# Another Look at Average Formulas of Nevanlinna Counting Functions of Holomorphic Self-maps of the Unit Disk

• Published : 2008.03.31

#### Abstract

This is an extended version of the paper [K] of the author. The average formulas on the circles and disks around arbitrary points of Nevanlinna counting functions of holomorphic self-maps of the unit disk, given in terms of the boundary values of the selfmaps, are shown to give another characterization of the whole class or a special subclass of inner functions in terms of Nevanlinna counting function in addition to the previous applications to Rudin's orthogonal functions.

#### References

1. C. J. Bishop, Orthogonal functions in $H^{\infty}$, Pacific J. Math, 228(2005), 1-31.
2. O. Frostman, Potentiel d´equilibre et capacite des ensembles, Lunds Univ. Mat. Sem. 3, 1935.
3. J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, 96. Academic Press, New York-London 1981.
4. P. Gorkin and K. Izuch, Some counter examples in subalgebras of $L^{\infty}$(D), Indiana Univ. Math. J., 40(1991), 1301-1313. https://doi.org/10.1512/iumj.1991.40.40058
5. H. O. Kim, Averages of Nevanlinna counting functions of holomorphic self-maps of the unit disk, Hokkaido Math. J., 33(2004), 697-706. https://doi.org/10.14492/hokmj/1285851918
6. R. Mortini and A. Nicolau, Frostman shifts of inner functions, J. D'Analyse Math., 92(2004), 285-326. https://doi.org/10.1007/BF02787765
7. W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill Book Co., New York-Auckland-Dusseldorf 1976.
8. W. Rudin, Real and Complex Analysis, 3d ed., McGraw-Hill Book Co., New York 1987.
9. W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam 1969.
10. W. Rudin,A generalization of a theorem of Frostman, Math. Scand., 21(1967), 136-143. https://doi.org/10.3891/acta.chem.scand.21-0136
11. C. Sundberg, Measures induced by analytic functions and a problem of Walter Rudin, J. AMS, 16(2003), 68-90.
12. J. H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics. Springer-Verlag, New York 1993.

#### Cited by

1. Nevanlinna counting function and Carleson function of analytic maps vol.351, pp.2, 2011, https://doi.org/10.1007/s00208-010-0596-1
2. Complete Nevanlinna counting functions of boundary-preserving Nevanlinna functions vol.60, pp.1, 2015, https://doi.org/10.1080/17476933.2014.898275