic function, Jack’s Lemma, Angular derivative

References

• H. P. Boas, Julius and Julia: Mastering the Art of the Schwarz lemma, Amer. Math. Monthly, 117 (2010), 770-785.
• D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129 (2001), 3275-3278.
• V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci. 122 (2004), 3623-3629.
• V. N. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci. 207 (2015), 825-831.
• G. M. Golusin, Geometric Theory of Functions of Complex Variable [inRussian], 2nd edn., Moscow 1966.
• I. S. Jack, Functions starlike and convex of order alfa , J. London Math. Soc. 3(1971), 469-474.
• M. Jeong, The Schwarz lemma and its applications at a boundary point, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 21 (2014), 275-284.
• D. M. Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary, J. Amer. Math. Soc. 7 (1994), 661-676.
• X. Tang and T. Liu, The Schwarz Lemma at the Boundary of the Egg Domain p1, p2 B in Cn , Canad. Math. Bull. 58 (2015), 381-392.
• X. Tang, T. Liu and J. Lu, Schwarz lemma at the boundary of the unit polydisk in Cn , Sci. China Math. 58 (2015), 1-14.
• M. Mateljević, Rigidity of holomorphic mappings & Schwarz and Jack lemma, DOI: 10.13140/RG.2.2.34140.90249, In press.
• R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), 3513–3517.
• T. Aliyev Azeroğlu and B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Variables and Elliptic Equations 58 (2013), 571–577.
• B. N. Örnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), 2053–2059.
• B. N. Örnek, Estimate for p-valently Functions at the Boundary, Bulletin of International Mathematical Virtual Institute, 7 (2017), 327-338.
• Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
• M. Elin, F. Jacobzon, M. Levenshtein, D. Shoikhet, The Schwarz lemma: Rigidity and Dynamics, Harmonic and Complex Analysis and its Applications. Springer International Publishing, (2014), 135-230.