Gelfer functions, integral means, bounded mean oscillation, and univalency
Author:
Shinji Yamashita
Journal:
Trans. Amer. Math. Soc. 321 (1990), 245259
MSC:
Primary 30C45; Secondary 30C50
MathSciNet review:
1010891
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Abstract: A Gelfer function is a holomorphic function in such that and for all , in . The family of Gelfer functions contains the family of holomorphic functions in with and Re in . If is holomorphic in and if the mean of on the circle is dominated by that of a function of as , then . This has two recent and seemingly different results as corollaries. A core of the proof is the fact that if . Besides the properties obtained concerning itself, we shall investigate some families of functions where the roles played by in Univalent Function Theory are replaced by those of . Some exact estimates are obtained.
 [1]
Albert
Baernstein II, Univalence and bounded mean oscillation,
Michigan Math. J. 23 (1976), no. 3, 217–223
(1977). MR
0444935 (56 #3281)
 [2]
Jochen
Becker, Löwnersche Differentialgleichung und quasikonform
fortsetzbare schlichte Funktionen, J. Reine Angew. Math.
255 (1972), 23–43 (German). MR 0299780
(45 #8828)
 [3]
Colin
Bennett and Manfred
Stoll, Derivatives of analytic functions and bounded mean
oscillation, Arch. Math. (Basel) 47 (1986),
no. 5, 438–442. MR 870281
(88a:30074), http://dx.doi.org/10.1007/BF01189985
 [4]
Johnny
E. Brown, Derivatives of closetoconvex functions, integral means
and bounded mean oscillation, Math. Z. 178 (1981),
no. 3, 353–358. MR 635204
(82j:30046), http://dx.doi.org/10.1007/BF01214872
 [5]
Joseph
A. Cima and Karl
E. Petersen, Some analytic functions whose boundary values have
bounded mean oscillation, Math. Z. 147 (1976),
no. 3, 237–247. MR 0404631
(53 #8431)
 [6]
Joseph
A. Cima and Glenn
Schober, Analytic functions with bounded mean oscillation and
logarithms of 𝐻^{𝑝} functions, Math. Z.
151 (1976), no. 3, 295–300. MR 0425128
(54 #13085)
 [7]
Nikolaos
Danikas, Über die BMOANorm von
𝑙𝑜𝑔(1𝑧), Arch. Math. (Basel)
42 (1984), no. 1, 74–75 (German). MR 751474
(86f:42015), http://dx.doi.org/10.1007/BF01198131
 [8]
Peter
L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and
Applied Mathematics, Vol. 38, Academic Press, New YorkLondon, 1970. MR 0268655
(42 #3552)
 [9]
Peter
L. Duren, Univalent functions, Grundlehren der Mathematischen
Wissenschaften [Fundamental Principles of Mathematical Sciences],
vol. 259, SpringerVerlag, New York, 1983. MR 708494
(85j:30034)
 [10]
S. A. Gelfer, (C. A. [ill]), [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] functions, assuming no pair of values and .)
 [11]
Daniel
Girela, Integral means and BMOAnorms of logarithms of univalent
functions, J. London Math. Soc. (2) 33 (1986),
no. 1, 117–132. MR 829393
(87k:30026), http://dx.doi.org/10.1112/jlms/s233.1.117
 [12]
Daniel
Girela, BMO, 𝐴₂weights and univalent
functions, Analysis 7 (1987), no. 2,
129–143. MR
885120 (88e:30050), http://dx.doi.org/10.1524/anly.1987.7.2.129
 [13]
A. W. Goodman, Univalent functions. I, II, Mariner, Tampa, Florida, 1983.
 [14]
A. Z. Grinshpan (A. [ill]), [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] transl.: On the coefficients of univalent functions assuming no pair of values and , Math. Notes 11(1972), 311.
 [15]
J.
A. Hummel, A variational method for Gel′fer functions,
J. Analyse Math. 30 (1976), 271–280. MR 0440025
(55 #12906)
 [16]
James
A. Jenkins, On BieberbachEilenberg
functions, Trans. Amer. Math. Soc. 76 (1954), 389–396. MR 0062831
(16,24e), http://dx.doi.org/10.1090/S00029947195400628316
 [17]
A.
J. Lohwater, G.
Piranian, and W.
Rudin, The derivative of a schlicht function, Math. Scand.
3 (1955), 103–106. MR 0072218
(17,249b)
 [18]
T.
H. MacGregor, Functions whose derivative has a
positive real part, Trans. Amer. Math. Soc.
104 (1962),
532–537. MR 0140674
(25 #4090), http://dx.doi.org/10.1090/S00029947196201406747
 [19]
Shinji
Yamashita, Almost locally univalent functions, Monatsh. Math.
81 (1976), no. 3, 235–240. MR 0407263
(53 #11042)
 [20]
Shinji
Yamashita, Schlicht holomorphic functions and the Riccati
differential equation, Math. Z. 157 (1977),
no. 1, 19–22. MR 0486487
(58 #6216)
 [21]
Shinji
Yamashita, F. Riesz’s decomposition of a subharmonic
function, applied to BMOA, Boll. Un. Mat. Ital. A (6)
3 (1984), no. 1, 103–109 (English, with Italian
summary). MR
739196 (85g:31001)
 [22]
Shinji
Yamashita, A gap series with growth conditions and its
applications, Math. Scand. 60 (1987), no. 1,
9–18. MR
908825 (88m:30002)
 [1]
 A. Baernstein II, Univalence and bounded mean oscillation, Michigan Math. J. 23(1976), 217223. MR 0444935 (56:3281)
 [2]
 J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 225(1972), 2343. MR 0299780 (45:8828)
 [3]
 C. Bennett and M. Stoll, Derivatives of analytic functions and bounded mean oscillation, Arch. Math. 47(1986), 438442. MR 870281 (88a:30074)
 [4]
 J. E. Brown, Derivatives of closetoconvex functions, integral means and bounded mean oscillation, Math. Z. 178(1981), 353358. MR 635204 (82j:30046)
 [5]
 J. A. Cima and K. E. Petersen, Some analytic functions whose boundary values have bounded mean oscillation, Math. Z. 147(1976), 237247. MR 0404631 (53:8431)
 [6]
 J. A. Cima and G. Schober, Analytic functions with bounded mean oscillation and logarithms of functions, Math. Z. 151(1976), 295300. MR 0425128 (54:13085)
 [7]
 N. Danikas, Über die BMOANorm von , Arch. Math. 42(1984), 7475. MR 751474 (86f:42015)
 [8]
 P. L. Duren, Theory of spaces, Pure and Appl. Math., vol. 38, Academic Press, New YorkSan FranciscoLondon, 1970. MR 0268655 (42:3552)
 [9]
 , Univalent functions, Grundlehren Math. Wiss., vol. 259, SpringerVerlag, New YorkBerlinHeidelbergTokyo, 1983. MR 708494 (85j:30034)
 [10]
 S. A. Gelfer, (C. A. [ill]), [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] functions, assuming no pair of values and .)
 [11]
 D. Girela, Integral means and BMOAnorms of logarithms of univalent functions, J. London Math. Soc. (2) 33(1986), 117132. MR 829393 (87k:30026)
 [12]
 , BMO, weights and univalent functions, Analysis 7(1987), 129143. MR 885120 (88e:30050)
 [13]
 A. W. Goodman, Univalent functions. I, II, Mariner, Tampa, Florida, 1983.
 [14]
 A. Z. Grinshpan (A. [ill]), [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] transl.: On the coefficients of univalent functions assuming no pair of values and , Math. Notes 11(1972), 311.
 [15]
 J. A. Hummel, A variational method for Gelfer functions, J. Analyse Math. 30(1976), 271280. MR 0440025 (55:12906)
 [16]
 J. A. Jenkins, On BieberbachEilenberg functions. Trans. Amer. Math. Soc. 76(1954), 389396. MR 0062831 (16:24e)
 [17]
 A. J. Lohwater, G. Piranian, and W. Rudin, The derivative of a schlicht function, Math. Scand, 3(1955), 103106. MR 0072218 (17:249b)
 [18]
 T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104(1962), 532537. MR 0140674 (25:4090)
 [19]
 S. Yamashita, Almost locally univalent functions, Monatsh. Math. 81(1976), 235240. MR 0407263 (53:11042)
 [20]
 , Schlicht holomorphic functions and the Riccati differential equation, Math. Z. 157(1977), 1922. MR 0486487 (58:6216)
 [21]
 , F. Riesz's decomposition of a subharmonic function, applied to BMOA, Boll. Un. Mat. Ital. (6) 3A(1984), 103109. MR 739196 (85g:31001)
 [22]
 , A gap series with growth conditions and its applications, Math. Scand. 60(1987), 918. MR 908825 (88m:30002)
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DOI:
http://dx.doi.org/10.1090/S0002994719901010891X
PII:
S 00029947(1990)1010891X
Article copyright:
© Copyright 1990
American Mathematical Society
