# Article

Full entry | PDF   (5.7 MB)
Keywords:
harmonic mapping; partial sum; univalent mapping; convex mapping; starlike mapping; close-to-convex mapping; harmonic convolution; direction convexity preserving map
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.
References:
[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. MR 3265059 | Zbl 1301.30011
[2] Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. A I, Math. 9 (1984), 3-25. DOI 10.5186/aasfm.1984.0905 | MR 0752388 | Zbl 0506.30007
[3] Dorff, M.: Convolutions of planar harmonic convex mappings. Complex Variables, Theory Appl. 45 (2001), 263-271. DOI 10.1080/17476930108815381 | MR 1909021 | Zbl 1023.30019
[4] Dorff, M., Nowak, M., Wo{ł}oszkiewicz, M.: Convolutions of harmonic convex mappings. Complex Var. Elliptic Equ. 57 (2012), 489-503. DOI 10.1080/17476933.2010.487211 | MR 2903478 | Zbl 1250.31001
[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. MR 2963968 | Zbl 1303.30010
[6] Duren, P. L.: Harmonic Mappings in the Plane. Cambridge Tracts in Mathematics 156 Cambridge University Press, Cambridge (2004). MR 2048384 | Zbl 1055.31001
[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
[8] Duren, P. L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften 259. A Series of Comprehensive Studies in Mathematics Springer, New York (1983). MR 0708494 | Zbl 0514.30001
[9] Fournier, R., Silverman, H.: Radii problems for generalized sections of convex functions. Proc. Am. Math. Soc. 112 (1991), 101-107. DOI 10.1090/S0002-9939-1991-1047000-3 | MR 1047000 | Zbl 0725.30007
[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. DOI 10.1007/BF02790196 | MR 0801293 | Zbl 0576.30010
[11] Hengartner, W., Schober, G.: On Schlicht mappings to domains convex in one direction. Comment. Math. Helv. 45 (1970), 303-314. DOI 10.1007/BF02567334 | MR 0277703 | Zbl 0203.07604
[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. MR 0737754 | Zbl 0538.30016
[13] Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42 (1936), 689-692. DOI 10.1090/S0002-9904-1936-06397-4 | MR 1563404 | Zbl 0015.15903
[14] Li, L., Ponnusamy, S.: Convolutions of slanted half-plane harmonic mappings. Analysis, München 33 (2013), 159-176. DOI 10.1524/anly.2013.1170 | MR 3082279 | Zbl 1286.30011
[15] Li, L., Ponnusamy, S.: Disk of convexity of sections of univalent harmonic functions. J. Math. Anal. Appl. 408 (2013), 589-596. DOI 10.1016/j.jmaa.2013.06.021 | MR 3085055 | Zbl 1307.31002
[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. DOI 10.1016/j.na.2013.05.016 | MR 3073331 | Zbl 1279.30038
[17] Li, L., Ponnusamy, S.: Solution to an open problem on convolutions of harmonic mappings. Complex Var. Elliptic Equ. 58 (2013), 1647-1653. DOI 10.1080/17476933.2012.702111 | MR 3170725 | Zbl 1287.30004
[18] MacGregor, T. H.: Functions whose derivative has a positive real part. Trans. Am. Math. Soc. 104 (1962), 532-537. DOI 10.1090/S0002-9947-1962-0140674-7 | MR 0140674 | Zbl 0106.04805
[19] Miki, Y.: A note on close-to-convex functions. J. Math. Soc. Japan 8 (1956), 256-268. DOI 10.2969/jmsj/00830256 | MR 0091343 | Zbl 0072.29402
[20] Obradovi{ć}, M., Ponnusamy, S.: Starlikeness of sections of univalent functions. Rocky Mt. J. Math. 44 (2014), 1003-1014. DOI 10.1216/RMJ-2014-44-3-1003 | MR 3264494 | Zbl 1298.30012
[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. MR 3155688 | Zbl 1320.30031
[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. DOI 10.1134/S0037446611020121 | MR 2841555
[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
[24] Pommerenke, C.: Univalent Functions. With a chapter on quadratic differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrbücher. Band 25 Vandenhoeck & Ruprecht, Göttingen (1975). MR 0507768 | Zbl 0298.30014
[25] Ponnusamy, S.: Pólya-Schoenberg conjecture for Carathéodory functions. J. Lond. Math. Soc., (2) Ser. 51 (1995), 93-104. DOI 10.1112/jlms/51.1.93 | MR 1310724 | Zbl 0814.30017
[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. MR 3220953 | Zbl 1318.30039
[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. DOI 10.1016/j.na.2013.09.009 | MR 3130518 | Zbl 1291.30096
[28] Robertson, M. S.: The partial sums of multivalently star-like functions. Ann. Math. (2) 42 (1941), 829-838. DOI 10.2307/1968770 | MR 0004905
[29] Robertson, M. S.: Analytic functions star-like in one direction. Am. J. Math. 58 (1936), 465-472. DOI 10.2307/2370963 | MR 1507169 | Zbl 0014.12002
[30] Royster, W. C., Ziegler, M.: Univalent functions convex in one direction. Publ. Math. Debrecen 23 (1976), 339-345. MR 0425101 | Zbl 0365.30005
[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. MR 1199156 | Zbl 0696.30019
[32] Ruscheweyh, S.: Extension of Szegő's theorem on the sections of univalent functions. SIAM J. Math. Anal. 19 (1988), 1442-1449. DOI 10.1137/0519107 | MR 0965264 | Zbl 0661.30012
[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. DOI 10.5186/aasfm.1989.1427 | MR 0997971
[34] Silverman, H.: Radii problems for sections of convex functions. Proc. Am. Math. Soc. 104 (1988), 1191-1196. DOI 10.1090/S0002-9939-1988-0942638-3 | MR 0942638 | Zbl 0692.30013
[35] Singh, R.: Radius of convexity of partial sums of a certain power series. J. Aust. Math. Soc. 11 (1970), 407-410. DOI 10.1017/S1446788700007874 | MR 0281902 | Zbl 0215.12502
[36] Szegő, G.: Zur Theorie der schlichten Abbildungen. Math. Ann. 100 German (1928), 188-211. DOI 10.1007/BF01448843 | MR 1512482

Partner of