Geodesic
Convexities of Gaussian integral means and weighted integral means for analytic functions
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 525-543
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/CMJ.2018.0432-17
Classification : 30H10, 30H20
Keywords: Gaussian integral means; weighted integral means; analytic function; \nobreak convexity 
@article{10_21136_CMJ_2018_0432_17,
     author = {Li, Haiying and Liu, Taotao},
     title = {Convexities of {Gaussian} integral means and weighted integral means for analytic functions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {525--543},
     year = {2019},
     volume = {69},
     number = {2},
     doi = {10.21136/CMJ.2018.0432-17},
     mrnumber = {3959963},
     zbl = {07088803},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0432-17/}
}
TY  - JOUR
AU  - Li, Haiying
AU  - Liu, Taotao
TI  - Convexities of Gaussian integral means and weighted integral means for analytic functions
JO  - Czechoslovak Mathematical Journal
PY  - 2019
SP  - 525
EP  - 543
VL  - 69
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0432-17/
DO  - 10.21136/CMJ.2018.0432-17
LA  - en
ID  - 10_21136_CMJ_2018_0432_17
ER  - 
%0 Journal Article
%A Li, Haiying
%A Liu, Taotao
%T Convexities of Gaussian integral means and weighted integral means for analytic functions
%J Czechoslovak Mathematical Journal
%D 2019
%P 525-543
%V 69
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0432-17/
%R 10.21136/CMJ.2018.0432-17
%G en
%F 10_21136_CMJ_2018_0432_17
Li, Haiying; Liu, Taotao. Convexities of Gaussian integral means and weighted integral means for analytic functions. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 525-543. doi: 10.21136/CMJ.2018.0432-17

[1] Al-Abbadi, M. H., Darus, M.: Angular estimates for certain analytic univalent functions. Int. J. Open Problems Complex Analysis 2 (2010), 212-220.

[2] Cho, H. R., Zhu, K.: Fock-Sobolev spaces and their Carleson measures. J. Funct. Anal. 263 (2012), 2483-2506. | DOI | MR | JFM

[3] Duren, P. L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften 259, Springer, New York (1983). | MR | JFM

[4] Nehari, Z.: The Schwarzian derivative and schlicht functions. Bull. Am. Math. Soc. 55 (1949), 545-551. | DOI | MR | JFM

[5] Nunokawa, M.: On some angular estimates of analytic functions. Math. Jap. 41 (1995), 447-452. | MR | JFM

[6] Peng, W., Wang, C., Zhu, K.: Convexity of area integral means for analytic functions. Complex Var. Elliptic Equ. 62 (2017), 307-317. | DOI | MR | JFM

[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. | DOI | MR | JFM

[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. | DOI | MR | JFM

[9] Wang, C., Zhu, K.: Logarithmic convexity of area integral means for analytic functions. Math. Scand. 114 (2014), 149-160. | DOI | MR | JFM

[10] Xiao, J., Xu, W.: Weighted integral means of mixed areas and lengths under holomorphic mappings. Anal. Theory Appl. 30 (2014), 1-19. | DOI | MR | JFM

[11] Xiao, J., Zhu, K.: Volume integral means of holomorphic functions. Proc. Am. Math. Soc. 139 (2011), 1455-1465. | DOI | MR | JFM

[12] Zhu, K.: Analysis on Fock Spaces. Graduate Texts in Mathematics 263, Springer, New York (2012). | DOI | MR | JFM

Cité par Sources :