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

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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 
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     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/}
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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

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