| Title:
|
Convexities of Gaussian integral means and weighted integral means for analytic functions (English) |
| Author:
|
Li, Haiying |
| Author:
|
Liu, Taotao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
525-543 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We first show that the Gaussian integral means of $f\colon \mathbb {C}\to \mathbb {C}$ (with respect to the area measure ${\rm e}^{-\alpha |z|^{2}} {\rm d} A(z)$) is a convex function of $r$ on $(0,\infty )$ when $\alpha \leq 0$. We then prove that the weighted integral means $A_{\alpha ,\beta }(f,r)$ and $L_{\alpha ,\beta }(f,r)$ of the mixed area and the mixed length of $f(r\mathbb {D})$ and $\partial f(r\mathbb {D})$, respectively, also have the property of convexity in the case of $\alpha \leq 0$. Finally, we show with examples that the range $\alpha \leq 0$ is the best possible. (English) |
| Keyword:
|
Gaussian integral means |
| Keyword:
|
weighted integral means |
| Keyword:
|
analytic function |
| Keyword:
|
\nobreak convexity |
| MSC:
|
30H10 |
| MSC:
|
30H20 |
| idZBL:
|
Zbl 07088803 |
| idMR:
|
MR3959963 |
| DOI:
|
10.21136/CMJ.2018.0432-17 |
| . |
| Date available:
|
2019-05-24T09:02:10Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147743 |
| . |
| Reference:
|
[1] Al-Abbadi, M. H., Darus, M.: Angular estimates for certain analytic univalent functions.Int. J. Open Problems Complex Analysis 2 (2010), 212-220. |
| Reference:
|
[2] Cho, H. R., Zhu, K.: Fock-Sobolev spaces and their Carleson measures.J. Funct. Anal. 263 (2012), 2483-2506. Zbl 1264.46017, MR 2964691, 10.1016/j.jfa.2012.08.003 |
| Reference:
|
[3] Duren, P. L.: Univalent Functions.Grundlehren der Mathematischen Wissenschaften 259, Springer, New York (1983). Zbl 0514.30001, MR 0708494 |
| Reference:
|
[4] Nehari, Z.: The Schwarzian derivative and schlicht functions.Bull. Am. Math. Soc. 55 (1949), 545-551. Zbl 0035.05104, MR 0029999, 10.1090/S0002-9904-1949-09241-8 |
| Reference:
|
[5] Nunokawa, M.: On some angular estimates of analytic functions.Math. Jap. 41 (1995), 447-452. Zbl 0822.30014, MR 1326978 |
| Reference:
|
[6] Peng, W., Wang, C., Zhu, K.: Convexity of area integral means for analytic functions.Complex Var. Elliptic Equ. 62 (2017), 307-317. Zbl 1376.30041, MR 3598979, 10.1080/17476933.2016.1218857 |
| Reference:
|
[7] Wang, C., Xiao, J.: Gaussian integral means of entire functions.Complex Anal. Oper. Theory 8 (2014), 1487-1505 addendum ibid. 10 495-503 2016. Zbl 1303.30024, MR 3261708, 10.1007/s11785-013-0339-x |
| Reference:
|
[8] Wang, C., Xiao, J., Zhu, K.: Logarithmic convexity of area integral means for analytic functions II.J. Aust. Math. Soc. 98 (2015), 117-128. Zbl 1316.30050, MR 3294311, 10.1017/S1446788714000457 |
| Reference:
|
[9] Wang, C., Zhu, K.: Logarithmic convexity of area integral means for analytic functions.Math. Scand. 114 (2014), 149-160. Zbl 1294.30104, MR 3178110, 10.7146/math.scand.a-16643 |
| Reference:
|
[10] Xiao, J., Xu, W.: Weighted integral means of mixed areas and lengths under holomorphic mappings.Anal. Theory Appl. 30 (2014), 1-19. Zbl 1313.32024, MR 3197626, 10.4208/ata.2014.v30.n1.1 |
| Reference:
|
[11] Xiao, J., Zhu, K.: Volume integral means of holomorphic functions.Proc. Am. Math. Soc. 139 (2011), 1455-1465. Zbl 1215.32002, MR 2748439, 10.1090/S0002-9939-2010-10797-9 |
| Reference:
|
[12] Zhu, K.: Analysis on Fock Spaces.Graduate Texts in Mathematics 263, Springer, New York (2012). Zbl 1262.30003, MR 2934601, 10.1007/978-1-4419-8801-0 |
| . |
