Injectivity of sections of convex harmonic mappings and convolution theorems

Li, Liulan ; Ponnusamy, Saminathan

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 331-350

Abstract (VO)

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.

Détail

DOI : 10.1007/s10587-016-0259-9

Classification : 30C45
Keywords: harmonic mapping; partial sum; univalent mapping; convex mapping; starlike mapping; close-to-convex mapping; harmonic convolution; direction convexity preserving map

Bibliographie

[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 | Zbl

[2] Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. A I, Math. 9 (1984), 3-25. | DOI | MR | Zbl

[3] Dorff, M.: Convolutions of planar harmonic convex mappings. Complex Variables, Theory Appl. 45 (2001), 263-271. | DOI | MR | Zbl

[4] Dorff, M., Nowak, M., Wo{ł}oszkiewicz, M.: Convolutions of harmonic convex mappings. Complex Var. Elliptic Equ. 57 (2012), 489-503. | DOI | MR | Zbl

[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 | Zbl

[6] Duren, P. L.: Harmonic Mappings in the Plane. Cambridge Tracts in Mathematics 156 Cambridge University Press, Cambridge (2004). | MR | Zbl

[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

[8] Duren, P. L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften 259. A Series of Comprehensive Studies in Mathematics Springer, New York (1983). | MR | Zbl

[9] Fournier, R., Silverman, H.: Radii problems for generalized sections of convex functions. Proc. Am. Math. Soc. 112 (1991), 101-107. | DOI | MR | Zbl

[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 | MR | Zbl

[11] Hengartner, W., Schober, G.: On Schlicht mappings to domains convex in one direction. Comment. Math. Helv. 45 (1970), 303-314. | DOI | MR | Zbl

[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 | Zbl

[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 | MR | Zbl

[14] Li, L., Ponnusamy, S.: Convolutions of slanted half-plane harmonic mappings. Analysis, München 33 (2013), 159-176. | DOI | MR | Zbl

[15] Li, L., Ponnusamy, S.: Disk of convexity of sections of univalent harmonic functions. J. Math. Anal. Appl. 408 (2013), 589-596. | DOI | MR | Zbl

[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 | MR | Zbl

[17] Li, L., Ponnusamy, S.: Solution to an open problem on convolutions of harmonic mappings. Complex Var. Elliptic Equ. 58 (2013), 1647-1653. | DOI | MR | Zbl

[18] MacGregor, T. H.: Functions whose derivative has a positive real part. Trans. Am. Math. Soc. 104 (1962), 532-537. | DOI | MR | Zbl

[19] Miki, Y.: A note on close-to-convex functions. J. Math. Soc. Japan 8 (1956), 256-268. | DOI | MR | Zbl

[20] Obradovi{ć}, M., Ponnusamy, S.: Starlikeness of sections of univalent functions. Rocky Mt. J. Math. 44 (2014), 1003-1014. | DOI | MR | Zbl

[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 | Zbl

[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 | MR

[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

[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 | Zbl

[25] Ponnusamy, S.: Pólya-Schoenberg conjecture for Carathéodory functions. J. Lond. Math. Soc., (2) Ser. 51 (1995), 93-104. | DOI | MR | Zbl

[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 | Zbl

[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 | MR | Zbl

[28] Robertson, M. S.: The partial sums of multivalently star-like functions. Ann. Math. (2) 42 (1941), 829-838. | DOI | MR

[29] Robertson, M. S.: Analytic functions star-like in one direction. Am. J. Math. 58 (1936), 465-472. | DOI | MR | Zbl

[30] Royster, W. C., Ziegler, M.: Univalent functions convex in one direction. Publ. Math. Debrecen 23 (1976), 339-345. | MR | Zbl

[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 | Zbl

[32] Ruscheweyh, S.: Extension of Szegő's theorem on the sections of univalent functions. SIAM J. Math. Anal. 19 (1988), 1442-1449. | DOI | MR | Zbl

[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 | MR

[34] Silverman, H.: Radii problems for sections of convex functions. Proc. Am. Math. Soc. 104 (1988), 1191-1196. | DOI | MR | Zbl

[35] Singh, R.: Radius of convexity of partial sums of a certain power series. J. Aust. Math. Soc. 11 (1970), 407-410. | DOI | MR | Zbl

[36] Szegő, G.: Zur Theorie der schlichten Abbildungen. Math. Ann. 100 German (1928), 188-211. | DOI | MR