×

Classes of harmonic functions defined by subordination. (English) Zbl 1345.31001

Summary: New classes of univalent harmonic functions are introduced. We give sufficient coefficient conditions for these classes. These coefficient conditions are shown to be also necessary if certain restrictions are imposed on the coefficients of these harmonic functions. By using extreme points theory we also obtain coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. Radii of convexity and starlikeness of the classes are also considered.

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bulletin of the American Mathematical Society, 42, 10, 689-692 (1936) · Zbl 0015.15903 · doi:10.1090/s0002-9904-1936-06397-4
[2] Clunie, J.; Sheil-Small, T., Harmonic univalent functions, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, 9, 3-25 (1984) · Zbl 0506.30007 · doi:10.5186/aasfm.1984.0905
[3] Sheil-Small, T., Constants for planar harmonic mappings, Journal London Mathematical Society, 2, 42, 237-248 (1990) · Zbl 0731.30012
[4] Janowski, W., Some extremal problems for certain families of analytic functions I, Annales Polonici Mathematici, 28, 297-326 (1973) · Zbl 0275.30009
[5] Jahangiri, J. M., Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Annales Universitatis Mariae Curie-Skłodowska A, 52, 2, 57-66 (1998) · Zbl 1009.30011
[6] Jahangiri, J. M., Harmonic functions starlike in the unit disk, Journal of Mathematical Analysis and Applications, 235, 2, 470-477 (1999) · Zbl 0940.30003 · doi:10.1006/jmaa.1999.6377
[7] Silverman, H., Harmonic univalent functions with negative coefficients, Journal of Mathematical Analysis and Applications, 220, 1, 283-289 (1998) · Zbl 0908.30013 · doi:10.1006/jmaa.1997.5882
[8] Krein, M.; Milman, D., On the extreme points of regularly convex sets, Studia Mathematica, 9, 1, 133-138 (1940) · JFM 66.0533.01
[9] Hallenbeck, D. J.; MacGregor, T. H., Linear Problems and Convexity Techniques in Geometric Function Theory. Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman Advanced Publishing Program (1984), Boston, Mass, USA: Pitman, Boston, Mass, USA · Zbl 0581.30001
[10] Dziok, J., On janowski harmonic functions, Journal of Applied Analysis, 21, 2 (2015) · Zbl 1327.31002
[11] Montel, P., Sur les families de functions analytiques qui admettent des valeurs exceptionelles dans un domaine, Annales Scientifiques de l École Normale Supérieure, 23, 487-535 (1912) · JFM 43.0509.05
[12] Littlewood, J. E., On inequalities in theory of functions, Proceedings London Mathematical Society, s2-23, 1, 481-519 (1925) · JFM 51.0247.03 · doi:10.1112/plms/s2-23.1.481
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.