Pub Date : 2023-12-18DOI: 10.30970/ms.60.2.132-137
V. MatematychniStudii., No 60, M. Buyankara, M.
Let the function $fleft( z right) =z+sum_{k=2}^{infty}a{_{k}}z {^{k}}in A$ be locally univalent for $z in mathbb{D}%:={z in mathbb{C}:{|}z {|}<1}$ and $0leqalpha<1$.Then, $f$textit{ }$in $ $M(alpha )$ if and only if begin{equation*}ReBig( left( 1-z ^{2}right) frac{f(z )}{z }Big) >alpha,quad z in mathbb{D}.end{equation*}%Due to their geometrical characteristics, this class has a significantimpact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant begin{equation*}leftvert H_{2}left( 2right) left( fright) rightvert =leftverta_{2}a_{4}-{a_{3}^{2}}rightvert end{equation*}and some Toeplitz determinants begin{equation*}leftvert {T}_{3}left( 1right) left( fright) rightvert =leftvert 1-2%{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}rightvert, leftvert {T}_{3}left( 2right) left( fright) rightvert =leftvert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}rightvert end{equation*}of a subclass of analytic functions $M(alpha )$ in the open unit disk $%mathbb{D}$.
让函数 $fleft( z right) =z+sum_{k=2}^{infty}a{_{k}}z {^{k}}in A$ 对于 $z in mathbb{D}%:={z in mathbb{C}:{|}z {|}alpha,quad z in mathbb{D} 是局部等价的。end{equation*}%由于它们的几何特征，这一类函数对几何函数理论有着重要影响。在这篇文章中，我们得到了第二个汉克尔行列式的尖锐边界 H_{2}left( 2right) left( fright) rightvert =leftverta_{2}a_{4}-{a_{3}^{2}}（right）vert =leftvert 1-2%{a_{2}^{2}+2{a_{2}^{2}a_{3}-{a_{3}^{2} （right）vert、leftvert {T}_{3}left( 2right) left( fright) rightvert =leftvert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}} （end{equation*}）是开放单位盘 $%mathbb{D}$ 中解析函数 $M(alpha )$ 的子类。
{"title":"Hankel and Toeplitz determinants for a subclass of analytic functions","authors":"V. MatematychniStudii., No 60, M. Buyankara, M.","doi":"10.30970/ms.60.2.132-137","DOIUrl":"https://doi.org/10.30970/ms.60.2.132-137","url":null,"abstract":"Let the function $fleft( z right) =z+sum_{k=2}^{infty}a{_{k}}z {^{k}}in A$ be locally univalent for $z in mathbb{D}%:={z in mathbb{C}:{|}z {|}<1}$ and $0leqalpha<1$.Then, $f$textit{ }$in $ $M(alpha )$ if and only if begin{equation*}ReBig( left( 1-z ^{2}right) frac{f(z )}{z }Big) >alpha,quad z in mathbb{D}.end{equation*}%Due to their geometrical characteristics, this class has a significantimpact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant begin{equation*}leftvert H_{2}left( 2right) left( fright) rightvert =leftverta_{2}a_{4}-{a_{3}^{2}}rightvert end{equation*}and some Toeplitz determinants begin{equation*}leftvert {T}_{3}left( 1right) left( fright) rightvert =leftvert 1-2%{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}rightvert, leftvert {T}_{3}left( 2right) left( fright) rightvert =leftvert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}rightvert end{equation*}of a subclass of analytic functions $M(alpha )$ in the open unit disk $%mathbb{D}$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138994601","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 : 2023-12-18DOI: 10.30970/ms.60.2.208-222
O. G. Nakonechnyi, P. Zinko
We investigate problems of estimating solutions of linear operator equations with random parameters under conditions of uncertainty. We establish that the guaranteed rms estimates of the matrices are found as solutions of special optimization problems under certain observations of the system state. As the output signals of the system, we have observations that are described by linear functions from the solutions of such equations with random right-hand sides, which have unknown second moments. Under the condition that the observation second moments of the right-hand parts and errors belong to certain sets, it is proved that the guaranteed estimates are expressed through solutions of operator equation systems. When the linear operator is given by the scalar product of rectangular matrices, a quasi-minimax estimate and its error are constructed. It is shown that the quasi-minimax estimation error tends to zero when the number of observations tends to infinity. An example of calculating the guaranteed rms estimate of the matrix's trace, which is a solution of a matrix equation with a random parameter, is given.
{"title":"Estimates of matrix solutions of operator equations with random parameters under uncertainties","authors":"O. G. Nakonechnyi, P. Zinko","doi":"10.30970/ms.60.2.208-222","DOIUrl":"https://doi.org/10.30970/ms.60.2.208-222","url":null,"abstract":"We investigate problems of estimating solutions of linear operator equations with random parameters under conditions of uncertainty. We establish that the guaranteed rms estimates of the matrices are found as solutions of special optimization problems under certain observations of the system state. As the output signals of the system, we have observations that are described by linear functions from the solutions of such equations with random right-hand sides, which have unknown second moments. Under the condition that the observation second moments of the right-hand parts and errors belong to certain sets, it is proved that the guaranteed estimates are expressed through solutions of operator equation systems. When the linear operator is given by the scalar product of rectangular matrices, a quasi-minimax estimate and its error are constructed. It is shown that the quasi-minimax estimation error tends to zero when the number of observations tends to infinity. An example of calculating the guaranteed rms estimate of the matrix's trace, which is a solution of a matrix equation with a random parameter, is given.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139173323","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 : 2023-12-18DOI: 10.30970/ms.60.2.145-161
H. R. Jayarama, S. Bhoosnurmath, C. N. Chaithra, S. Naveenkumar
In the paper, we discuss the distribution of uniqueness and its elements over the extended complex plane from different polynomials of view. We obtain some new results regarding the structure and position of uniqueness. These new results have immense applications like classifying different expressions to be or not to be unique. The principal objective of the paper is to study the uniqueness of meromorphic functions when sharing a small function $a(z)$ IM with restricted finite order and its nonlinear differential polynomials. The lemma on the logarithmic derivative by Halburb and Korhonen (Journal of Mathematical Analysis and Applications, textbf{314} (2006), 477--87) is the starting point of this kind of research. In this direction, the current focus in this field involves exploring unique results for the differential-difference polynomials of meromorphic functions, covering both derivatives and differences. Liu et al. (Applied Mathematics A Journal of Chinese Universities, textbf{27} (2012), 94--104) have notably contributed to this research. Their research establishes that when $n leq k + 2$ for a finite-order transcendental entire function $f$ the differential-difference polynomial$[f^{n}f(z+c)]^{(k)} - alpha(z)$ has infinitely many zeros. Here, $alpha(z)$ is characterized by its smallness relatively to $f$. Additionally, for two distinct meromorphic functions $f$ and $g$, both of finite order, if the differential-difference polynomials $[f^{n}f(z+c)]^{(k)}$ and $[g^{n}g(z+c)]^{(k)}$ share the value $1$ in the same set, then $f(z)=c_1e^{dz},$ $g(z)=c_2e^{-dz}.$ We prove two results, which significantly generalize the results of Dyavanal and Mathai (Ukrainian Math. J., textbf{71} (2019), 1032--1042), and Zhang and Xu (Comput. Math. Appl., textbf{61} (2011), 722-730) and citing a proper example we have shown that the result is true only for a particular case. Finally, we present the compact version of the same result as an improvement.
{"title":"Uniqueness of Meromorphic Functions With Nonlinear Differential Polynomials Sharing a Small Function IM","authors":"H. R. Jayarama, S. Bhoosnurmath, C. N. Chaithra, S. Naveenkumar","doi":"10.30970/ms.60.2.145-161","DOIUrl":"https://doi.org/10.30970/ms.60.2.145-161","url":null,"abstract":"In the paper, we discuss the distribution of uniqueness and its elements over the extended complex plane from different polynomials of view. We obtain some new results regarding the structure and position of uniqueness. These new results have immense applications like classifying different expressions to be or not to be unique. The principal objective of the paper is to study the uniqueness of meromorphic functions when sharing a small function $a(z)$ IM with restricted finite order and its nonlinear differential polynomials. The lemma on the logarithmic derivative by Halburb and Korhonen (Journal of Mathematical Analysis and Applications, textbf{314} (2006), 477--87) is the starting point of this kind of research. In this direction, the current focus in this field involves exploring unique results for the differential-difference polynomials of meromorphic functions, covering both derivatives and differences. Liu et al. (Applied Mathematics A Journal of Chinese Universities, textbf{27} (2012), 94--104) have notably contributed to this research. Their research establishes that when $n leq k + 2$ for a finite-order transcendental entire function $f$ the differential-difference polynomial$[f^{n}f(z+c)]^{(k)} - alpha(z)$ has infinitely many zeros. Here, $alpha(z)$ is characterized by its smallness relatively to $f$. Additionally, for two distinct meromorphic functions $f$ and $g$, both of finite order, if the differential-difference polynomials $[f^{n}f(z+c)]^{(k)}$ and $[g^{n}g(z+c)]^{(k)}$ share the value $1$ in the same set, then $f(z)=c_1e^{dz},$ $g(z)=c_2e^{-dz}.$ We prove two results, which significantly generalize the results of Dyavanal and Mathai (Ukrainian Math. J., textbf{71} (2019), 1032--1042), and Zhang and Xu (Comput. Math. Appl., textbf{61} (2011), 722-730) and citing a proper example we have shown that the result is true only for a particular case. Finally, we present the compact version of the same result as an improvement.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139175506","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 : 2023-12-18DOI: 10.30970/ms.60.2.115-131
M. M. Sheremeta
The properties of Laplace-Stieltjes-type integrals $I(r)=int_{0}^{infty}a(x)f(xr)dF(x)$ are studied, where $F$ is a non-negative non-decreasing unbounded continuous on the right function on $[0,,+infty)$,$f(z)=sum_{k=0}^{infty}f_kz^k$ is an entire transcendental function with $f_kge 0$ for all $kge0$, and a function $a(x)ge 0$ on $[0,,+infty)$ is such that the Lebesgue-Stieltjes integral $int_{0}^{K}a(x)f(xr)dF(x)$ exists for every $rge 0$ and$K in [0,,+infty)$.For the maximum of the integrand $mu(r)=sup{a(x)f(xr)colon xge 0}$ it is proved that if$$varliminflimits_{xto+infty}frac{f^{-1}left(1/a(x)right)}{x}=R_{mu}$$ then $mu(r)<+infty$ for $rR_{mu}$. The relationship between $R_{mu}$ and the radius $R_c$ of convergence of the integral $I(r)$ was found. The concept of the central point $nu(r)$ of the maximum of the integrand is introduced and the formula for finding $ln mu(r)$ over $nu(r)$ is proved.Under certain conditions on the function $F$, estimates of $I(r)$ in terms of $mu(r)$ are obtained, and in the case when $R_{mu}=+infty$,in terms of generalized orders, a relation is established between the growth $mu(r)$ and $I(r)$ and the decrease of the function $a(x)$.
{"title":"Properties of Laplace-Stieltjes-type integrals","authors":"M. M. Sheremeta","doi":"10.30970/ms.60.2.115-131","DOIUrl":"https://doi.org/10.30970/ms.60.2.115-131","url":null,"abstract":"The properties of Laplace-Stieltjes-type integrals $I(r)=int_{0}^{infty}a(x)f(xr)dF(x)$ are studied, where $F$ is a non-negative non-decreasing unbounded continuous on the right function on $[0,,+infty)$,$f(z)=sum_{k=0}^{infty}f_kz^k$ is an entire transcendental function with $f_kge 0$ for all $kge0$, and a function $a(x)ge 0$ on $[0,,+infty)$ is such that the Lebesgue-Stieltjes integral $int_{0}^{K}a(x)f(xr)dF(x)$ exists for every $rge 0$ and$K in [0,,+infty)$.For the maximum of the integrand $mu(r)=sup{a(x)f(xr)colon xge 0}$ it is proved that if$$varliminflimits_{xto+infty}frac{f^{-1}left(1/a(x)right)}{x}=R_{mu}$$ then $mu(r)<+infty$ for $rR_{mu}$. The relationship between $R_{mu}$ and the radius $R_c$ of convergence of the integral $I(r)$ was found. The concept of the central point $nu(r)$ of the maximum of the integrand is introduced and the formula for finding $ln mu(r)$ over $nu(r)$ is proved.Under certain conditions on the function $F$, estimates of $I(r)$ in terms of $mu(r)$ are obtained, and in the case when $R_{mu}=+infty$,in terms of generalized orders, a relation is established between the growth $mu(r)$ and $I(r)$ and the decrease of the function $a(x)$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139173764","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 : 2023-12-18DOI: 10.30970/ms.60.2.191-200
V. Baksa, A. I. Bandura
There is introduced a concept of index for the Fueter regular function of the quaternionic variables. There are considered three approaches (Fueter, Sudbery and Mariconda) constructing the Fueter regular function from a holomorphic function of complex variable. Using Mariconda's approach there are constucted some analogs of such elementary functions as the exponent, the sine and the cosine. For the Mariconda analogs we proved that they have bounded index and their indices equal 1, 2, 2, respectively. Using recent results on sum of entire functions whose derivatives are of bounded index it is established that the Fueter regular function constructed by Mariconda's approach is of bounded index, if the derivatives of its addends have bounded index. Also there was examined a function of the form $H(q)=f_1(x_0+ix_1)+jf_2(x_2+ix_3)$, where $f_1$ and $f_2$ are entire functions of complex variable. For the function $H$ it is proved its Fueter regularity and index boundedness if the first order derivatives of $f_1$ and $f_2$ have bounded index. Moreover, the index of the function $H$ does not exceed the maximum of indices of the functions $f'_1$ and $f'_2$ increased by $1$.
{"title":"On an attempt to introduce a notion of bounded index for the Fueter regular functions of the quaternionic variable","authors":"V. Baksa, A. I. Bandura","doi":"10.30970/ms.60.2.191-200","DOIUrl":"https://doi.org/10.30970/ms.60.2.191-200","url":null,"abstract":"There is introduced a concept of index for the Fueter regular function of the quaternionic variables. There are considered three approaches (Fueter, Sudbery and Mariconda) constructing the Fueter regular function from a holomorphic function of complex variable. Using Mariconda's approach there are constucted some analogs of such elementary functions as the exponent, the sine and the cosine. For the Mariconda analogs we proved that they have bounded index and their indices equal 1, 2, 2, respectively. Using recent results on sum of entire functions whose derivatives are of bounded index it is established that the Fueter regular function constructed by Mariconda's approach is of bounded index, if the derivatives of its addends have bounded index. Also there was examined a function of the form $H(q)=f_1(x_0+ix_1)+jf_2(x_2+ix_3)$, where $f_1$ and $f_2$ are entire functions of complex variable. For the function $H$ it is proved its Fueter regularity and index boundedness if the first order derivatives of $f_1$ and $f_2$ have bounded index. Moreover, the index of the function $H$ does not exceed the maximum of indices of the functions $f'_1$ and $f'_2$ increased by $1$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139176182","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 : 2023-12-18DOI: 10.30970/ms.60.2.180-190
S. M. Bak, G. Kovtonyuk
The paper deals with the Fermi--Pasta--Ulam type systems that describe an infinite systems of nonlinearly coupled particles with nonlocal interaction on a two dimensional lattice. It is assumed that each particle interacts nonlinearly with several neighbors horizontally and vertically on both sides. The main result concerns the existence of traveling waves solutions with periodic relative displacement profiles. We obtain sufficient conditions for the existence of such solutions with the aid of critical point method and a suitable version of the Mountain Pass Theorem for functionals satisfying the Cerami condition instead of the Palais--Smale condition. We prove that under natural assumptions there exist monotone traveling waves.
{"title":"Periodic traveling waves in Fermi–Pasta–Ulam type systems with nonlocal interaction on 2d-lattice","authors":"S. M. Bak, G. Kovtonyuk","doi":"10.30970/ms.60.2.180-190","DOIUrl":"https://doi.org/10.30970/ms.60.2.180-190","url":null,"abstract":"The paper deals with the Fermi--Pasta--Ulam type systems that describe an infinite systems of nonlinearly coupled particles with nonlocal interaction on a two dimensional lattice. It is assumed that each particle interacts nonlinearly with several neighbors horizontally and vertically on both sides. The main result concerns the existence of traveling waves solutions with periodic relative displacement profiles. We obtain sufficient conditions for the existence of such solutions with the aid of critical point method and a suitable version of the Mountain Pass Theorem for functionals satisfying the Cerami condition instead of the Palais--Smale condition. We prove that under natural assumptions there exist monotone traveling waves.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139176047","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 : 2023-12-18DOI: 10.30970/ms.60.2.201-207
A. Mukharjee
The pre-open sets are a generalization of open sets of topological spaces. In this paper, we introduce and study a notion of po-paracompact spaces via pre-open sets on topological spaces. We see that po-paracompact spaces are equivalent to nearly paracompact spaces. However, we find new characterizations to nearly paracompact spaces when we study it in the sense of poparacompact spaces. We see that a topological space is nearly paracompact if and only if each regularly open cover of the topological space has a locally finite pre-open refinement. We also show that four statements involving pre-open sets on an almost regular topological space are equivalent. A result on a subspace of a topological space is also obtained in term of pre-open sets.
{"title":"A new approach to nearly paracompact spaces","authors":"A. Mukharjee","doi":"10.30970/ms.60.2.201-207","DOIUrl":"https://doi.org/10.30970/ms.60.2.201-207","url":null,"abstract":"The pre-open sets are a generalization of open sets of topological spaces. In this paper, we introduce and study a notion of po-paracompact spaces via pre-open sets on topological spaces. We see that po-paracompact spaces are equivalent to nearly paracompact spaces. However, we find new characterizations to nearly paracompact spaces when we study it in the sense of poparacompact spaces. We see that a topological space is nearly paracompact if and only if each regularly open cover of the topological space has a locally finite pre-open refinement. We also show that four statements involving pre-open sets on an almost regular topological space are equivalent. A result on a subspace of a topological space is also obtained in term of pre-open sets.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138964504","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 : 2023-12-18DOI: 10.30970/ms.60.2.173-179
O. Buhrii, O. Kholyavka, T. M. Bokalo
In this paper, we study the problem for a nonlinear hyperbolic Stokes system of the second order with an integral term.Sufficient conditions for the uniqueness of the weak solution of this problem are found in a bounded domain. The nonlinear term of the system contains a variable exponent of nonlinearity, which is a function of spatial variables.The problem is studied in ordinary Sobolev spaces and generalized Lebesgue spaces, which is quite natural in this case.
{"title":"Nonlocal hyperbolic Stokes system with variable exponent of nonlinearity","authors":"O. Buhrii, O. Kholyavka, T. M. Bokalo","doi":"10.30970/ms.60.2.173-179","DOIUrl":"https://doi.org/10.30970/ms.60.2.173-179","url":null,"abstract":"In this paper, we study the problem for a nonlinear hyperbolic Stokes system of the second order with an integral term.Sufficient conditions for the uniqueness of the weak solution of this problem are found in a bounded domain. The nonlinear term of the system contains a variable exponent of nonlinearity, which is a function of spatial variables.The problem is studied in ordinary Sobolev spaces and generalized Lebesgue spaces, which is quite natural in this case.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138965142","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 : 2023-12-18DOI: 10.30970/ms.60.2.138-144
I. Denega, Yaroslav V. Zabolotnyi
In 1934 Lavrentiev solved the problem of maximum ofproduct of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form$$T_{n}:={prodlimits_{k=1}^nr(B_{k},a_{k})}{bigg(prodlimits_{1leqslant k
1934 年，拉夫连季耶夫解决了两个不重叠的简单相连域的共形半径的最大积问题。在三个或更多点的情况下，许多学者考虑了形式为$$T_{n}的更一般莫比乌斯不变量的估计值：={prodlimits_{k=1}^nr(B_{k},a_{k})}{bigg(prodlimits_{1leqslant k
{"title":"Application of upper estimates for products of inner radii to distortion theorems for univalent functions","authors":"I. Denega, Yaroslav V. Zabolotnyi","doi":"10.30970/ms.60.2.138-144","DOIUrl":"https://doi.org/10.30970/ms.60.2.138-144","url":null,"abstract":"In 1934 Lavrentiev solved the problem of maximum ofproduct of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form$$T_{n}:={prodlimits_{k=1}^nr(B_{k},a_{k})}{bigg(prodlimits_{1leqslant k<pleqslant n}|a_{k}-a_{p}|bigg)^{-frac{2}{n-1}}},$$where $r(B,a)$ denotes the inner radius of the domain $B$ with respect to the point $a$ (for an infinitely distant point under the corresponding factor we understand the unit).In 1951 Goluzin for $n=3$ obtained an accurate evaluation for $T_{3}$.In 1980 Kuzmina showedthat the problem of the evaluation of $T_{4}$ isreduced to the smallest capacity problem in the certain continuumfamily and obtained the exact inequality for $T_{4}$.No other ultimate results in this problem for $n geqslant 5$ are known at present.In 2021 cite{Bakhtin2021,BahDen22} effective upper estimates are obtained for $T_{n}$, $n geqslant 2$.Among the possible applications of the obtained results in other tasks of the function theory are the so-called distortion theorems.In the paper we consider an application of upper estimates for products of inner radii to distortion theorems for univalent functionsin disk $U$, which map it onto a star-shaped domains relative to the origin.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139174463","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}
Let $R$ be a prime ring and $L$ a non-central Lie ideal of $R.$ The purpose of this paper is to describe generalized derivations of $R$ satisfying some algebraic identities locally on $L.$ More precisely, we consider two generalized derivations $F_1$ and $F_2$ of a prime ring $R$ satisfying one of the following identities:1. $F_1(x)circ y +x circ F_2(y) =0,$2. $[F_1(x),y] + F_2([x,y]) =0,$for all $x,y$ in a non-central Lie ideal $L$ of $R.$ Furthermore, as an application, we study continuous generalized derivations satisfying similar algebraic identities with power values on nonvoidopen subsets of a prime Banach algebra $A$. Our topological approach is based on Baire'scategory theorem and some properties from functional analysis.
{"title":"Generalized derivations acting on Lie ideals in prime rings and Banach algebras","authors":"A. Hermas, L. Oukhtite, L. Taoufiq","doi":"10.30970/ms.60.1.3-11","DOIUrl":"https://doi.org/10.30970/ms.60.1.3-11","url":null,"abstract":"Let $R$ be a prime ring and $L$ a non-central Lie ideal of $R.$ The purpose of this paper is to describe generalized derivations of $R$ satisfying some algebraic identities locally on $L.$ More precisely, we consider two generalized derivations $F_1$ and $F_2$ of a prime ring $R$ satisfying one of the following identities:1. $F_1(x)circ y +x circ F_2(y) =0,$2. $[F_1(x),y] + F_2([x,y]) =0,$for all $x,y$ in a non-central Lie ideal $L$ of $R.$ Furthermore, as an application, we study continuous generalized derivations satisfying similar algebraic identities with power values on nonvoidopen subsets of a prime Banach algebra $A$. Our topological approach is based on Baire'scategory theorem and some properties from functional analysis.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099710","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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