Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
Bloch function; meromorphic function; Landau's reduction; Taylor coefficient
Bloch function; meromorphic function; Landau's reduction; Taylor coefficient
Summary:
It is known that if $f$ is holomorphic in the open unit disc ${\mathbb D}$ of the complex plane and if, for some $c>0$, $|f(z)|\leq 1/(1-|z|^2)^c$, $z\in {\mathbb D}$, then $|f'(z)|\leq 2(c+1)/(1-|z|^2)^{c+1}$. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in ${\mathbb D}$. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.
References:
[1] Ahlfors, L. V., Grunsky, H.: Über die Blochsche Konstante. Math. Z. 42 (1937), 671-673 German. DOI 10.1007/BF01160101 | MR 1545698 | Zbl 0016.30902
[2] Anderson, J. M., Clunie, J., Pommerenke, C.: On Bloch functions and normal functions. J. Reine Angew. Math. 270 (1974), 12-37. DOI 10.1515/crll.1974.270.12 | MR 0361090 | Zbl 0292.30030
[3] Bhowmik, B., Sen, S.: Improved Bloch and Landau constants for meromorphic functions. Can. Math. Bull. 66 (2023), 1269-1273. DOI 10.4153/S0008439523000346 | MR 4658218 | Zbl 1526.30037
[4] Bhowmik, B., Sen, S.: Landau and Bloch constants for meromorphic functions. Monatsh. Math. 201 (2023), 359-373. DOI 10.1007/s00605-023-01839-w | MR 4581493 | Zbl 1515.30072
[5] Bloch, A.: Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation. C.R. Acad. Sci. Paris 178 (1924), 2051-2052 French \99999JFM99999 50.0217.02. MR 1508386
[6] Bloch, A.: Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation. Ann. Fac. Sci. Univ. Toulouse, III. Ser. 17 (1925), 1-22 French \99999JFM99999 52.0324.02. DOI 10.5802/afst.335 | MR 1508386
[7] Bonk, M.: Extremalprobleme bei Bloch-Funktionen: Ph. D. Thesis. Technische Universität Braunschweig, Braunschweig (1988), German. Zbl 0663.30030
[8] Chen, H., Gauthier, P. M.: On Bloch's constant. J. Anal. Math. 69 (1996), 275-291. DOI 10.1007/BF02787110 | MR 1428103 | Zbl 0864.30025
[9] Duren, P. L., Shaprio, H. S., Shields, A. L.: Singular measures and domains not of Smirnov type. Duke Math. J. 33 (1966), 247-254. DOI 10.1215/S0012-7094-66-03328-X | MR 0199359 | Zbl 0174.37501
[10] Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Pure and Applied Mathematics 255. Marcel Dekker, New York (2003). DOI 10.1201/9780203911624 | MR 2017933 | Zbl 1042.30001
[11] Kayumov, I. R., Wirths, K.-J.: Coefficient problems for Bloch functions. Anal. Math. Phys. 9 (2019), 1069-1085. DOI 10.1007/s13324-019-00303-z | MR 4014857 | Zbl 1430.30017
[12] Kayumov, I. R., Wirths, K.-J.: On the sum of squares of the coefficients of Bloch functions. Monatsh. Math. 190 (2019), 123-135. DOI 10.1007/s00605-019-01321-6 | MR 3998335 | Zbl 1420.30015
[13] Kayumov, I. R., Wirths, K.-J.: Inequalities of Carlson type for $\alpha$-Bloch functions. Mediterr. J. Math. 17 (2020), Article ID 83, 9 pages. DOI 10.1007/s00009-020-01519-1 | MR 4099644 | Zbl 1441.30049
[14] Liu, M.-C.: On the derivative of some analytic functions. Math. Z. 132 (1973), 205-208. DOI 10.1007/BF01213864 | MR 0323999 | Zbl 0257.30001
[15] Pommerenke, C.: On Bloch functions. J. Lond. Math. Soc., II. Ser. 2 (1970), 689-695. DOI 10.1112/jlms/2.Part_4.689 | MR 0284574 | Zbl 0199.39803
[16] Problem N in a list of problems on Analytic Function Theory drawn up by members of a Symposium held at the University of Kentucky May 27--June 1, 1965. Bull. Am. Math. Soc. 71 (1965), 855-857. DOI 10.1090/S0002-9904-1965-11414-8 | MR 1566376
[17] Raupach, E.: Eine Abschätzungsmethode für die reellwertigen Lösungen der Differentialgleichung $\Delta \alpha=-4/(1-z\bar{z})^2$. Bonn. Math. Schr. 9 (1960), 124 pages German. MR 0125234 | Zbl 0124.04201
[18] Robertson, M. S.: A distortion theorem for analytic functions. Proc. Am. Math. Soc. 28 (1971), 551-556. DOI 10.1090/S0002-9939-1971-0281901-6 | MR 0281901 | Zbl 0222.30022
[19] Wirths, K.-J.: Über holomorphe Funktionen, die einer Wachstumsbeschränkung unterliegen. Arch. Math. 30 (1978), 606-612 German. DOI 10.1007/BF01226108 | MR 0492223 | Zbl 0373.30016
[20] Zhaou, R.: On $\alpha$-Bloch functions and VMOA. Acta Math. Sci. 16 (1996), 349-360. DOI 10.1016/S0252-9602(17)30811-1 | MR 1415692 | Zbl 0938.30024
Search
Partner of
