Estimates for analytic functions concerned with Hankel determinant

B. N. Örnek

doi:10.37863/umzh.v73i9.907

B. N. Örnek Amasya Univ., Turkey

DOI: https://doi.org/10.37863/umzh.v73i9.907

Keywords: Fekete-Szegˆ functional, Hankel determinant, Jackís lemma, Analytic function, Schwarz lemma

Abstract

UDC 517.5

We give an upper bound of Hankel determinant of the first order $(H_{2}(1))$ for the classes of an analytic function.
In addition, an evaluation with the Hankel determinant from below will be given for the second angular derivative of $f(z)$ analytic function.
For new inequalities, the results of Jack's lemma and Hankel determinant were used.
Moreover, in a class of analytic functions on the unit disc, assuming the existence of an angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

References

T. Akyel, B. N. Örnek, Sharpened forms of the generalized Schwarz inequality on the boundary, Proc. Indian Acad. Sci. Math. Sci., 126, № 1, 69 – 78 (2016), https://doi.org/10.1007/s12044-015-0255-2 DOI: https://doi.org/10.1007/s12044-015-0255-2

T. A. Azeroǧlu, B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Var. and Elliptic Equat., 58, 571 – 577 (2013), https://doi.org/10.1080/17476933.2012.718338 DOI: https://doi.org/10.1080/17476933.2012.718338

H. P. Boas, Julius and Julia: mastering the art of the Schwarz lemma, Amer. Math. Monthly, 117, № 9, 770 – 785 (2010), https://doi.org/10.4169/000298910X521643 DOI: https://doi.org/10.4169/000298910x521643

V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci., 122, 3623 – 3629 (2004), https://doi.org/10.1023/B:JOTH.0000035237.43977.39 DOI: https://doi.org/10.1023/B:JOTH.0000035237.43977.39

G. M. Golusin, Geometric theory of functions of complex variable (in Russian), 2nd ed., Moscow (1966).

I. S. Jack, Functions starlike and convex of order $alpha$, J. London Math. Soc., 3, 469 – 474 (1971), https://doi.org/10.1112/jlms/s2-3.3.469 DOI: https://doi.org/10.1112/jlms/s2-3.3.469

M. Mateljevi´c, Rigidity of holomorphic mappings & Schwarz and Jack lemma; https://doi.org/10.13140/RG.2.2.34140.90249.

P. R. Mercer, Sharpened versions of the Schwarz lemma, J. Math. Anal. and Appl., 205, 508 – 511 (1997), https://doi.org/10.1006/jmaa.1997.5217 DOI: https://doi.org/10.1006/jmaa.1997.5217

P. R. Mercer, Boundary Schwarz inequalities arising from Rogosinski’s lemma, J. Class. Anal., 12, 93 – 97 (2018), https://doi.org/10.7153/jca-2018-12-08 DOI: https://doi.org/10.7153/jca-2018-12-08

P. R. Mercer, An improved Schwarz lemma at the boundary, Open Math., 16, 1140 – 1144 (2018), https://doi.org/10.1515/math-2018-0096 DOI: https://doi.org/10.1515/math-2018-0096

R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc., 128, 3513 – 3517 (2000), https://doi.org/10.1090/S0002-9939-00-05463-0 DOI: https://doi.org/10.1090/S0002-9939-00-05463-0

B. N.Örnek, T. Du¨zenli, Bound estimates for the derivative of driving point impedance functions, Filomat, 32, № 18, 6211 – 6218 (2018), https://doi.org/10.2298/fil1818211o DOI: https://doi.org/10.2298/FIL1818211O

B. N. Örnek, T. Du¨zenli, Boundary analysis for the derivative of driving point impedance functions, IEEE Trans. Circuits and Syst. Pt. II: Express Briefs, 65, № 9, 1149 – 1153 (2018) DOI: https://doi.org/10.1109/TCSII.2018.2809539

B. N. O¨ rnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc., 50, № 6, 2053 – 2059 (2013), https://doi.org/10.4134/BKMS.2013.50.6.2053 DOI: https://doi.org/10.4134/BKMS.2013.50.6.2053

Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin (1992), https://doi.org/10.1007/978-3-662-02770-7 DOI: https://doi.org/10.1007/978-3-662-02770-7

Ch. Pommerenke, On the Hankel determinants of univalent functions, Matematika, 14, 108 – 112 (1967), https://doi.org/10.1112/S002557930000807X DOI: https://doi.org/10.1112/S002557930000807X

J. Sok´ol, D. K. Thomas, The second Hankel determinant for alpha-convex functions, Lith. Math. J., 58, №2, 212 – 218 (2018); https://doi.org/10.1007/s10986-018-9397-0. DOI: https://doi.org/10.1007/s10986-018-9397-0

G. Szeg¨o, M. Fekete, Eine Bemerkung ¨uber ungerade schlichte Funktionen, J. London Math. Soc., 2, 85 – 89 (1933), https://doi.org/10.1112/jlms/s1-8.2.85 DOI: https://doi.org/10.1112/jlms/s1-8.2.85

D. K. Thomas, J. W. Noonan, On the second Hankel determinant of areally mean $p$-valent functions, Trans. Amer. Math. Soc., 223, 337 – 346 (1976), https://doi.org/10.2307/1997533 DOI: https://doi.org/10.2307/1997533