SOME RESULTS ON JACK’S LEMMA FOR ANALYTIC FUNCTIONS

Year 2022, , 31 - 40, 31.12.2022

Bülent Nafi Örnek

https://doi.org/10.54559/jauist.1207927

Abstract

In this paper, an upper bound will be found for the second coefficient in the Taylor expansion of the analytical function $p(z)$ using the Jack lemma. Also, the modulus of the angular derivative of the $I_{f}(z)=\frac{zp^{\prime }(z)}{p(z)}$ function on the unit disc will be estimated from below.

Keywords

Analytic function, Jack’s lemma, Schwarz lemma.

References

• [1] Akyel. T. (2022). Estimates for λ-Spirallike Functions of Complex Order on the Boundary, Ukrainian Mathematical Journal, 74, 1-14.
• [2] Azeroğlu, T. A. and Örnek, B. N. (2013). A refined Schwarz inequality on the boundary, Complex Variab. Elliptic Equa., 58, 571-577.
• [3] Boas, H. P. (2010). Julius and Julia: Mastering the Art of the Schwarz lemma, Amer. Math. Monthly, 117, 770-785.
• [4] Dubinin, V. N. (2004). The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci., 122, 3623-3629.
• [5] Golusin G. M. (1996). Geometric Theory of Functions of Complex Variable [in Russian], 2nd edn., Moscow.
• [6] Jack, I. S. (1971). Functions starlike and convex of order α, J. London Math. Soc., 3, 469-474.
• [7] Mateljevic, M., Mutavdžć, N. and Örnek B. N. (2022), Estimates for some classes of holomorphic functions in the unit disc, Appl. Anal. Discrete Math., 16, 111-131.
• [8] Mercer, P. R. (2018). Boundary Schwarz inequalities arising from Rogosinski’s lemma, Journal of Classical Analysis, 12, 93-97.
• [9] Mercer, P. R. (2018). An improved Schwarz Lemma at the boundary, Open Mathematics, 16, 1140-1144.
• [10] Osserman, R. (2000). A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc., 128, 3513-3517.
• [11] Örnek, B. N. (2016). The Carathéodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc, Journal of Mathematical Physics, Analysis, Geometry, 12(4), 287-301.
• [12] Örnek, B. N. and Düzenli, T. (2018). Boundary Analysis for the Derivative of Driving Point Impedance Functions, IEEE Transactions on Circuits and Systems II: Express Briefs, 65(9), 1149-1153.
• [13] Örnek B. N., Aydemir S. B., Düzenli T. and Özak B. (2022). Some remarks on activation function design in complex extreme learning using Schwarz lemma, Neurocomputing, 492, 23-33.
• [14] Pommerenke, Ch. (1992). Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin. [15] Unkelbach, H. (1938). Über die Randverzerrung bei konformer Abbildung, Math. Z., 43, 739-742.