×

Geometric properties of nonlinear integral transforms of certain analytic functions. (English) Zbl 1063.30009

Let \(H\) denote the class of all analytic functions in the open unit disk \(D=\{z:| z| <1\}\) and \(A\) denote the class of functions \(f\in H\) normalised by a Taylor series. By using norm estimates of the pre-Schwarzian derivatives for certain analytic functions defined by a nonlinear integral transform, the authors gave several interesting geometric properties of the integral transform.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] Aksent’ev, L. A., and Nezhmetdinov, I. R.: Sufficient conditions for univalence of certain integral representations. Trudy Sem. Kraev. Zadacham, 18 , 3-11 (1982), (in Russian); English translation in Amer. Math. Soc. Transl., 136 (2), 1-9 (1987). · Zbl 0616.30008
[2] Alexander, J. W.: Functions which map the interior of the unit circle upon simple regions. Ann. of Math., 17 , 12-22 (1915). · JFM 45.0672.02
[3] Becker, J.: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. J. Reine Angew. Math., 255 , 23-43 (1972). · Zbl 0239.30015
[4] Becker, J., and Pommerenke, Ch.: Schlichtheitskriterien und Jordangebiete. J. Reine Angew. Math., 354 , 74-94 (1984). · Zbl 0539.30009
[5] Duren, P. L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, New York (1983). · Zbl 0514.30001
[6] Hag, K., and Hag, P.: John disks and the pre-Schwarzian derivative. Ann. Acad. Sci. Fenn. Math., 26 , 205-224 (2001). · Zbl 1002.30005
[7] Hornich, H.: Ein Banachraum analytischer Funktionen in Zusammenhang mit den schlichten Funktionen. Monatsh. Math., 73 , 36-45 (1969). · Zbl 0167.42702
[8] Kim, Y. C., Ponnusamy, S., and Sugawa, T.: Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives. (Preprint). · Zbl 1066.30013
[9] Kim, Y. C., and Sugawa, T.: Growth and coefficient estimates for uniformly locally univalent functions on the unit disk. Rocky Mountain J. Math., 32 , 179-200 (2002). · Zbl 1034.30009
[10] Kim, Y. C., and Sugawa, T.: Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions. (2004). (Preprint). · Zbl 1112.30012
[11] Kim, Y. J., and Merkes, E. P.: On an integral of powers of a spirallike function. Kyungpook Math. J., 12 , 249-253 (1972). · Zbl 0252.30017
[12] Krzy. z, J. G.: Convolution and quasiconformal extension. Comment. Math. Helv., 51 , 99-104 (1976). · Zbl 0323.30022
[13] Merkes, E. P., and Wright, D. J.: On the univalence of a certain integral. Proc. Amer. Math. Soc., 27 , 97-100 (1971). · Zbl 0221.30012
[14] Obradović, M., and Ponnusamy, S.: New criteria and distortion theorems for univalent functions. Complex Variables Theory Appl., 44 , 173-191 (2001). · Zbl 1023.30015
[15] Obradović, M., Ponnusamy, S., Singh, V., and Vasundhra, P.: Univalency, starlikeness and convexity applied to certain classes of rational functions. Analysis (Munich), 22 (3), 2252-242 (2002). · Zbl 1010.30011
[16] Ozaki, S., and Nunokawa, M.: The Schwarzian derivative and univalent functions. Proc. Amer. Math. Soc., 33 , 392-394 (1972). · Zbl 0233.30011
[17] Singh, V., and Chichra, P. N.: An extension of Becker’s criterion of univalence. J. Indian Math. Soc., 41 , 353-361 (1977). · Zbl 0441.30018
[18] Yamashita, S.: Almost locally univalent functions. Monatsh. Math., 81 , 235-240 (1976). · Zbl 0331.30015
[19] Yamashita, S.: Norm estimates for function starlike or convex of order alpha. Hokkaido Math. J., 28 , 217-230 (1999). · Zbl 0926.30009
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.