| Title:
|
Injectivity of sections of convex harmonic mappings and convolution theorems (English) |
| Author:
|
Li, Liulan |
| Author:
|
Ponnusamy, Saminathan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
331-350 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline {g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal {P}_H^0(\alpha )$ and $\mathcal {G}_H^0(\beta )$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal {P}_H^0(\alpha )$ and $F\in \mathcal {G}_H^0(\beta )$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha $ and $\beta $ are satisfied. In the second part we study the harmonic sections (partial sums) $$ s_{n, n}(f)(z)=s_n(h)(z)+\overline {s_n(g)(z)}, $$ where $f=h+\overline {g}\in {\mathcal H}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline {g}\in {\mathcal H}_0$ is a univalent harmonic convex mapping, then $s_{n, n}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\geq 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\geq 2$ and $n\neq 3$. Moreover, we show that the section $s_{3,3}(f)$ of $f\in {\mathcal C}_H^0$ is not convex in the disk $|z|<1/4$ but it is convex in a smaller disk. (English) |
| Keyword:
|
harmonic mapping |
| Keyword:
|
partial sum |
| Keyword:
|
univalent mapping |
| Keyword:
|
convex mapping |
| Keyword:
|
starlike mapping |
| Keyword:
|
close-to-convex mapping |
| Keyword:
|
harmonic convolution |
| Keyword:
|
direction convexity preserving map |
| MSC:
|
30C45 |
| idZBL:
|
Zbl 06604470 |
| idMR:
|
MR3519605 |
| DOI:
|
10.1007/s10587-016-0259-9 |
| . |
| Date available:
|
2016-06-16T12:42:07Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145727 |
| . |
| Reference:
|
[1] Bharanedhar, S. V., Ponnusamy, S.: Uniform close-to-convexity radius of sections of functions in the close-to-convex family.J. Ramanujan Math. Soc. 29 (2014), 243-251. Zbl 1301.30011, MR 3265059 |
| Reference:
|
[2] Clunie, J., Sheil-Small, T.: Harmonic univalent functions.Ann. Acad. Sci. Fenn., Ser. A I, Math. 9 (1984), 3-25. Zbl 0506.30007, MR 0752388, 10.5186/aasfm.1984.0905 |
| Reference:
|
[3] Dorff, M.: Convolutions of planar harmonic convex mappings.Complex Variables, Theory Appl. 45 (2001), 263-271. Zbl 1023.30019, MR 1909021, 10.1080/17476930108815381 |
| Reference:
|
[4] Dorff, M., Nowak, M., Wo{ł}oszkiewicz, M.: Convolutions of harmonic convex mappings.Complex Var. Elliptic Equ. 57 (2012), 489-503. Zbl 1250.31001, MR 2903478, 10.1080/17476933.2010.487211 |
| Reference:
|
[5] Dorff, M. J., Rolf, J. S.: Anamorphosis, mapping problems, and harmonic univalent functions.Explorations in Complex Analysis Classr. Res. Mater. Ser. The Mathematical Association of America, Washington (2012), 197-269 M. A. Brilleslyper et al. Zbl 1303.30010, MR 2963968 |
| Reference:
|
[6] Duren, P. L.: Harmonic Mappings in the Plane.Cambridge Tracts in Mathematics 156 Cambridge University Press, Cambridge (2004). Zbl 1055.31001, MR 2048384 |
| Reference:
|
[7] Duren, P. L.: A survey of harmonic mappings in the plane.Texas Tech. Univ., Math. Series, Visiting Scholars Lectures 18 (1990-1992), 1-15. MR 2048384 |
| Reference:
|
[8] Duren, P. L.: Univalent Functions.Grundlehren der Mathematischen Wissenschaften 259. A Series of Comprehensive Studies in Mathematics Springer, New York (1983). Zbl 0514.30001, MR 0708494 |
| Reference:
|
[9] Fournier, R., Silverman, H.: Radii problems for generalized sections of convex functions.Proc. Am. Math. Soc. 112 (1991), 101-107. Zbl 0725.30007, MR 1047000, 10.1090/S0002-9939-1991-1047000-3 |
| Reference:
|
[10] Goodman, A. W., Schoenberg, I. J.: On a theorem of Szegő on univalent convex maps of the unit circle.J. Anal. Math. 44 (1985), 200-204. Zbl 0576.30010, MR 0801293, 10.1007/BF02790196 |
| Reference:
|
[11] Hengartner, W., Schober, G.: On Schlicht mappings to domains convex in one direction.Comment. Math. Helv. 45 (1970), 303-314. Zbl 0203.07604, MR 0277703, 10.1007/BF02567334 |
| Reference:
|
[12] Iliev, L.: Classical extremal problems for univalent functions.Complex Analysis Banach Center Publ. 11. Polish Academy of Sciences, Institute of Mathematics. PWN-Polish Scientific Publishers, Warsaw J. Ławrynowicz et al. (1983), 89-110. Zbl 0538.30016, MR 0737754 |
| Reference:
|
[13] Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings.Bull. Am. Math. Soc. 42 (1936), 689-692. Zbl 0015.15903, MR 1563404, 10.1090/S0002-9904-1936-06397-4 |
| Reference:
|
[14] Li, L., Ponnusamy, S.: Convolutions of slanted half-plane harmonic mappings.Analysis, München 33 (2013), 159-176. Zbl 1286.30011, MR 3082279, 10.1524/anly.2013.1170 |
| Reference:
|
[15] Li, L., Ponnusamy, S.: Disk of convexity of sections of univalent harmonic functions.J. Math. Anal. Appl. 408 (2013), 589-596. Zbl 1307.31002, MR 3085055, 10.1016/j.jmaa.2013.06.021 |
| Reference:
|
[16] Li, L., Ponnusamy, S.: Injectivity of sections of univalent harmonic mappings.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 89 (2013), 276-283. Zbl 1279.30038, MR 3073331, 10.1016/j.na.2013.05.016 |
| Reference:
|
[17] Li, L., Ponnusamy, S.: Solution to an open problem on convolutions of harmonic mappings.Complex Var. Elliptic Equ. 58 (2013), 1647-1653. Zbl 1287.30004, MR 3170725, 10.1080/17476933.2012.702111 |
| Reference:
|
[18] MacGregor, T. H.: Functions whose derivative has a positive real part.Trans. Am. Math. Soc. 104 (1962), 532-537. Zbl 0106.04805, MR 0140674, 10.1090/S0002-9947-1962-0140674-7 |
| Reference:
|
[19] Miki, Y.: A note on close-to-convex functions.J. Math. Soc. Japan 8 (1956), 256-268. Zbl 0072.29402, MR 0091343, 10.2969/jmsj/00830256 |
| Reference:
|
[20] Obradovi{ć}, M., Ponnusamy, S.: Starlikeness of sections of univalent functions.Rocky Mt. J. Math. 44 (2014), 1003-1014. Zbl 1298.30012, MR 3264494, 10.1216/RMJ-2014-44-3-1003 |
| Reference:
|
[21] Obradovi{ć}, M., Ponnusamy, S.: Injectivity and starlikeness of sections of a class of univalent functions.Complex Analysis and Dynamical Systems V. Proc. of the 5th Int. Conf. On complex analysis and dynamical systems Israel, 2011. Contemp. Math. 591 AMS Providence, Ramat Gan: Bar-Ilan University (2013), 195-203 M. Agranovsky et al. Zbl 1320.30031, MR 3155688 |
| Reference:
|
[22] Obradovi{ć}, M., Ponnusami, S.: Partial sums and the radius problem for a class of conformal mappings.Sib. Math. J. 52 (2011), 291-302 translation from Sibirsk. Mat. Zh. 52 (2011), 371-384. MR 2841555, 10.1134/S0037446611020121 |
| Reference:
|
[23] Obradovich, M., Ponnusami, S., Wirths, K.-J.: Coefficient characterizations and sections for some univalent functions.Sib. Math. J. 54 (2013), 679-696 translation from Sib. Mat. Zh. 54 (2013), 852-870. MR 3137152 |
| Reference:
|
[24] Pommerenke, C.: Univalent Functions. With a chapter on quadratic differentials by Gerd Jensen.Studia Mathematica/Mathematische Lehrbücher. Band 25 Vandenhoeck & Ruprecht, Göttingen (1975). Zbl 0298.30014, MR 0507768 |
| Reference:
|
[25] Ponnusamy, S.: Pólya-Schoenberg conjecture for Carathéodory functions.J. Lond. Math. Soc., (2) Ser. 51 (1995), 93-104. Zbl 0814.30017, MR 1310724, 10.1112/jlms/51.1.93 |
| Reference:
|
[26] Ponnusamy, S., Rasila, A.: Planar harmonic and quasiregular mappings.Topics in Modern Function Theory. Based on mini-courses of the CMFT workshop On computational methods and function theory, Guwahati, India, 2008 Ramanujan Math. Soc. Lect. Notes Ser. 19 Ramanujan Mathematical Society, Mysore (2013), 267-333 S. Ruscheweyh et al. Zbl 1318.30039, MR 3220953 |
| Reference:
|
[27] Ponnusamy, S., Sahoo, S. K., Yanagihara, H.: Radius of convexity of partial sums of functions in the close-to-convex family.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 95 (2014), 219-228. Zbl 1291.30096, MR 3130518, 10.1016/j.na.2013.09.009 |
| Reference:
|
[28] Robertson, M. S.: The partial sums of multivalently star-like functions.Ann. Math. (2) 42 (1941), 829-838. MR 0004905, 10.2307/1968770 |
| Reference:
|
[29] Robertson, M. S.: Analytic functions star-like in one direction.Am. J. Math. 58 (1936), 465-472. Zbl 0014.12002, MR 1507169, 10.2307/2370963 |
| Reference:
|
[30] Royster, W. C., Ziegler, M.: Univalent functions convex in one direction.Publ. Math. Debrecen 23 (1976), 339-345. Zbl 0365.30005, MR 0425101 |
| Reference:
|
[31] Ruscheweyh, S.: Convolutions in geometric function theory: Recent results and open problems.Univalent Functions, Fractional Calculus, and Their Applications Ellis Horwood Series in Mathematics and Its Applications Horwood, Chichester, Halsted Press (1989), 267-282 H. M. Srivastava et al. Zbl 0696.30019, MR 1199156 |
| Reference:
|
[32] Ruscheweyh, S.: Extension of Szegő's theorem on the sections of univalent functions.SIAM J. Math. Anal. 19 (1988), 1442-1449. Zbl 0661.30012, MR 0965264, 10.1137/0519107 |
| Reference:
|
[33] Ruscheweyh, S., Salinas, L. C.: On the preservation of direction-convexity and the \kern0ptGoodman-Saff conjecture.Ann. Acad. Sci. Fenn., Ser. A I, Math. 14 (1989), 63-73. MR 0997971, 10.5186/aasfm.1989.1427 |
| Reference:
|
[34] Silverman, H.: Radii problems for sections of convex functions.Proc. Am. Math. Soc. 104 (1988), 1191-1196. Zbl 0692.30013, MR 0942638, 10.1090/S0002-9939-1988-0942638-3 |
| Reference:
|
[35] Singh, R.: Radius of convexity of partial sums of a certain power series.J. Aust. Math. Soc. 11 (1970), 407-410. Zbl 0215.12502, MR 0281902, 10.1017/S1446788700007874 |
| Reference:
|
[36] Szegő, G.: Zur Theorie der schlichten Abbildungen.Math. Ann. 100 German (1928), 188-211. MR 1512482, 10.1007/BF01448843 |
| . |
