Gakhov set in the Merkes theorem on convex combinations

T. V. Zharkova   ; A. V. Kazantsev

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 2, pp. 34-42

Résumé

The Merkes theorem deduces the starlikeness of any convex combination $f_\lambda$ of identity mapping and a holomorphic convex function $f$ in the unit disk with $f''(0)=0$. Under the same conditions, all of the functions $f_\lambda$ (except the mappings onto a strip when $\lambda=1$) are proved to belong also to the Gakhov set characterized by the property of (no more than) uniqueness of the root of the Gakhov equation. These results allow for the analogies for the exterior of the unit disk. The behavior of convex combinations is studied on the functions of the Alexander classes. For the exhaustion of each such class by the “level curves”, the “stopping moment” is found which corresponds to the exit out of the Gakhov set, and all of the trajectories of such an exit are described.

Détail

Keywords: hyperbolic derivative, conformal radius, Gakhov set, Gakhov equation, classes of convex and starlike functions, Alexander classes.

Bibliographie

[1] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966, 628 pp. | MR | Zbl

[2] Merkes E. P., “On the convex sum of certain univalent functions and the identity function”, Revista Colombiana de Matemáticas, 21 (1987), 5–12 | MR

[3] Kazantsev A. V., Chetyre etyuda na temu F. D. Gakhova, Uchebnoe posobie, Mar. un-t, Ioshkar-Ola, 2012, 64 pp.

[4] Haegi H. R., “Extremalprobleme und Ungleichungen konformer Gebietsgrössen”, Compositio Math., 8:2 (1950), 81–111 | MR | Zbl

[5] Gakhov F. D., “Ob obratnykh kraevykh zadachakh”, Dokl. AN SSSR, 86:4 (1952), 649–652 | Zbl

[6] Aksentev L. A., “Svyaz vneshnei obratnoi kraevoi zadachi s vnutrennim radiusom oblasti”, Izv. vuzov. Matem., 1984, no. 2, 3–11 | MR | Zbl

[7] Kinder M. I., “Issledovanie uravneniya F. D. Gakhova v sluchae mnogosvyaznykh oblastei”, Trudy seminara po kraevym zadacham, 22, Kazan. un-t, Kazan, 1985, 104–116 | MR | Zbl

[8] Aksentev L. A., Kazantsev A. V., “Novoe svoistvo klassa Nekhari i ego primenenie”, Trudy seminara po kraevym zadacham, 25, Kazan. un-t, Kazan, 1990, 33–51 | MR | Zbl

[9] Aksentev L. A., Kazantsev A. V., Kinder M. I., Kiselev A. V., “O klassakh edinstvennosti vneshnei obratnoi kraevoi zadachi”, Trudy seminara po kraevym zadacham, 24, Kazan. un-t, Kazan, 1990, 39–62 | MR

[10] Alexander J. W., “Functions which map the interior of the unit circle upon simple regions”, Ann. of Math. Ser. 2, 17:1 (1915), 12–22 | DOI | MR | Zbl

[11] Aksentev L. A., Kazantsev A. V., “Svoistva poverkhnosti konformnogo radiusa i razreshimost vneshnei obratnoi kraevoi zadachi”, Algebra i analiz, Tez. dokl. mezhdunar. nauch. konf., posv. 100-letiyu so dnya rozhd. N. G. Chebotareva (Kazan, 5–11 iyunya 1994 g.), Ch. 2, Kazan. un-t, Kazan, 1994, 13–14

[12] Aksentev L. A., Avkhadiev F. G., “Ob odnom klasse odnolistnykh funktsii”, Izv. vuzov. Matem., 1970, no. 10, 12–20 | MR | Zbl

[13] Kazantsev A. V., “O vykhode iz mnozhestva Gakhova, kontroliruemom usloviyami podchinennosti”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 156, no. 1, 2014, 31–43

[14] Goluzin G. M., “Nekotorye otsenki koeffitsientov odnolistnykh funktsii”, Matem. sbornik, 3(45):2 (1938), 321–330 | Zbl

[15] Kudryashov S. N., “O chisle reshenii vneshnikh obratnykh kraevykh zadach”, Izv. vuzov. Matem., 1969, no. 8, 30–32 | MR | Zbl

[16] Aksentev L. A., Khokhlov Yu. E., Shirokova E. A., “O edinstvennosti resheniya vneshnei obratnoi kraevoi zadachi”, Matem. zametki, 24:3 (1978), 319–330 | MR | Zbl