Totally monotone functions with applications to the Bergman space
Authors: B. Korenblum, R. O’Neil, K. Richards and K. Zhu
Journal: Trans. Amer. Math. Soc. 337 (1993), 795-806
MSC: Primary 30D15; Secondary 26A48, 30C80, 30H05
MathSciNet review: 1118827
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Abstract: Using a theorem of S. Bernstein  we prove a special case of the following maximum principle for the Bergman space conjectured by B. Korenblum : There exists a number such that if and are analytic functions on the open unit disk with on then , where is the norm with respect to area measure on . We prove the above conjecture when either or is a monomial; in this case we show that the optimal constant is greater than or equal to .
-  Serge Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), no. 1, 1–66 (French). MR 1555269, https://doi.org/10.1007/BF02547400
-  L. Carleson, Private communication with the first-named author.
-  Boris Korenblum, A maximum principle for the Bergman space, Publ. Mat. 35 (1991), no. 2, 479–486. MR 1201570, https://doi.org/10.5565/PUBLMAT_35291_12
-  B. I. Korenblyum, On two theorems from the theory of absolutely monotonic functions, Uspehi Matem. Nauk (N.S.) 6 (1951), no. 4(44), 172–175. MR 0043852
- S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1-66. MR 1555269
- L. Carleson, Private communication with the first-named author.
- B. Korenblum, A maximum principle for the Bergman space, Publ. Math. 35 (1991), 479-486. MR 1201570 (93j:30018)
- -, On two theorems from the theory of absolutely monotone functions, Uspekhi Mat. Nauk4 (44) (1952), 172-175. (Russian) MR 0043852 (13:329e)