Geodesic
Koenigs Function and Fractional Iteration of Functions Analytic in the Unit Disk with Real Coefficients and Fixed Points
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 67-71.

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The present paper deals with the problem of fractional iteration of functions analytic in the unit disk, with real Taylor's coefficients. It is assumed that there exist interior and boundary fixed points. The solution is given in terms of the Koenigs function. 
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O. S. Kudryavtseva. Koenigs Function and Fractional Iteration of Functions Analytic in the Unit Disk with Real Coefficients and Fixed Points. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 67-71. http://geodesic.mathdoc.fr/item/ISU_2013_13_1_a16/

[1] Berkson E., Porta H., “Semigroups of analytic functions and composition operators”, Michigan Math. J., 25:1 (1978), 101–115 | DOI | MR | Zbl

[2] Contreras M. D., Díaz-Madrigal S., Pommerenke Ch., “Fixed points and boundary behaviour of the Koenigs function”, Ann. Acad. Sci. Fenn. Math., 29:2 (2004), 471–488 | MR | Zbl

[3] Schröeder E., “Über itierte Funktionen”, Math. Ann. J., 3 (1871), 296–322 | DOI

[4] Königs G., “Recherches sur les intégrales des certaines equations fonctionelles”, Ann. Sci. Ecole Norm. Sup., 1:3 (1884), 3–41 | MR

[5] Baker I. N., “Fractional iteration near a fixpoint of multiplier 1”, J. Australian Math. Soc., 4:2 (1964), 143–148 | DOI | MR | Zbl

[6] Karlin S., McGregor J., “Embedding iterates of analytic functions with two fixed points into continuous groups”, Trans. Amer. Math. Soc., 132:1 (1968), 137–145 | DOI | MR | Zbl

[7] Cowen C. C., “Iteration and the solution of functional equations for functions analytic in the unit disk”, Trans. Amer. Math. Soc., 265:1 (1981), 69–95 | DOI | MR | Zbl

[8] Goryainov V. V., “Fractional iterates of functions analytic in the unit disk, with given fixed points”, Math. USSR Sb., 74:1 (1993), 29–-46 | DOI | MR

[9] Valiron G., Analytic Functions, Gostekhizdat, Moscow, 1957, 235 pp.

[10] Goryainov V. V., Kudryavtseva O. S., “One-parameter semigroups of analytic functions, fixed points and the Koenigs function”, Sb. Math., 202:7 (2011), 971–-1000 | DOI | DOI | MR | Zbl

[11] Kudryavtseva O. S., “Fractional iteration of functions analytic in the unit disk, with real coefficients”, Vestn. Volgograd. Gos. Un-ta. Ser. 1. Mathematics. Physics, 2011, no. 2(15), 50–62 | MR

[12] Goluzin G. M., Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, R. I., 1969, 676 pp. | MR | MR | Zbl

[13] Duren P. L., Univalent functions, Springer-Verlag, New York–Berlin–Heidelberg–Tokyo, 1983, 383 pp. | MR | Zbl

[14] Pommerenke Ch., Univalent functions, Vandenhoeck and Ruprecht, Göttingen, 1975, 376 pp. | MR | Zbl