Let $H$ be a subgroup of the plane affine group ${rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family ${ B_{m,n}(x,y) }$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ { x^m y^n } $ and ${ B_{m,n}(x,y) }$ have textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family ${ B_{m,n}(x,y) }$.
{"title":"Quasi-monomials with respect to subgroups of the plane affine group","authors":"N. Samaruk","doi":"10.30970/ms.59.1.3-11","DOIUrl":"https://doi.org/10.30970/ms.59.1.3-11","url":null,"abstract":"Let $H$ be a subgroup of the plane affine group ${rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family ${ B_{m,n}(x,y) }$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ { x^m y^n } $ and ${ B_{m,n}(x,y) }$ have textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family ${ B_{m,n}(x,y) }$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42107140","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}
It is well-known that one of open problems in the theory of Leibniz algebras is to find asuitable generalization of Lie’s third theorem which associates a (local) Lie group to any Liealgebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Liegroup for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution ofthis problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibnizalgebras. A digroup is a nonempty set equipped with two binary associative operations, aunary operation and a nullary operation satisfying additional axioms relating these operations.Digroups generalize groups and have close relationships with the dimonoids and dialgebras,the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied themethod of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitraryrank, in particular, they considered a monogenic case separately. In this paper, we give a simplerand more convenient digroup model of the free monogenic digroup. We construct a new classof digroups which are based on commutative groups and show how the free monogenic groupcan be obtained from the free monogenic digroup by a suitable factorization.
{"title":"A new model of the free monogenic digroup","authors":"Y. Zhuchok, G. Pilz","doi":"10.30970/ms.59.1.12-19","DOIUrl":"https://doi.org/10.30970/ms.59.1.12-19","url":null,"abstract":"It is well-known that one of open problems in the theory of Leibniz algebras is to find asuitable generalization of Lie’s third theorem which associates a (local) Lie group to any Liealgebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Liegroup for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution ofthis problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibnizalgebras. A digroup is a nonempty set equipped with two binary associative operations, aunary operation and a nullary operation satisfying additional axioms relating these operations.Digroups generalize groups and have close relationships with the dimonoids and dialgebras,the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied themethod of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitraryrank, in particular, they considered a monogenic case separately. In this paper, we give a simplerand more convenient digroup model of the free monogenic digroup. We construct a new classof digroups which are based on commutative groups and show how the free monogenic groupcan be obtained from the free monogenic digroup by a suitable factorization.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47912568","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}
The problem of optimal control of the system described by the oblique derivative problem forthe elliptic equation of the second order is studied. Cases of internal and boundary managementare considered. The quality criterion is given by the sum of volume and surface integrals.The coefficients of the equation and the boundary condition allow power singularities of arbitraryorder in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special H"{o}lder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will bethe solution to the given problem. The necessary and sufficient conditions for the existence of the optimal solution of the systemdescribed by the boundary value problem for the elliptic equation with degeneracy have been established.
{"title":"Optimal control in the boundary value problem for elliptic equations with degeneration","authors":"I. Pukal’skii, B. Yashan","doi":"10.30970/ms.59.1.76-85","DOIUrl":"https://doi.org/10.30970/ms.59.1.76-85","url":null,"abstract":"The problem of optimal control of the system described by the oblique derivative problem forthe elliptic equation of the second order is studied. Cases of internal and boundary managementare considered. The quality criterion is given by the sum of volume and surface integrals.The coefficients of the equation and the boundary condition allow power singularities of arbitraryorder in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special H\"{o}lder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will bethe solution to the given problem. \u0000The necessary and sufficient conditions for the existence of the optimal solution of the systemdescribed by the boundary value problem for the elliptic equation with degeneracy have been established.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43927543","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}
In the paper we consider a random analytic function of the form$$f(z,omega )=sumlimits_{n=0}^{+infty}varepsilon_n(omega_1)xi_n(omega_2)a_nz^n.$$Here $(varepsilon_n)$ is a sequence of inde-pendent Steinhausrandom variables, $(xi_n)$ is a sequence of indepen-dent standard complex Gaussianrandom variables, and a sequence of numbers $a_ninmathbb{C}$such that$a_0neq0, varlimsuplimits_{nto+infty}sqrt[n]{|a_n|}=1, sup{|a_n|colon ninmathbb{N}}=+infty.$We investigate asymptotic estimates of theprobability $p_0(r)=ln^-P{omegacolon f(z,omega )$ hasno zeros inside $rmathbb{D}}$ as $ruparrow1$ outside some set $E$ of finite logarithmic measure. Denote$N(r):=#{ncolon |a_n|r^n>1},$ $ s(r):=2sum_{n=0}^{+infty}ln^+(|a_n|r^{n}),$$ alpha:=varliminflimits_{ruparrow1}frac{ln N(r)}{lnfrac{1}{1-r}}.$ The article, in particular, proves the following statements:noi 1) if $alpha>4$ thencenterline{$displaystyle lim_{begin{substack} {ruparrow1 rnotin E}end{substack}}frac{ln(p_0(r)- s(r))}{ln N(r)}=1$;} noi2) if $alpha=+infty$ thencenterline{$displaystyle 0leqvarliminf_{begin{substack} {ruparrow1 rnotin E}end{substack}}frac{ln(p_0(r)- s(r))}{ln s(r)},quad varlimsup_{begin{substack} {ruparrow1 rnotin E}end{substack}}frac{ln(p_0(r)- s(r))}{ln s(r)}leqfrac1{2}.$} noiHere $E$ is a set of finite logarithmic measure. The obtained asymptotic estimates are in a certain sense best possible.Also we give an answer to an open question from !cite[p. 119]{Nishry2013} for such random functions.
{"title":"Analytic Gaussian functions in the unit disc: probability of zeros absence","authors":"A. Kuryliak, O. Skaskiv","doi":"10.30970/ms.59.1.29-45","DOIUrl":"https://doi.org/10.30970/ms.59.1.29-45","url":null,"abstract":"In the paper we consider a random analytic function of the form$$f(z,omega )=sumlimits_{n=0}^{+infty}varepsilon_n(omega_1)xi_n(omega_2)a_nz^n.$$Here $(varepsilon_n)$ is a sequence of inde-pendent Steinhausrandom variables, $(xi_n)$ is a sequence of indepen-dent standard complex Gaussianrandom variables, and a sequence of numbers $a_ninmathbb{C}$such that$a_0neq0, varlimsuplimits_{nto+infty}sqrt[n]{|a_n|}=1, sup{|a_n|colon ninmathbb{N}}=+infty.$We investigate asymptotic estimates of theprobability $p_0(r)=ln^-P{omegacolon f(z,omega )$ hasno zeros inside $rmathbb{D}}$ as $ruparrow1$ outside some set $E$ of finite logarithmic measure. Denote$N(r):=#{ncolon |a_n|r^n>1},$ $ s(r):=2sum_{n=0}^{+infty}ln^+(|a_n|r^{n}),$$ alpha:=varliminflimits_{ruparrow1}frac{ln N(r)}{lnfrac{1}{1-r}}.$ The article, in particular, proves the following statements:noi 1) if $alpha>4$ thencenterline{$displaystyle lim_{begin{substack} {ruparrow1 rnotin E}end{substack}}frac{ln(p_0(r)- s(r))}{ln N(r)}=1$;} \u0000noi2) if $alpha=+infty$ thencenterline{$displaystyle 0leqvarliminf_{begin{substack} {ruparrow1 rnotin E}end{substack}}frac{ln(p_0(r)- s(r))}{ln s(r)},quad varlimsup_{begin{substack} {ruparrow1 rnotin E}end{substack}}frac{ln(p_0(r)- s(r))}{ln s(r)}leqfrac1{2}.$} \u0000noiHere $E$ is a set of finite logarithmic measure. The obtained asymptotic estimates are in a certain sense best possible.Also we give an answer to an open question from !cite[p. 119]{Nishry2013} for such random functions.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46619737","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}
Initial-boundary value problems for parabolic and elliptic-parabolic (that is degenerated parabolic) equations in unbounded domains with respect to the spatial variables were studied by many authors. It is well known that in order to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic and elliptic-parabolic equations in unbounded domains we need some restrictions on behavior of solution as $|x|to +infty$ (for example, growth restriction of solution as $|x|to +infty$, or the solution to belong to some functional spaces).Note, that we need some restrictions on the data-in behavior as$|x|to +infty$ for the initial-boundary value problemsfor equations considered above to be solvable.
However, there are nonlinear parabolic equations for whichthe corresponding initial-boundary value problems are uniquely solvable withoutany conditions at infinity.
We prove the unique solvability of the initial-boundary value problemwithout conditions at infinity for some of the higher-orders anisotropic parabolic equationswith variable exponents of the nonlinearity. A priori estimate of the weak solutionsof this problem was also obtained. As far as we know, the initial-boundary value problem for the higher-orders anisotropic elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains were not considered before.
{"title":"Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity","authors":"M. M. Bokalo, O. V. Domanska","doi":"10.30970/ms.59.1.86-105","DOIUrl":"https://doi.org/10.30970/ms.59.1.86-105","url":null,"abstract":"Initial-boundary value problems for parabolic and elliptic-parabolic (that is degenerated parabolic) equations in unbounded domains with respect to the spatial variables were studied by many authors. It is well known that in order to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic and elliptic-parabolic equations in unbounded domains we need some restrictions on behavior of solution as $|x|to +infty$ (for example, growth restriction of solution as $|x|to +infty$, or the solution to belong to some functional spaces).Note, that we need some restrictions on the data-in behavior as$|x|to +infty$ for the initial-boundary value problemsfor equations considered above to be solvable.
 However, there are nonlinear parabolic equations for whichthe corresponding initial-boundary value problems are uniquely solvable withoutany conditions at infinity.
 We prove the unique solvability of the initial-boundary value problemwithout conditions at infinity for some of the higher-orders anisotropic parabolic equationswith variable exponents of the nonlinearity. A priori estimate of the weak solutionsof this problem was also obtained. As far as we know, the initial-boundary value problem for the higher-orders anisotropic elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains were not considered before.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135723436","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}
The logarithmic coefficients play an important role for different estimates in the theory of univalent functions.Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the second Hankel determinant of these coefficients, that is $H_{2,1}(F_f/2)$ was paid attention. We recall that if $f$ and $F$ are two analytic functions in $mathbb{D}$, the function $f$ is subordinate to $F$, written $f(z)prec F(z)$, if there exists an analytic function $omega$ in $mathbb{D}$ with $omega(0)=0$ and $|omega(z)|<1$, such that $f(z)=Fleft(omega(z)right)$ for all $zinmathbb{D}$. It is well-known that if $F$ is univalent in $mathbb{D}$, then $f(z)prec F(z)$ if and only if $f(0)=F(0)$ and $f(mathbb{D})subset F(mathbb{D})$.A function $finmathcal{A}$ is starlike with respect to symmetric points in $mathbb{D}$ iffor every $r$ close to $1,$ $r < 1$ and every $z_0$ on $|z| = r$ the angular velocity of $f(z)$about $f(-z_0)$ is positive at $z = z_0$ as $z$ traverses the circle $|z| = r$ in the positivedirection. In the current study, we obtain the sharp bounds of the second Hankel determinant of the logarithmic coefficients for families $mathcal{S}_s^*(psi)$ and $mathcal{C}_s(psi)$ where were defined by the concept subordination and $psi$ is considered univalent in $mathbb{D}$ with positive real part in $mathbb{D}$ and satisfies the condition $psi(0)=1$. Note that $fin mathcal{S}_s^*(psi)$ if[dfrac{2zf^prime(z)}{f(z)-f(-z)}precpsi(z),quad zinmathbb{D}]and $fin mathcal{C}_s(psi)$ if[dfrac{2(zf^prime(z))^prime}{f^prime(z)+f^prime(-z)}precpsi(z),quad zinmathbb{D}.]It is worthwhile mentioning that the given bounds in this paper extend and develop some related recent results in the literature. In addition, the results given in these theorems can be used for determining the upper bound of $leftvert H_{2,1}(F_f/2)rightvert$ for other popular families.
{"title":"Sharp bounds of logarithmic coefficient problems for functions with respect to symmetric points","authors":"N. H. Mohammed","doi":"10.30970/ms.59.1.68-75","DOIUrl":"https://doi.org/10.30970/ms.59.1.68-75","url":null,"abstract":"The logarithmic coefficients play an important role for different estimates in the theory of univalent functions.Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the second Hankel determinant of these coefficients, that is $H_{2,1}(F_f/2)$ was paid attention. We recall that if $f$ and $F$ are two analytic functions in $mathbb{D}$, the function $f$ is subordinate to $F$, written $f(z)prec F(z)$, if there exists an analytic function $omega$ in $mathbb{D}$ with $omega(0)=0$ and $|omega(z)|<1$, such that $f(z)=Fleft(omega(z)right)$ for all $zinmathbb{D}$. It is well-known that if $F$ is univalent in $mathbb{D}$, then $f(z)prec F(z)$ if and only if $f(0)=F(0)$ and $f(mathbb{D})subset F(mathbb{D})$.A function $finmathcal{A}$ is starlike with respect to symmetric points in $mathbb{D}$ iffor every $r$ close to $1,$ $r < 1$ and every $z_0$ on $|z| = r$ the angular velocity of $f(z)$about $f(-z_0)$ is positive at $z = z_0$ as $z$ traverses the circle $|z| = r$ in the positivedirection. In the current study, we obtain the sharp bounds of the second Hankel determinant of the logarithmic coefficients for families $mathcal{S}_s^*(psi)$ and $mathcal{C}_s(psi)$ where were defined by the concept subordination and $psi$ is considered univalent in $mathbb{D}$ with positive real part in $mathbb{D}$ and satisfies the condition $psi(0)=1$. Note that $fin mathcal{S}_s^*(psi)$ if[dfrac{2zf^prime(z)}{f(z)-f(-z)}precpsi(z),quad zinmathbb{D}]and $fin mathcal{C}_s(psi)$ if[dfrac{2(zf^prime(z))^prime}{f^prime(z)+f^prime(-z)}precpsi(z),quad zinmathbb{D}.]It is worthwhile mentioning that the given bounds in this paper extend and develop some related recent results in the literature. In addition, the results given in these theorems can be used for determining the upper bound of $leftvert H_{2,1}(F_f/2)rightvert$ for other popular families.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42384541","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 : 2023-01-23DOI: 10.30970/ms.58.2.212-221
A. I. Bandura, T. M. Salo, O. B. Skaskiv
In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the ErdH{o}s-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one v{I}.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the ErdH{o}s-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case.
{"title":"Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems","authors":"A. I. Bandura, T. M. Salo, O. B. Skaskiv","doi":"10.30970/ms.58.2.212-221","DOIUrl":"https://doi.org/10.30970/ms.58.2.212-221","url":null,"abstract":"In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the ErdH{o}s-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one v{I}.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the ErdH{o}s-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136296682","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 : 2023-01-23DOI: 10.30970/ms.58.2.182-200
M. Sheremeta, O. Skaskiv
The article introduces the concepts of pseudostarlikeness and pseudoconvexity in the direction of absolutely converges in $Pi_0={sinmathbb{C}^pcolon text{Re},s<0}$, $pinmathbb{N},$ the multiple Dirichlet series of the form$$ F(s)=e^{(h,s)}+sumnolimits_{|(n)|=|(n^0)|}^{+infty}f_{(n)}exp{(lambda_{(n)},s)}, quad s=(s_1,...,s_p)in {mathbb C}^p,quad pgeq 1,$$where $ lambda_{(n^0)}>h$, $text{Re},s<0Longleftrightarrow (text{Re},s_1<0,...,text{Re},s_p<0)$,$h=(h_1,...,h_p)in {mathbb R}^p_+$, $(n)=(n_1,...,n_p)in {mathbb N}^p$, $(n^0)=(n^0_1,...,n^0_p)in {mathbb N}^p$, $|(n)|=n_1+...+n_p$ and the sequences$lambda_{(n)}=(lambda^{(1)}_{n_1},...,lambda^{(p)}_{n_p})$ are such that $0c$ if $a_jge c_j$ for all $1le jle p$ and there exists at least one $j$ such that $a_j> c_j$. Let ${bf b}=(b_1,...,b_p)$ and $partial_{{bf b}}F( {s})=sumlimits_{j=1}^p b_jdfrac{partial F( {s})}{partial {s}_j}$ be the derivative of $F$ in the direction ${bf b}$. In this paper, in particular, the following assertions were obtained: 1) If ${bf b}>0$ and$sumlimits_{|(n)|=k_0}^{+infty}(lambda_{(n)},{bf b})|f_{(n)}|le (h,{bf b})$then $partial_{{bf b}}F( {s})not=0$ in $Pi_0:={scolon text{Re},s<0}$, i.e. $F$ is conformal in $Pi_0$ in the direction ${bf b}$ (Proposition 1).2) We say that function $F$ is pseudostarlike of the order $alphain [0,,(h,{bf b}))$ and the type$beta >0$ in the direction ${bf b}$ if$Big|frac{partial_{{bf b}}F( {s})}{F(s)}-(h, {bf b})Big|0$. In order that the function $F$ ispseudostarlike of the order $alpha$ and the type $beta$ in the direction ${bf b}> 0$, it is sufficient and in the case, when all $f_{(n)}le 0$, it is necessary that$sumlimits_{|(n)|=k_0}^{+infty}{((1+beta)lambda_{(n)}-(1-beta)h,{bf b})-2betaalpha}|f_{(n)}|le 2beta ((h,{bf b})-alpha)$ (Theorem 1).
{"title":"Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series","authors":"M. Sheremeta, O. Skaskiv","doi":"10.30970/ms.58.2.182-200","DOIUrl":"https://doi.org/10.30970/ms.58.2.182-200","url":null,"abstract":"The article introduces the concepts of pseudostarlikeness and pseudoconvexity in the direction of absolutely converges in $Pi_0={sinmathbb{C}^pcolon text{Re},s<0}$, $pinmathbb{N},$ the multiple Dirichlet series of the form$$ F(s)=e^{(h,s)}+sumnolimits_{|(n)|=|(n^0)|}^{+infty}f_{(n)}exp{(lambda_{(n)},s)}, quad s=(s_1,...,s_p)in {mathbb C}^p,quad pgeq 1,$$where $ lambda_{(n^0)}>h$, $text{Re},s<0Longleftrightarrow (text{Re},s_1<0,...,text{Re},s_p<0)$,$h=(h_1,...,h_p)in {mathbb R}^p_+$, $(n)=(n_1,...,n_p)in {mathbb N}^p$, $(n^0)=(n^0_1,...,n^0_p)in {mathbb N}^p$, $|(n)|=n_1+...+n_p$ and the sequences$lambda_{(n)}=(lambda^{(1)}_{n_1},...,lambda^{(p)}_{n_p})$ are such that $0c$ if $a_jge c_j$ for all $1le jle p$ and there exists at least one $j$ such that $a_j> c_j$. Let ${bf b}=(b_1,...,b_p)$ and $partial_{{bf b}}F( {s})=sumlimits_{j=1}^p b_jdfrac{partial F( {s})}{partial {s}_j}$ be the derivative of $F$ in the direction ${bf b}$. In this paper, in particular, the following assertions were obtained: 1) If ${bf b}>0$ and$sumlimits_{|(n)|=k_0}^{+infty}(lambda_{(n)},{bf b})|f_{(n)}|le (h,{bf b})$then $partial_{{bf b}}F( {s})not=0$ in $Pi_0:={scolon text{Re},s<0}$, i.e. $F$ is conformal in $Pi_0$ in the direction ${bf b}$ (Proposition 1).2) We say that function $F$ is pseudostarlike of the order $alphain [0,,(h,{bf b}))$ and the type$beta >0$ in the direction ${bf b}$ if$Big|frac{partial_{{bf b}}F( {s})}{F(s)}-(h, {bf b})Big|<betaBig|frac{partial_{{bf b}}F( {s})}{F(s)}-(2alpha-(h, {bf b}))Big|,quad sin Pi_0.$Let $0le alpha<(h,{bf b})$ and $beta>0$. In order that the function $F$ ispseudostarlike of the order $alpha$ and the type $beta$ in the direction ${bf b}> 0$, it is sufficient and in the case, when all $f_{(n)}le 0$, it is necessary that$sumlimits_{|(n)|=k_0}^{+infty}{((1+beta)lambda_{(n)}-(1-beta)h,{bf b})-2betaalpha}|f_{(n)}|le 2beta ((h,{bf b})-alpha)$ (Theorem 1).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41886376","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 : 2023-01-16DOI: 10.30970/ms.58.2.201-211
Sung Guen Kim
Let $nin mathbb{N}, ngeq 2.$ An element $x=(x_1, ldots, x_n)in E^n$ is called a {em norming point} of $Tin {mathcal L}(^n E)$ if $|x_1|=cdots=|x_n|=1$ and$|T(x)|=|T|,$ where ${mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $Tin {mathcal L}(^n E)$ we define the {em norming set} of $T$ centerline{$qopnamerelax o{Norm}(T)=Big{(x_1, ldots, x_n)in E^n: (x_1, ldots, x_n)~mbox{is a norming point of}~TBig}.$} By $i=(i_1,i_2,ldots,i_m)$ we denote the multi-index. In this paper we show the following: noi (a) Let $n, mgeq 2$ and let $l_1^n=mathbb{R}^n$ with the $l_1$-norm. Let $T=big(a_{i}big)_{1leq i_kleq n}in {mathcal L}(^ml_{1}^n)$ with $|T|=1.$Define $S=big(b_{i}big)_{1leq i_kleq n}in {mathcal L}(^n l_1^m)$ be such that $b_{i}=a_{i}$ if$|a_{i}|=1$ and $b_{i}=1$ if$|a_{i}|<1.$ Let $A={1, ldots, n}times cdotstimes{1, ldots, n}$ and $M={iin A: |a_{i}|<1}.$Then, centerline{$qopnamerelax o{Norm}(T)=bigcup_{(i_1, ldots, i_m)in M}Big{Big(big(t_1^{(1)}, ldots, t_{{i_1}-1}^{(1)}, 0, t_{{i_1}+1}^{(1)}, ldots, t_{n}^{(1)}big), big(t_1^{(2)}, ldots, t_{n}^{(2)}big), ldots, big(t_1^{(m)}, ldots, t_{n}^{(m)}big)Big),$} centerline{$Big(big(t_1^{(1)}, ldots, t_{n}^{(1)}big), big(t_1^{(2)}, ldots, t_{{i_2}-1}^{(2)}, 0, t_{{i_2}+1}^{(2)}, ldots, t_{n}^{(2)}big), big(t_1^{(3)}, ldots, t_{n}^{(3)}big), ldots, big(t_1^{(m)}, ldots, t_{n}^{(m)}big)Big),ldots$} centerline{$ldots, Big(big(t_1^{(1)}, ldots, t_{n}^{(1)}big), ldots, big(t_1^{(m-1)}, ldots, t_{n}^{(m-1)}big), big(t_1^{(m)}, ldots, t_{{i_m}-1}^{(m)}, 0, t_{{i_m}+1}^{(m)}, ldots, t_{n}^{(m)}big)Big)colon$} centerline{$ Big(big(t_1^{(1)}, ldots, t_{n}^{(1)}big), ldots, big(t_1^{(m)}, ldots, t_{n}^{(m)}big)Big)in qopnamerelax o{Norm}(S)Big}.$} This statement extend the results of [9]. noi (b) Using the result (a), we describe the norming sets of every $Tin {mathcal L}(^3l_{1}^2).$
{"title":"Remarks on the norming sets of ${mathcal L}(^ml_{1}^n)$ and description of the norming sets of ${mathcal L}(^3l_{1}^2)$","authors":"Sung Guen Kim","doi":"10.30970/ms.58.2.201-211","DOIUrl":"https://doi.org/10.30970/ms.58.2.201-211","url":null,"abstract":"Let $nin mathbb{N}, ngeq 2.$ An element $x=(x_1, ldots, x_n)in E^n$ is called a {em norming point} of $Tin {mathcal L}(^n E)$ if $|x_1|=cdots=|x_n|=1$ and$|T(x)|=|T|,$ where ${mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $Tin {mathcal L}(^n E)$ we define the {em norming set} of $T$ \u0000centerline{$qopnamerelax o{Norm}(T)=Big{(x_1, ldots, x_n)in E^n: (x_1, ldots, x_n)~mbox{is a norming point of}~TBig}.$} \u0000By $i=(i_1,i_2,ldots,i_m)$ we denote the multi-index. In this paper we show the following: \u0000noi (a) Let $n, mgeq 2$ and let $l_1^n=mathbb{R}^n$ with the $l_1$-norm. Let $T=big(a_{i}big)_{1leq i_kleq n}in {mathcal L}(^ml_{1}^n)$ with $|T|=1.$Define $S=big(b_{i}big)_{1leq i_kleq n}in {mathcal L}(^n l_1^m)$ be such that $b_{i}=a_{i}$ if$|a_{i}|=1$ and $b_{i}=1$ if$|a_{i}|<1.$ \u0000Let $A={1, ldots, n}times cdotstimes{1, ldots, n}$ and $M={iin A: |a_{i}|<1}.$Then, \u0000centerline{$qopnamerelax o{Norm}(T)=bigcup_{(i_1, ldots, i_m)in M}Big{Big(big(t_1^{(1)}, ldots, t_{{i_1}-1}^{(1)}, 0, t_{{i_1}+1}^{(1)}, ldots, t_{n}^{(1)}big), big(t_1^{(2)}, ldots, t_{n}^{(2)}big), ldots, big(t_1^{(m)}, ldots, t_{n}^{(m)}big)Big),$} \u0000centerline{$Big(big(t_1^{(1)}, ldots, t_{n}^{(1)}big), big(t_1^{(2)}, ldots, t_{{i_2}-1}^{(2)}, 0, t_{{i_2}+1}^{(2)}, ldots, t_{n}^{(2)}big), big(t_1^{(3)}, ldots, t_{n}^{(3)}big), ldots, big(t_1^{(m)}, ldots, t_{n}^{(m)}big)Big),ldots$} \u0000centerline{$ldots, Big(big(t_1^{(1)}, ldots, t_{n}^{(1)}big), ldots, big(t_1^{(m-1)}, ldots, t_{n}^{(m-1)}big), big(t_1^{(m)}, ldots, t_{{i_m}-1}^{(m)}, 0, t_{{i_m}+1}^{(m)}, ldots, t_{n}^{(m)}big)Big)colon$} \u0000centerline{$ Big(big(t_1^{(1)}, ldots, t_{n}^{(1)}big), ldots, big(t_1^{(m)}, ldots, t_{n}^{(m)}big)Big)in qopnamerelax o{Norm}(S)Big}.$} \u0000This statement extend the results of [9]. \u0000noi (b) Using the result (a), we describe the norming sets of every $Tin {mathcal L}(^3l_{1}^2).$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46894785","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 : 2023-01-16DOI: 10.30970/ms.58.2.142-158
V. Kostov
The partial theta function is the sum of the series medskipcenterline{$displaystyletheta (q,x):=sumnolimits _{j=0}^{infty}q^{j(j+1)/2}x^j$,}medskipnoi where $q$is a real or complex parameter ($|q|<1$). Its name is due to similaritieswith the formula for the Jacobi theta function$Theta (q,x):=sum _{j=-infty}^{infty}q^{j^2}x^j$. The function $theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, the theory of (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently in the study of section-hyperbolic polynomials,i.~e. real polynomials with all coefficients positive,with all roots real negative and all whose sections (i.~e. truncations)are also real-rooted.For each $q$ fixed,$theta$ is an entire function of order $0$ in the variable~$x$. When$q$ is real and $qin (0,0.3092ldots )$, $theta (q,.)$ is a function of theLaguerre-P'olyaclass $mathcal{L-P}I$. More generally, for $q in (0,1)$, the function $theta (q,.)$ is the product of a realpolynomialwithout real zeros and a function of the class $mathcal{L-P}I$. Thus it isan entire function withinfinitely-many negative, with no positive and with finitely-many complexconjugate zeros. The latter are known to belongto an explicitly defined compact domain containing $0$ andindependent of $q$ while the negative zeros tend to infinity as ageometric progression with ratio $1/q$. A similar result is true for$qin (-1,0)$ when there are also infinitely-many positive zeros.We consider thequestion how close to the origin the zeros of the function $theta$ can be.In the generalcase when $q$ is complex it is truethat their moduli are always larger than $1/2|q|$. We consider the case when $q$ is real and prove that for any $qin (0,1)$,the function $theta (q,.)$ has no zeros on the set $$displaystyle {xinmathbb{C}colon |x|leq 3} cap {xinmathbb{C}colon {rm Re} xleq 0}cap {xinmathbb{C}colon |{rm Im} x|leq 3/sqrt{2}}$$which containsthe closure left unit half-disk and is more than $7$ times larger than it.It is unlikely this result to hold true for the whole of the lefthalf-disk of radius~$3$. Similar domains do not exist for $qin (0,1)$, Re$xgeq 0$, for$qin (-1,0)$, Re$xgeq 0$ and for $qin (-1,0)$, Re$xleq 0$. We show alsothat for $qin (0,1)$, the function $theta (q,.)$ has no real zeros $geq -5$ (but one can find zeros larger than $-7.51$).
{"title":"A domain free of the zeros of the partial theta function","authors":"V. Kostov","doi":"10.30970/ms.58.2.142-158","DOIUrl":"https://doi.org/10.30970/ms.58.2.142-158","url":null,"abstract":"The partial theta function is the sum of the series medskipcenterline{$displaystyletheta (q,x):=sumnolimits _{j=0}^{infty}q^{j(j+1)/2}x^j$,}medskipnoi where $q$is a real or complex parameter ($|q|<1$). Its name is due to similaritieswith the formula for the Jacobi theta function$Theta (q,x):=sum _{j=-infty}^{infty}q^{j^2}x^j$. The function $theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, the theory of (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently in the study of section-hyperbolic polynomials,i.~e. real polynomials with all coefficients positive,with all roots real negative and all whose sections (i.~e. truncations)are also real-rooted.For each $q$ fixed,$theta$ is an entire function of order $0$ in the variable~$x$. When$q$ is real and $qin (0,0.3092ldots )$, $theta (q,.)$ is a function of theLaguerre-P'olyaclass $mathcal{L-P}I$. More generally, for $q in (0,1)$, the function $theta (q,.)$ is the product of a realpolynomialwithout real zeros and a function of the class $mathcal{L-P}I$. Thus it isan entire function withinfinitely-many negative, with no positive and with finitely-many complexconjugate zeros. The latter are known to belongto an explicitly defined compact domain containing $0$ andindependent of $q$ while the negative zeros tend to infinity as ageometric progression with ratio $1/q$. A similar result is true for$qin (-1,0)$ when there are also infinitely-many positive zeros.We consider thequestion how close to the origin the zeros of the function $theta$ can be.In the generalcase when $q$ is complex it is truethat their moduli are always larger than $1/2|q|$. We consider the case when $q$ is real and prove that for any $qin (0,1)$,the function $theta (q,.)$ has no zeros on the set $$displaystyle {xinmathbb{C}colon |x|leq 3} cap {xinmathbb{C}colon {rm Re} xleq 0}cap {xinmathbb{C}colon |{rm Im} x|leq 3/sqrt{2}}$$which containsthe closure left unit half-disk and is more than $7$ times larger than it.It is unlikely this result to hold true for the whole of the lefthalf-disk of radius~$3$. Similar domains do not exist for $qin (0,1)$, Re$xgeq 0$, for$qin (-1,0)$, Re$xgeq 0$ and for $qin (-1,0)$, Re$xleq 0$. We show alsothat for $qin (0,1)$, the function $theta (q,.)$ has no real zeros $geq -5$ (but one can find zeros larger than $-7.51$).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44639597","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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