Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.18
C. Pintea
"We denote by ${rm Hess}^+(f)$ the set of all points $pinmathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:mathbb{R}^nlongrightarrowmathbb{R}$ is positive definite. In this paper we prove several properties of real-valued functions of several variables by showing the connectedness of their level sets for sufficiently high levels, under the boundedness assumption on the critical set. In the case of three variables we also prove the convexity of the levels surfaces for sufficiently high levels, under the additional boundedness assumption on the ${rm Hess}^+$ complement. The selection of the {em a priori} convex levels, among the connected regular ones, is done through the positivity of the Gauss curvature function which ensure an ovaloidal shape of the levels to be selected. The ovaloidal shape of a level set makes a diffeomorphism out of the associated Gauss map. This outcome Gauss map diffeomorphism is then extended to a smooth homeomorphism which is used afterwards to construct one-parameter families of smooth homeomorphisms of Loewner chain flavor. "
“我们用${rm Hess}^+(f)$表示mathbb{R}^n$中所有点$p $的集合，使得$C^2$-光滑函数$f:mathbb{R}^n lonightarrow mathbb{R}$的Hessian矩阵$H_p(f)$是正定的。在临界集上的有界性假设下，通过证明多变量实值函数在足够高水平下的水平集的连通性，证明了多变量实值函数的几个性质。在三个变量的情况下，在${rm Hess}^+$补上的附加有界性假设下，我们也证明了足够高的水平面的凸性。{em a priori}凸层的选择，在连接的规则层中，是通过高斯曲率函数的正性来完成的，这确保了要选择的层的椭圆形。水平集的卵形使相关的高斯映射产生微分同构。这一结果将高斯映射微分同构推广到光滑同胚，并用于构造Loewner链型光滑同胚的单参数族。
{"title":"The level sets of functions with bounded criticalsets and bounded Hess+ complements","authors":"C. Pintea","doi":"10.24193/subbmath.2022.2.18","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.18","url":null,"abstract":"\"We denote by ${rm Hess}^+(f)$ the set of all points $pinmathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f:mathbb{R}^nlongrightarrowmathbb{R}$ is positive definite. In this paper we prove several properties of real-valued functions of several variables by showing the connectedness of their level sets for sufficiently high levels, under the boundedness assumption on the critical set. In the case of three variables we also prove the convexity of the levels surfaces for sufficiently high levels, under the additional boundedness assumption on the ${rm Hess}^+$ complement. The selection of the {em a priori} convex levels, among the connected regular ones, is done through the positivity of the Gauss curvature function which ensure an ovaloidal shape of the levels to be selected. The ovaloidal shape of a level set makes a diffeomorphism out of the associated Gauss map. This outcome Gauss map diffeomorphism is then extended to a smooth homeomorphism which is used afterwards to construct one-parameter families of smooth homeomorphisms of Loewner chain flavor. \"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81571693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.16
R. Hernández
"Using the Loewner Chain Theory, we obtain a new criterion of univalence in Cn in terms of the Schwarzian derivative for locally biholomorphic mappings. We as well derive explicitly the formula of this Schwarzian derivative using the numerical method of approximation of zeros due by Halley."
{"title":"A criterion of univalence in Cn in terms of the Schwarzian derivative","authors":"R. Hernández","doi":"10.24193/subbmath.2022.2.16","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.16","url":null,"abstract":"\"Using the Loewner Chain Theory, we obtain a new criterion of univalence in Cn in terms of the Schwarzian derivative for locally biholomorphic mappings. We as well derive explicitly the formula of this Schwarzian derivative using the numerical method of approximation of zeros due by Halley.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89060323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.05
Oliver Roth
"We survey a number of recent generalizations and sharpenings of Nehari's extension of Schwarz' lemma for holomorphic self{maps of the unit disk. In particular, we discuss the case of in nitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary."
{"title":"The Nehari-Schwarz lemma and infinitesimal boundary rigidity of bounded holomorphic functions","authors":"Oliver Roth","doi":"10.24193/subbmath.2022.2.05","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.05","url":null,"abstract":"\"We survey a number of recent generalizations and sharpenings of Nehari's extension of Schwarz' lemma for holomorphic self{maps of the unit disk. In particular, we discuss the case of in nitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79731767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.14
T. Akyel, M. Lanza de Cristoforis
"Let $Omega^{i}$, $Omega^{o}$ be bounded open connected subsets of ${mathbb{R}}^{n}$ that contain the origin. Let $Omega(epsilon)equiv Omega^{o}setminusepsilonoverline{Omega^i}$ for small $epsilon>0$. Then we consider a linear transmission problem for the Helmholtz equation in the pair of domains $epsilon Omega^i$ and $Omega(epsilon)$ with Neumann boundary conditions on $partialOmega^o$. Under appropriate conditions on the wave numbers in $epsilon Omega^i$ and $Omega(epsilon)$ and on the parameters involved in the transmission conditions on $epsilon partialOmega^i$, the transmission problem has a unique solution $(u^i(epsilon,cdot), u^o(epsilon,cdot))$ for small values of $epsilon>0$. Here $u^i(epsilon,cdot) $ and $u^o(epsilon,cdot) $ solve the Helmholtz equation in $epsilon Omega^i$ and $Omega(epsilon)$, respectively. Then we prove that if $xiinoverline{Omega^i}$ and $xiin mathbb{R}^nsetminus Omega^i$ then the rescaled solutions $u^i(epsilon,epsilonxi) $ and $u^o(epsilon,epsilonxi)$ can be expanded into a convergent power expansion of $epsilon$, $kappa_nepsilonlogepsilon$, $delta_{2,n}log^{-1}epsilon$, $ kappa_nepsilonlog^2epsilon $ for $epsilon$ small enough. Here $kappa_{n}=1$ if $n$ is even and $kappa_{n}=0$ if $n$ is odd and $delta_{2,2}equiv 1$ and $delta_{2,n}equiv 0$ if $ngeq 3$."
{"title":"\"Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation. A functional analytic approach\"","authors":"T. Akyel, M. Lanza de Cristoforis","doi":"10.24193/subbmath.2022.2.14","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.14","url":null,"abstract":"\"Let $Omega^{i}$, $Omega^{o}$ be bounded open connected subsets of ${mathbb{R}}^{n}$ that contain the origin. Let $Omega(epsilon)equiv Omega^{o}setminusepsilonoverline{Omega^i}$ for small $epsilon>0$. Then we consider a linear transmission problem for the Helmholtz equation in the pair of domains $epsilon Omega^i$ and $Omega(epsilon)$ with Neumann boundary conditions on $partialOmega^o$. Under appropriate conditions on the wave numbers in $epsilon Omega^i$ and $Omega(epsilon)$ and on the parameters involved in the transmission conditions on $epsilon partialOmega^i$, the transmission problem has a unique solution $(u^i(epsilon,cdot), u^o(epsilon,cdot))$ for small values of $epsilon>0$. Here $u^i(epsilon,cdot) $ and $u^o(epsilon,cdot) $ solve the Helmholtz equation in $epsilon Omega^i$ and $Omega(epsilon)$, respectively. Then we prove that if $xiinoverline{Omega^i}$ and $xiin mathbb{R}^nsetminus Omega^i$ then the rescaled solutions $u^i(epsilon,epsilonxi) $ and $u^o(epsilon,epsilonxi)$ can be expanded into a convergent power expansion of $epsilon$, $kappa_nepsilonlogepsilon$, $delta_{2,n}log^{-1}epsilon$, $ kappa_nepsilonlog^2epsilon $ for $epsilon$ small enough. Here $kappa_{n}=1$ if $n$ is even and $kappa_{n}=0$ if $n$ is odd and $delta_{2,2}equiv 1$ and $delta_{2,n}equiv 0$ if $ngeq 3$.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91391548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.19
N. H. Mohammed, N. Cho, E. A. Adegani, T. Bulboacă
"The error function takes place in a wide range in the elds of mathe- matics, mathematical physics and natural sciences. The aim of the current paper is to investigate certain properties such as univalence and close-to-convexity of normalized imaginary error function, which its region is symmetric with respect to the real axis. Some other outcomes are also obtained."
{"title":"Geometric properties of normalized imaginary error function","authors":"N. H. Mohammed, N. Cho, E. A. Adegani, T. Bulboacă","doi":"10.24193/subbmath.2022.2.19","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.19","url":null,"abstract":"\"The error function takes place in a wide range in the elds of mathe- matics, mathematical physics and natural sciences. The aim of the current paper is to investigate certain properties such as univalence and close-to-convexity of normalized imaginary error function, which its region is symmetric with respect to the real axis. Some other outcomes are also obtained.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75357303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.12
M. Cristea
"We generalize some cluster sets theorems of Tsuji and Iversen from plane holomorphic mappings to the class of ring mappings."
“我们将Tsuji和Iversen的一些聚类集定理从平面全纯映射推广到环映射类。”
{"title":"On some cluster sets problems","authors":"M. Cristea","doi":"10.24193/subbmath.2022.2.12","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.12","url":null,"abstract":"\"We generalize some cluster sets theorems of Tsuji and Iversen from plane holomorphic mappings to the class of ring mappings.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"116 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87880651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.01
Ian D. Graham, H. Hamada, G. Kohr, M. Kohr
"In this paper, we survey recent results obtained by the authors on the preservations of the first elements of (g-) Loewner chains and the Bloch mappings by the Roper-Suffridge type extension operators, the Muir type extension operators and the Pfaltzgraff-Suffridge type extension operators into the mappings on the domains in the complex Banach spaces."
{"title":"\"g-Loewner chains, Bloch functions and extension operators into the family of locally biholomorphic mappings in infinite dimensional spaces\"","authors":"Ian D. Graham, H. Hamada, G. Kohr, M. Kohr","doi":"10.24193/subbmath.2022.2.01","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.01","url":null,"abstract":"\"In this paper, we survey recent results obtained by the authors on the preservations of the first elements of (g-) Loewner chains and the Bloch mappings by the Roper-Suffridge type extension operators, the Muir type extension operators and the Pfaltzgraff-Suffridge type extension operators into the mappings on the domains in the complex Banach spaces.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"144 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75954281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.10
C. Chu
"We construct a natural linear isomorphism between the little Bloch space B0 and the Banach space c0 of complex null sequences. This paper is written for the special issue of Studia Universitatis Babes-Bolyai Mathematica in memory of Professor Gabriela Kohr."
{"title":"A note on Bloch functions","authors":"C. Chu","doi":"10.24193/subbmath.2022.2.10","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.10","url":null,"abstract":"\"We construct a natural linear isomorphism between the little Bloch space B0 and the Banach space c0 of complex null sequences. This paper is written for the special issue of Studia Universitatis Babes-Bolyai Mathematica in memory of Professor Gabriela Kohr.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85174816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.03
Filippo Bracci, P. Gumenyuk
In this paper we study the class of ``shearing'' holomorphic maps of the unit ball of the form $(z,w)mapsto (z+g(w), w)$. Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a Runge domain in $C^n$ which however cannot be embedded into a Loewner chain whose range is $C^n$.
{"title":"Shearing maps and a Runge map of the unit ball which does not embed into a Loewner chain with range Cn","authors":"Filippo Bracci, P. Gumenyuk","doi":"10.24193/subbmath.2022.2.03","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.03","url":null,"abstract":"In this paper we study the class of ``shearing'' holomorphic maps of the unit ball of the form $(z,w)mapsto (z+g(w), w)$. Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a Runge domain in $C^n$ which however cannot be embedded into a Loewner chain whose range is $C^n$.","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"133 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76679631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.24193/subbmath.2022.2.11
P. Curt
"In this paper, two subclasses of biholomorphic starlike mappings named Janowski starlike and Janowski almost starlike with complex parameters are introduced and studied. We determine $M$ such that holomorphic mappings $f$ which satisfy the condition $|Df(z)-I|le M$, $zin B^n$, are Janowski starlike, respectively Janowski almost starlike. We also derive sufficient conditions for normalized holomorphic mappings (expressed in terms of their coefficient bounds) to belong to one of the subclasses of mappings mentioned above."
{"title":"Janowski subclasses of starlike mappings","authors":"P. Curt","doi":"10.24193/subbmath.2022.2.11","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.11","url":null,"abstract":"\"In this paper, two subclasses of biholomorphic starlike mappings named Janowski starlike and Janowski almost starlike with complex parameters are introduced and studied. We determine $M$ such that holomorphic mappings $f$ which satisfy the condition $|Df(z)-I|le M$, $zin B^n$, are Janowski starlike, respectively Janowski almost starlike. We also derive sufficient conditions for normalized holomorphic mappings (expressed in terms of their coefficient bounds) to belong to one of the subclasses of mappings mentioned above.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82318688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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